10 82: Difference between revisions

Revision as of 17:20, 29 August 2005

10_81

10_83

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10 82 Quick Notes

10 82 Further Notes and Views

Knot presentations

Planar diagram presentation	X₁₆₂₇ X_7,16,8,17 X₃₉₄₈ X_15,3,16,2 X_5,15,6,14 X_9,5,10,4 X_11,18,12,19 X_13,20,14,1 X_17,10,18,11 X_19,12,20,13
Gauss code	-1, 4, -3, 6, -5, 1, -2, 3, -6, 9, -7, 10, -8, 5, -4, 2, -9, 7, -10, 8
Dowker-Thistlethwaite code	6 8 14 16 4 18 20 2 10 12
Conway Notation	[.4.2]

Minimum Braid Representative:

Length is 10, width is 3.

Braid index is 3.

A Morse Link Presentation:

Three dimensional invariants

Symmetry type	Chiral
Unknotting number	1
3-genus	4
Bridge index	3
Super bridge index	Missing
Nakanishi index	1
Maximal Thurston-Bennequin number	[-9][-3]
Hyperbolic Volume	12.4315
A-Polynomial	See Data:10 82/A-polynomial

[edit Notes for 10 82's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1}
Topological 4 genus	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1}
Concordance genus	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [2,4]}
Rasmussen s-Invariant	-2

[edit Notes for 10 82's four dimensional invariants]

Polynomial invariants

Alexander polynomial	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -t^4+4 t^3-8 t^2+12 t-13+12 t^{-1} -8 t^{-2} +4 t^{-3} - t^{-4} }
Conway polynomial	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -z^8-4 z^6-4 z^4+1}
2nd Alexander ideal (db, data sources)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{1\}}
Determinant and Signature	{ 63, -2 }
Jones polynomial	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^3-3 q^2+5 q-7+10 q^{-1} -10 q^{-2} +10 q^{-3} -8 q^{-4} +5 q^{-5} -3 q^{-6} + q^{-7} }
HOMFLY-PT polynomial (db, data sources)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -a^2 z^8+a^4 z^6-6 a^2 z^6+z^6+4 a^4 z^4-12 a^2 z^4+4 z^4+4 a^4 z^2-8 a^2 z^2+4 z^2+1}
Kauffman polynomial (db, data sources)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2 a^3 z^9+2 a z^9+4 a^4 z^8+8 a^2 z^8+4 z^8+4 a^5 z^7-a^3 z^7-2 a z^7+3 z^7 a^{-1} +4 a^6 z^6-8 a^4 z^6-27 a^2 z^6+z^6 a^{-2} -14 z^6+3 a^7 z^5-3 a^5 z^5-4 a^3 z^5-8 a z^5-10 z^5 a^{-1} +a^8 z^4-4 a^6 z^4+10 a^4 z^4+32 a^2 z^4-3 z^4 a^{-2} +14 z^4-4 a^7 z^3-2 a^5 z^3+5 a^3 z^3+10 a z^3+7 z^3 a^{-1} -a^8 z^2-5 a^4 z^2-13 a^2 z^2+z^2 a^{-2} -6 z^2+a^7 z+2 a^5 z-2 a z-z a^{-1} +1}
The A2 invariant	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{20}-q^{18}+q^{16}-2 q^{14}-q^{12}+q^{10}-q^8+4 q^6-q^4+2 q^2- q^{-2} + q^{-4} - q^{-6} + q^{-8} }
The G2 invariant	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{114}-2 q^{112}+4 q^{110}-6 q^{108}+4 q^{106}-2 q^{104}-4 q^{102}+12 q^{100}-18 q^{98}+22 q^{96}-19 q^{94}+8 q^{92}+7 q^{90}-22 q^{88}+35 q^{86}-38 q^{84}+35 q^{82}-25 q^{80}+6 q^{78}+16 q^{76}-37 q^{74}+57 q^{72}-60 q^{70}+47 q^{68}-18 q^{66}-20 q^{64}+52 q^{62}-67 q^{60}+53 q^{58}-17 q^{56}-28 q^{54}+53 q^{52}-53 q^{50}+16 q^{48}+36 q^{46}-76 q^{44}+82 q^{42}-56 q^{40}-q^{38}+64 q^{36}-107 q^{34}+116 q^{32}-84 q^{30}+30 q^{28}+38 q^{26}-85 q^{24}+106 q^{22}-88 q^{20}+49 q^{18}+4 q^{16}-52 q^{14}+75 q^{12}-60 q^{10}+23 q^8+28 q^6-66 q^4+68 q^2-36-21 q^{-2} +70 q^{-4} -95 q^{-6} +84 q^{-8} -38 q^{-10} -21 q^{-12} +69 q^{-14} -87 q^{-16} +78 q^{-18} -43 q^{-20} +2 q^{-22} +27 q^{-24} -41 q^{-26} +39 q^{-28} -26 q^{-30} +13 q^{-32} + q^{-34} -7 q^{-36} +7 q^{-38} -7 q^{-40} +4 q^{-42} -2 q^{-44} + q^{-46} }

Further Quantum Invariants

Further quantum knot invariants for 10_82.

A1 Invariants.

Weight	Invariant
1	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{15}-2 q^{13}+2 q^{11}-3 q^9+2 q^7+3 q-2 q^{-1} +2 q^{-3} -2 q^{-5} + q^{-7} }
2	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{42}-2 q^{40}-q^{38}+5 q^{36}-4 q^{34}-q^{32}+9 q^{30}-10 q^{28}-3 q^{26}+16 q^{24}-11 q^{22}-9 q^{20}+15 q^{18}-2 q^{16}-10 q^{14}+4 q^{12}+9 q^{10}-5 q^8-8 q^6+13 q^4+2 q^2-15+11 q^{-2} +8 q^{-4} -16 q^{-6} +3 q^{-8} +12 q^{-10} -9 q^{-12} -4 q^{-14} +7 q^{-16} - q^{-18} -2 q^{-20} + q^{-22} }
3	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{81}-2 q^{79}-q^{77}+2 q^{75}+4 q^{73}-q^{71}-7 q^{69}+3 q^{65}+3 q^{63}+2 q^{61}+2 q^{59}-11 q^{57}-14 q^{55}+17 q^{53}+35 q^{51}-14 q^{49}-57 q^{47}-2 q^{45}+73 q^{43}+26 q^{41}-73 q^{39}-48 q^{37}+52 q^{35}+65 q^{33}-29 q^{31}-63 q^{29}-3 q^{27}+56 q^{25}+27 q^{23}-42 q^{21}-46 q^{19}+30 q^{17}+56 q^{15}-18 q^{13}-66 q^{11}+8 q^9+75 q^7+7 q^5-74 q^3-27 q+73 q^{-1} +48 q^{-3} -52 q^{-5} -69 q^{-7} +27 q^{-9} +73 q^{-11} +4 q^{-13} -64 q^{-15} -31 q^{-17} +44 q^{-19} +42 q^{-21} -20 q^{-23} -38 q^{-25} +27 q^{-29} +9 q^{-31} -14 q^{-33} -7 q^{-35} +4 q^{-37} +4 q^{-39} - q^{-41} -2 q^{-43} + q^{-45} }
4	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{132}-2 q^{130}-q^{128}+2 q^{126}+q^{124}+7 q^{122}-7 q^{120}-9 q^{118}-4 q^{116}+33 q^{112}+6 q^{110}-14 q^{108}-35 q^{106}-41 q^{104}+51 q^{102}+62 q^{100}+42 q^{98}-58 q^{96}-154 q^{94}-21 q^{92}+114 q^{90}+213 q^{88}+53 q^{86}-268 q^{84}-263 q^{82}-12 q^{80}+387 q^{78}+371 q^{76}-151 q^{74}-483 q^{72}-370 q^{70}+268 q^{68}+618 q^{66}+214 q^{64}-355 q^{62}-603 q^{60}-105 q^{58}+463 q^{56}+440 q^{54}+22 q^{52}-441 q^{50}-337 q^{48}+87 q^{46}+337 q^{44}+261 q^{42}-118 q^{40}-310 q^{38}-169 q^{36}+154 q^{34}+310 q^{32}+75 q^{30}-260 q^{28}-285 q^{26}+85 q^{24}+355 q^{22}+193 q^{20}-275 q^{18}-423 q^{16}+11 q^{14}+423 q^{12}+378 q^{10}-192 q^8-542 q^6-203 q^4+314 q^2+542+95 q^{-2} -434 q^{-4} -420 q^{-6} -37 q^{-8} +439 q^{-10} +367 q^{-12} -57 q^{-14} -343 q^{-16} -346 q^{-18} +70 q^{-20} +315 q^{-22} +251 q^{-24} -7 q^{-26} -300 q^{-28} -198 q^{-30} +20 q^{-32} +206 q^{-34} +197 q^{-36} -50 q^{-38} -143 q^{-40} -127 q^{-42} +13 q^{-44} +125 q^{-46} +57 q^{-48} -6 q^{-50} -67 q^{-52} -41 q^{-54} +22 q^{-56} +21 q^{-58} +18 q^{-60} -8 q^{-62} -13 q^{-64} + q^{-66} + q^{-68} +4 q^{-70} - q^{-72} -2 q^{-74} + q^{-76} }
5	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{195}-2 q^{193}-q^{191}+2 q^{189}+q^{187}+4 q^{185}+q^{183}-9 q^{181}-13 q^{179}+13 q^{175}+27 q^{173}+23 q^{171}-17 q^{169}-57 q^{167}-57 q^{165}+4 q^{163}+80 q^{161}+116 q^{159}+49 q^{157}-106 q^{155}-210 q^{153}-128 q^{151}+111 q^{149}+318 q^{147}+293 q^{145}-55 q^{143}-487 q^{141}-552 q^{139}-82 q^{137}+625 q^{135}+948 q^{133}+433 q^{131}-687 q^{129}-1473 q^{127}-1043 q^{125}+511 q^{123}+1986 q^{121}+1932 q^{119}+64 q^{117}-2277 q^{115}-2977 q^{113}-1068 q^{111}+2116 q^{109}+3853 q^{107}+2369 q^{105}-1348 q^{103}-4236 q^{101}-3661 q^{99}+105 q^{97}+3920 q^{95}+4510 q^{93}+1313 q^{91}-2916 q^{89}-4646 q^{87}-2531 q^{85}+1529 q^{83}+4076 q^{81}+3163 q^{79}-140 q^{77}-2954 q^{75}-3194 q^{73}-944 q^{71}+1738 q^{69}+2712 q^{67}+1518 q^{65}-646 q^{63}-2036 q^{61}-1704 q^{59}-75 q^{57}+1448 q^{55}+1653 q^{53}+461 q^{51}-1101 q^{49}-1624 q^{47}-621 q^{45}+1024 q^{43}+1758 q^{41}+756 q^{39}-1145 q^{37}-2093 q^{35}-1008 q^{33}+1259 q^{31}+2572 q^{29}+1481 q^{27}-1224 q^{25}-3062 q^{23}-2147 q^{21}+894 q^{19}+3360 q^{17}+2922 q^{15}-197 q^{13}-3313 q^{11}-3622 q^9-757 q^7+2788 q^5+3991 q^3+1869 q-1786 q^{-1} -3894 q^{-3} -2782 q^{-5} +481 q^{-7} +3161 q^{-9} +3262 q^{-11} +881 q^{-13} -1975 q^{-15} -3102 q^{-17} -1881 q^{-19} +544 q^{-21} +2305 q^{-23} +2286 q^{-25} +724 q^{-27} -1135 q^{-29} -2005 q^{-31} -1471 q^{-33} -50 q^{-35} +1216 q^{-37} +1571 q^{-39} +892 q^{-41} -286 q^{-43} -1143 q^{-45} -1176 q^{-47} -443 q^{-49} +470 q^{-51} +977 q^{-53} +789 q^{-55} +99 q^{-57} -543 q^{-59} -724 q^{-61} -401 q^{-63} +111 q^{-65} +462 q^{-67} +437 q^{-69} +123 q^{-71} -188 q^{-73} -295 q^{-75} -181 q^{-77} +7 q^{-79} +145 q^{-81} +141 q^{-83} +39 q^{-85} -43 q^{-87} -66 q^{-89} -39 q^{-91} +28 q^{-95} +21 q^{-97} -7 q^{-101} -5 q^{-103} -2 q^{-105} + q^{-107} +4 q^{-109} - q^{-111} -2 q^{-113} + q^{-115} }
6	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{270}-2 q^{268}-q^{266}+2 q^{264}+q^{262}+4 q^{260}-2 q^{258}-q^{256}-13 q^{254}-9 q^{252}+13 q^{250}+14 q^{248}+27 q^{246}+10 q^{244}-7 q^{242}-64 q^{240}-67 q^{238}-5 q^{236}+38 q^{234}+109 q^{232}+108 q^{230}+63 q^{228}-126 q^{226}-212 q^{224}-146 q^{222}-33 q^{220}+191 q^{218}+311 q^{216}+273 q^{214}-127 q^{212}-407 q^{210}-397 q^{208}-167 q^{206}+342 q^{204}+709 q^{202}+622 q^{200}-250 q^{198}-1047 q^{196}-1190 q^{194}-507 q^{192}+985 q^{190}+2277 q^{188}+2202 q^{186}+25 q^{184}-2767 q^{182}-4327 q^{180}-3257 q^{178}+809 q^{176}+5653 q^{174}+7774 q^{172}+4452 q^{170}-2851 q^{168}-9942 q^{166}-11951 q^{164}-5764 q^{162}+5982 q^{160}+16039 q^{158}+16644 q^{156}+6076 q^{154}-10333 q^{152}-22654 q^{150}-21292 q^{148}-5371 q^{146}+16099 q^{144}+28920 q^{142}+24120 q^{140}+3309 q^{138}-21544 q^{136}-33763 q^{134}-24841 q^{132}+390 q^{130}+25729 q^{128}+35371 q^{126}+23053 q^{124}-3977 q^{122}-27889 q^{120}-33953 q^{118}-18819 q^{116}+6776 q^{114}+26990 q^{112}+29781 q^{110}+14185 q^{108}-8412 q^{106}-24021 q^{104}-23712 q^{102}-9831 q^{100}+8388 q^{98}+19677 q^{96}+17926 q^{94}+6138 q^{92}-7736 q^{90}-15046 q^{88}-12885 q^{86}-3334 q^{84}+6936 q^{82}+11631 q^{80}+8678 q^{78}+578 q^{76}-6798 q^{74}-9172 q^{72}-4928 q^{70}+2480 q^{68}+7842 q^{66}+7132 q^{64}+713 q^{62}-6470 q^{60}-9155 q^{58}-4510 q^{56}+4319 q^{54}+11134 q^{52}+9893 q^{50}+510 q^{48}-10612 q^{46}-15276 q^{44}-8896 q^{42}+4907 q^{40}+17207 q^{38}+18167 q^{36}+5956 q^{34}-11771 q^{32}-22836 q^{30}-18828 q^{28}-1546 q^{26}+18498 q^{24}+26989 q^{22}+17637 q^{20}-4042 q^{18}-23870 q^{16}-28776 q^{14}-15236 q^{12}+8937 q^{10}+27543 q^8+28812 q^6+11855 q^4-12251 q^2-28682-27467 q^{-2} -9150 q^{-4} +13933 q^{-6} +27945 q^{-8} +24702 q^{-10} +7438 q^{-12} -13416 q^{-14} -25505 q^{-16} -21891 q^{-18} -6415 q^{-20} +11529 q^{-22} +21381 q^{-24} +19027 q^{-26} +6386 q^{-28} -8517 q^{-30} -17022 q^{-32} -15857 q^{-34} -6564 q^{-36} +4678 q^{-38} +12564 q^{-40} +12917 q^{-42} +6863 q^{-44} -1485 q^{-46} -8096 q^{-48} -9915 q^{-50} -7158 q^{-52} -1041 q^{-54} +4453 q^{-56} +7237 q^{-58} +6518 q^{-60} +2914 q^{-62} -1556 q^{-64} -5070 q^{-66} -5497 q^{-68} -3631 q^{-70} -237 q^{-72} +2908 q^{-74} +4405 q^{-76} +3649 q^{-78} +1046 q^{-80} -1410 q^{-82} -3083 q^{-84} -2987 q^{-86} -1507 q^{-88} +567 q^{-90} +2011 q^{-92} +2086 q^{-94} +1344 q^{-96} -33 q^{-98} -1084 q^{-100} -1448 q^{-102} -927 q^{-104} -103 q^{-106} +485 q^{-108} +808 q^{-110} +590 q^{-112} +168 q^{-114} -271 q^{-116} -379 q^{-118} -278 q^{-120} -124 q^{-122} +95 q^{-124} +172 q^{-126} +144 q^{-128} +24 q^{-130} -30 q^{-132} -51 q^{-134} -57 q^{-136} -13 q^{-138} +15 q^{-140} +27 q^{-142} +3 q^{-144} + q^{-146} + q^{-148} -8 q^{-150} -2 q^{-152} + q^{-154} +4 q^{-156} - q^{-158} -2 q^{-160} + q^{-162} }

A2 Invariants.

Weight	Invariant
1,0	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{20}-q^{18}+q^{16}-2 q^{14}-q^{12}+q^{10}-q^8+4 q^6-q^4+2 q^2- q^{-2} + q^{-4} - q^{-6} + q^{-8} }
1,1	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{60}-4 q^{58}+10 q^{56}-20 q^{54}+34 q^{52}-54 q^{50}+78 q^{48}-100 q^{46}+122 q^{44}-140 q^{42}+156 q^{40}-174 q^{38}+187 q^{36}-202 q^{34}+212 q^{32}-202 q^{30}+172 q^{28}-108 q^{26}+8 q^{24}+116 q^{22}-256 q^{20}+384 q^{18}-498 q^{16}+576 q^{14}-604 q^{12}+588 q^{10}-518 q^8+414 q^6-269 q^4+112 q^2+50-190 q^{-2} +294 q^{-4} -350 q^{-6} +358 q^{-8} -328 q^{-10} +272 q^{-12} -202 q^{-14} +136 q^{-16} -86 q^{-18} +48 q^{-20} -22 q^{-22} +10 q^{-24} -4 q^{-26} + q^{-28} }
2,0	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{52}-q^{50}-q^{48}+q^{46}-q^{44}+q^{42}+2 q^{40}-q^{38}-q^{36}+2 q^{34}+4 q^{32}-5 q^{30}-6 q^{28}+4 q^{26}-8 q^{22}+q^{20}+9 q^{18}-q^{16}-q^{14}+4 q^{12}+2 q^{10}-4 q^8+6 q^4-5 q^2-1+7 q^{-2} + q^{-4} -6 q^{-6} + q^{-8} +5 q^{-10} -2 q^{-12} -4 q^{-14} + q^{-16} +3 q^{-18} - q^{-20} - q^{-22} + q^{-24} }

A3 Invariants.

Weight	Invariant
0,1,0	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{48}-2 q^{46}+4 q^{42}-6 q^{40}-q^{38}+9 q^{36}-7 q^{34}-4 q^{32}+15 q^{30}-5 q^{28}-8 q^{26}+12 q^{24}-5 q^{22}-9 q^{20}+2 q^{18}+q^{16}-q^{14}-3 q^{12}+9 q^{10}+6 q^8-10 q^6+6 q^4+7 q^2-12+3 q^{-2} +8 q^{-4} -9 q^{-6} +4 q^{-8} +4 q^{-10} -5 q^{-12} +3 q^{-14} -2 q^{-18} + q^{-20} }
1,0,0	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{25}-q^{23}+2 q^{21}-3 q^{19}+q^{17}-3 q^{15}+q^{13}+2 q^9+2 q^7+2 q^3-2 q+2 q^{-1} -2 q^{-3} +2 q^{-5} - q^{-7} + q^{-9} }
1,0,1	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{78}-4 q^{76}+8 q^{74}-8 q^{72}-2 q^{70}+21 q^{68}-38 q^{66}+37 q^{64}-10 q^{62}-33 q^{60}+69 q^{58}-72 q^{56}+41 q^{54}+14 q^{52}-65 q^{50}+73 q^{48}-32 q^{46}-51 q^{44}+117 q^{42}-119 q^{40}+44 q^{38}+97 q^{36}-211 q^{34}+254 q^{32}-198 q^{30}+65 q^{28}+68 q^{26}-171 q^{24}+188 q^{22}-159 q^{20}+106 q^{18}-66 q^{16}+69 q^{14}-77 q^{12}+88 q^{10}-36 q^8-59 q^6+174 q^4-246 q^2+244-160 q^{-2} +38 q^{-4} +84 q^{-6} -159 q^{-8} +175 q^{-10} -135 q^{-12} +57 q^{-14} +17 q^{-16} -60 q^{-18} +64 q^{-20} -41 q^{-22} +13 q^{-24} +6 q^{-26} -10 q^{-28} +8 q^{-30} -4 q^{-32} + q^{-34} }

A4 Invariants.

Weight

Invariant

0,1,0,0

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{58}-q^{56}-q^{54}+3 q^{52}-2 q^{50}-6 q^{48}+4 q^{46}+4 q^{44}-9 q^{42}+q^{40}+12 q^{38}+2 q^{36}-6 q^{34}+7 q^{32}+6 q^{30}-12 q^{28}-8 q^{26}+6 q^{24}-8 q^{22}-12 q^{20}+13 q^{18}+5 q^{16}-7 q^{14}+7 q^{12}+13 q^{10}-6 q^8-5 q^6+5 q^4+3 q^2-6+7 q^{-4} - q^{-6} +3 q^{-10} - q^{-12} -2 q^{-14} +2 q^{-16} - q^{-18} - q^{-20} + q^{-22} }

1,0,0,0

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{30}-q^{28}+2 q^{26}-2 q^{24}-q^{20}-2 q^{18}+q^{16}-q^{14}+3 q^{12}+3 q^8-q^6+2 q^4-q^2+ q^{-2} - q^{-4} +2 q^{-6} - q^{-8} + q^{-10} }

B2 Invariants.

Weight

Invariant

0,1

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{48}-2 q^{46}+4 q^{44}-6 q^{42}+8 q^{40}-11 q^{38}+13 q^{36}-15 q^{34}+16 q^{32}-15 q^{30}+11 q^{28}-6 q^{26}-2 q^{24}+9 q^{22}-17 q^{20}+24 q^{18}-29 q^{16}+33 q^{14}-31 q^{12}+29 q^{10}-22 q^8+16 q^6-6 q^4-q^2+8-13 q^{-2} +16 q^{-4} -17 q^{-6} +16 q^{-8} -14 q^{-10} +11 q^{-12} -7 q^{-14} +4 q^{-16} -2 q^{-18} + q^{-20} }

1,0

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{78}-2 q^{74}-2 q^{72}+2 q^{70}+5 q^{68}-7 q^{64}-6 q^{62}+5 q^{60}+11 q^{58}+2 q^{56}-11 q^{54}-10 q^{52}+7 q^{50}+17 q^{48}+2 q^{46}-15 q^{44}-9 q^{42}+11 q^{40}+12 q^{38}-7 q^{36}-15 q^{34}+12 q^{30}+q^{28}-12 q^{26}-5 q^{24}+10 q^{22}+7 q^{20}-6 q^{18}-6 q^{16}+9 q^{14}+11 q^{12}-4 q^{10}-14 q^8+q^6+16 q^4+8 q^2-14-16 q^{-2} +6 q^{-4} +18 q^{-6} +3 q^{-8} -15 q^{-10} -9 q^{-12} +9 q^{-14} +11 q^{-16} -2 q^{-18} -8 q^{-20} - q^{-22} +5 q^{-24} +2 q^{-26} -2 q^{-28} -2 q^{-30} + q^{-34} }

D4 Invariants.

Weight

Invariant

1,0,0,0

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{66}-2 q^{64}+2 q^{62}-3 q^{60}+5 q^{58}-7 q^{56}+6 q^{54}-8 q^{52}+11 q^{50}-11 q^{48}+10 q^{46}-11 q^{44}+15 q^{42}-9 q^{40}+7 q^{38}-5 q^{36}+2 q^{34}+4 q^{32}-10 q^{30}+9 q^{28}-19 q^{26}+20 q^{24}-23 q^{22}+24 q^{20}-25 q^{18}+26 q^{16}-19 q^{14}+21 q^{12}-15 q^{10}+11 q^8-5 q^6+2 q^4+2 q^2-6+11 q^{-2} -11 q^{-4} +13 q^{-6} -13 q^{-8} +15 q^{-10} -12 q^{-12} +10 q^{-14} -9 q^{-16} +7 q^{-18} -4 q^{-20} +2 q^{-22} -2 q^{-24} + q^{-26} }

G2 Invariants.

Weight

Invariant

1,0

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{114}-2 q^{112}+4 q^{110}-6 q^{108}+4 q^{106}-2 q^{104}-4 q^{102}+12 q^{100}-18 q^{98}+22 q^{96}-19 q^{94}+8 q^{92}+7 q^{90}-22 q^{88}+35 q^{86}-38 q^{84}+35 q^{82}-25 q^{80}+6 q^{78}+16 q^{76}-37 q^{74}+57 q^{72}-60 q^{70}+47 q^{68}-18 q^{66}-20 q^{64}+52 q^{62}-67 q^{60}+53 q^{58}-17 q^{56}-28 q^{54}+53 q^{52}-53 q^{50}+16 q^{48}+36 q^{46}-76 q^{44}+82 q^{42}-56 q^{40}-q^{38}+64 q^{36}-107 q^{34}+116 q^{32}-84 q^{30}+30 q^{28}+38 q^{26}-85 q^{24}+106 q^{22}-88 q^{20}+49 q^{18}+4 q^{16}-52 q^{14}+75 q^{12}-60 q^{10}+23 q^8+28 q^6-66 q^4+68 q^2-36-21 q^{-2} +70 q^{-4} -95 q^{-6} +84 q^{-8} -38 q^{-10} -21 q^{-12} +69 q^{-14} -87 q^{-16} +78 q^{-18} -43 q^{-20} +2 q^{-22} +27 q^{-24} -41 q^{-26} +39 q^{-28} -26 q^{-30} +13 q^{-32} + q^{-34} -7 q^{-36} +7 q^{-38} -7 q^{-40} +4 q^{-42} -2 q^{-44} + q^{-46} }

.

Computer Talk

The above data is available with the Mathematica package KnotTheory`, as shown in the (simulated) Mathematica session below. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting. This Mathematica session is also available (albeit only for the knot 5_2) as the notebook PolynomialInvariantsSession.nb.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:= AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of August 31, 2006, 11:25:27.5625.

In[3]:= K = Knot["10 82"];

In[4]:= Alexander[K][t]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]= Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -t^4+4 t^3-8 t^2+12 t-13+12 t^{-1} -8 t^{-2} +4 t^{-3} - t^{-4} }

In[5]:= Conway[K][z]

Out[5]= Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -z^8-4 z^6-4 z^4+1}

In[6]:= Alexander[K, 2][t]

KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.

Out[6]= Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{1\}}

In[7]:= {KnotDet[K], KnotSignature[K]}

Out[7]= { 63, -2 }

In[8]:= Jones[K][q]

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[8]= Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^3-3 q^2+5 q-7+10 q^{-1} -10 q^{-2} +10 q^{-3} -8 q^{-4} +5 q^{-5} -3 q^{-6} + q^{-7} }

In[9]:= HOMFLYPT[K][a, z]

KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.

Out[9]= Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -a^2 z^8+a^4 z^6-6 a^2 z^6+z^6+4 a^4 z^4-12 a^2 z^4+4 z^4+4 a^4 z^2-8 a^2 z^2+4 z^2+1}

In[10]:= Kauffman[K][a, z]

KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.

Out[10]= Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2 a^3 z^9+2 a z^9+4 a^4 z^8+8 a^2 z^8+4 z^8+4 a^5 z^7-a^3 z^7-2 a z^7+3 z^7 a^{-1} +4 a^6 z^6-8 a^4 z^6-27 a^2 z^6+z^6 a^{-2} -14 z^6+3 a^7 z^5-3 a^5 z^5-4 a^3 z^5-8 a z^5-10 z^5 a^{-1} +a^8 z^4-4 a^6 z^4+10 a^4 z^4+32 a^2 z^4-3 z^4 a^{-2} +14 z^4-4 a^7 z^3-2 a^5 z^3+5 a^3 z^3+10 a z^3+7 z^3 a^{-1} -a^8 z^2-5 a^4 z^2-13 a^2 z^2+z^2 a^{-2} -6 z^2+a^7 z+2 a^5 z-2 a z-z a^{-1} +1}

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {...}

Same Jones Polynomial (up to mirroring, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q\leftrightarrow q^{-1}} ): {...}

Vassiliev invariants

V₂ and V₃:

(0, 0)

V_2,1 through V_6,9:

V_2,1

V_3,1

V_4,1

V_4,2

V_4,3

V_5,1

V_5,2

V_5,3

V_5,4

V_6,1

V_6,2

V_6,3

V_6,4

V_6,5

V_6,6

V_6,7

V_6,8

V_6,9

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 32}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -96}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -32}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -64}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 176}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{208}{3}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\frac{32}{3}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{352}{3}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -32}

V_2,1 through V_6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V₂ and V₃.

Khovanov Homology

The coefficients of the monomials Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t^rq^j} are shown, along with their alternating sums Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \chi} (fixed Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j} , alternation over Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r} ). The squares with yellow highlighting are those on the "critical diagonals", where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j-2r=s+1} or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j-2r=s-1} , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s=} -2 is the signature of 10 82. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\		r
	\
j		\

-6

-5

-4

-3

-2

-1

0

1

2

3

4

χ

7

1

5

2

-2

3

1

2

1

4

2

-2

-1

6

3

-3

5

0

-5

5

0

-7

3

5

2

-9

2

5

-3

-11

1

3

2

-13

2

-2

-15

1

Integral Khovanov Homology

(db, data source)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i=-3}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i=-1}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-6}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-5}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{2}\oplus{\mathbb Z}_2}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-4}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{2}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-3}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-2}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{5}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-1}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{5}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=0}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{6}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=1}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{4}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=2}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=3}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}\oplus{\mathbb Z}_2^{2}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{2}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=4}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}_2}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}}

The Coloured Jones Polynomials

The Coloured Jones Polynomials (in the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n+1} -dimensional representation of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle sl(2)} )

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J_n}
2	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{10}-3 q^9+q^8+9 q^7-14 q^6-4 q^5+30 q^4-23 q^3-23 q^2+54 q-20-49 q^{-1} +71 q^{-2} -9 q^{-3} -70 q^{-4} +74 q^{-5} +5 q^{-6} -75 q^{-7} +60 q^{-8} +13 q^{-9} -58 q^{-10} +36 q^{-11} +11 q^{-12} -31 q^{-13} +17 q^{-14} +4 q^{-15} -12 q^{-16} +7 q^{-17} + q^{-18} -3 q^{-19} + q^{-20} }
3	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{21}-3 q^{20}+q^{19}+5 q^{18}+q^{17}-14 q^{16}-6 q^{15}+28 q^{14}+19 q^{13}-41 q^{12}-44 q^{11}+46 q^{10}+81 q^9-39 q^8-119 q^7+13 q^6+149 q^5+30 q^4-165 q^3-83 q^2+166 q+130-140 q^{-1} -183 q^{-2} +119 q^{-3} +211 q^{-4} -72 q^{-5} -250 q^{-6} +45 q^{-7} +259 q^{-8} +2 q^{-9} -276 q^{-10} -31 q^{-11} +263 q^{-12} +71 q^{-13} -247 q^{-14} -90 q^{-15} +203 q^{-16} +105 q^{-17} -153 q^{-18} -103 q^{-19} +103 q^{-20} +80 q^{-21} -54 q^{-22} -56 q^{-23} +28 q^{-24} +25 q^{-25} -11 q^{-26} -7 q^{-27} +10 q^{-28} -6 q^{-29} -8 q^{-30} +6 q^{-31} +10 q^{-32} -5 q^{-33} -8 q^{-34} +3 q^{-35} +3 q^{-36} + q^{-37} -3 q^{-38} + q^{-39} }
4	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{36}-3 q^{35}+q^{34}+5 q^{33}-3 q^{32}+q^{31}-17 q^{30}+6 q^{29}+31 q^{28}+2 q^{26}-80 q^{25}-20 q^{24}+92 q^{23}+63 q^{22}+70 q^{21}-192 q^{20}-160 q^{19}+76 q^{18}+156 q^{17}+317 q^{16}-183 q^{15}-346 q^{14}-142 q^{13}+54 q^{12}+610 q^{11}+75 q^{10}-282 q^9-387 q^8-362 q^7+613 q^6+361 q^5+142 q^4-315 q^3-838 q^2+230 q+347+671 q^{-1} +132 q^{-2} -1066 q^{-3} -287 q^{-4} +8 q^{-5} +1021 q^{-6} +702 q^{-7} -1021 q^{-8} -699 q^{-9} -426 q^{-10} +1169 q^{-11} +1170 q^{-12} -859 q^{-13} -969 q^{-14} -796 q^{-15} +1194 q^{-16} +1505 q^{-17} -624 q^{-18} -1125 q^{-19} -1119 q^{-20} +1053 q^{-21} +1697 q^{-22} -245 q^{-23} -1049 q^{-24} -1369 q^{-25} +629 q^{-26} +1593 q^{-27} +218 q^{-28} -631 q^{-29} -1346 q^{-30} +61 q^{-31} +1095 q^{-32} +466 q^{-33} -62 q^{-34} -942 q^{-35} -289 q^{-36} +457 q^{-37} +353 q^{-38} +270 q^{-39} -420 q^{-40} -273 q^{-41} +58 q^{-42} +102 q^{-43} +265 q^{-44} -99 q^{-45} -113 q^{-46} -41 q^{-47} -33 q^{-48} +132 q^{-49} -3 q^{-50} -13 q^{-51} -21 q^{-52} -44 q^{-53} +40 q^{-54} +3 q^{-55} +8 q^{-56} - q^{-57} -17 q^{-58} +7 q^{-59} - q^{-60} +3 q^{-61} + q^{-62} -3 q^{-63} + q^{-64} }
5	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{55}-3 q^{54}+q^{53}+5 q^{52}-3 q^{51}-3 q^{50}-2 q^{49}-5 q^{48}+8 q^{47}+26 q^{46}+4 q^{45}-31 q^{44}-41 q^{43}-32 q^{42}+31 q^{41}+108 q^{40}+106 q^{39}-27 q^{38}-179 q^{37}-220 q^{36}-83 q^{35}+215 q^{34}+417 q^{33}+287 q^{32}-154 q^{31}-571 q^{30}-595 q^{29}-108 q^{28}+598 q^{27}+929 q^{26}+536 q^{25}-383 q^{24}-1102 q^{23}-1021 q^{22}-135 q^{21}+962 q^{20}+1393 q^{19}+795 q^{18}-423 q^{17}-1376 q^{16}-1401 q^{15}-459 q^{14}+859 q^{13}+1665 q^{12}+1436 q^{11}+186 q^{10}-1382 q^9-2220 q^8-1566 q^7+444 q^6+2563 q^5+3042 q^4+999 q^3-2321 q^2-4246 q-2819+1451 q^{-1} +5150 q^{-2} +4654 q^{-3} -199 q^{-4} -5449 q^{-5} -6364 q^{-6} -1414 q^{-7} +5459 q^{-8} +7770 q^{-9} +2920 q^{-10} -5011 q^{-11} -8830 q^{-12} -4455 q^{-13} +4544 q^{-14} +9608 q^{-15} +5625 q^{-16} -3920 q^{-17} -10143 q^{-18} -6722 q^{-19} +3459 q^{-20} +10556 q^{-21} +7526 q^{-22} -2918 q^{-23} -10877 q^{-24} -8367 q^{-25} +2456 q^{-26} +11079 q^{-27} +9088 q^{-28} -1726 q^{-29} -11082 q^{-30} -9890 q^{-31} +827 q^{-32} +10747 q^{-33} +10478 q^{-34} +438 q^{-35} -9888 q^{-36} -10864 q^{-37} -1855 q^{-38} +8497 q^{-39} +10718 q^{-40} +3252 q^{-41} -6585 q^{-42} -9951 q^{-43} -4402 q^{-44} +4437 q^{-45} +8603 q^{-46} +4982 q^{-47} -2356 q^{-48} -6754 q^{-49} -4992 q^{-50} +622 q^{-51} +4837 q^{-52} +4407 q^{-53} +532 q^{-54} -3037 q^{-55} -3508 q^{-56} -1115 q^{-57} +1653 q^{-58} +2498 q^{-59} +1232 q^{-60} -696 q^{-61} -1640 q^{-62} -1061 q^{-63} +178 q^{-64} +944 q^{-65} +802 q^{-66} +90 q^{-67} -520 q^{-68} -546 q^{-69} -145 q^{-70} +237 q^{-71} +332 q^{-72} +155 q^{-73} -88 q^{-74} -198 q^{-75} -120 q^{-76} +30 q^{-77} +93 q^{-78} +73 q^{-79} +16 q^{-80} -43 q^{-81} -53 q^{-82} -6 q^{-83} +17 q^{-84} +12 q^{-85} +16 q^{-86} -3 q^{-87} -13 q^{-88} -2 q^{-89} +3 q^{-90} - q^{-91} +3 q^{-92} + q^{-93} -3 q^{-94} + q^{-95} }
6	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{78}-3 q^{77}+q^{76}+5 q^{75}-3 q^{74}-3 q^{73}-6 q^{72}+10 q^{71}-3 q^{70}+3 q^{69}+29 q^{68}-15 q^{67}-31 q^{66}-50 q^{65}+16 q^{64}+18 q^{63}+57 q^{62}+149 q^{61}+13 q^{60}-108 q^{59}-269 q^{58}-138 q^{57}-83 q^{56}+165 q^{55}+588 q^{54}+435 q^{53}+110 q^{52}-592 q^{51}-726 q^{50}-907 q^{49}-356 q^{48}+952 q^{47}+1486 q^{46}+1487 q^{45}+150 q^{44}-801 q^{43}-2351 q^{42}-2430 q^{41}-528 q^{40}+1390 q^{39}+3160 q^{38}+2606 q^{37}+1802 q^{36}-1595 q^{35}-3927 q^{34}-3673 q^{33}-2004 q^{32}+1294 q^{31}+3033 q^{30}+5316 q^{29}+2875 q^{28}-323 q^{27}-2954 q^{26}-4788 q^{25}-4200 q^{24}-3084 q^{23}+2559 q^{22}+4694 q^{21}+6288 q^{20}+5394 q^{19}+1266 q^{18}-4553 q^{17}-10970 q^{16}-8683 q^{15}-4599 q^{14}+5123 q^{13}+13899 q^{12}+16169 q^{11}+8088 q^{10}-8616 q^9-18535 q^8-22543 q^7-10353 q^6+10285 q^5+28258 q^4+28942 q^3+8648 q^2-15292 q-36555-33436 q^{-1} -8032 q^{-2} +27043 q^{-3} +45373 q^{-4} +32754 q^{-5} +1665 q^{-6} -37824 q^{-7} -52042 q^{-8} -32205 q^{-9} +13503 q^{-10} +50279 q^{-11} +52582 q^{-12} +23344 q^{-13} -28472 q^{-14} -60533 q^{-15} -52249 q^{-16} -3779 q^{-17} +46271 q^{-18} +63647 q^{-19} +41071 q^{-20} -16261 q^{-21} -61493 q^{-22} -64549 q^{-23} -17582 q^{-24} +39891 q^{-25} +68311 q^{-26} +52193 q^{-27} -6878 q^{-28} -60252 q^{-29} -71364 q^{-30} -26411 q^{-31} +35246 q^{-32} +70996 q^{-33} +59376 q^{-34} -459 q^{-35} -59542 q^{-36} -76726 q^{-37} -33819 q^{-38} +31002 q^{-39} +73370 q^{-40} +66752 q^{-41} +7641 q^{-42} -56589 q^{-43} -81421 q^{-44} -44089 q^{-45} +21451 q^{-46} +71209 q^{-47} +74062 q^{-48} +21515 q^{-49} -44801 q^{-50} -79670 q^{-51} -55378 q^{-52} +3232 q^{-53} +57328 q^{-54} +73753 q^{-55} +37124 q^{-56} -22204 q^{-57} -64074 q^{-58} -58169 q^{-59} -16982 q^{-60} +31733 q^{-61} +58619 q^{-62} +43188 q^{-63} +1708 q^{-64} -37044 q^{-65} -45851 q^{-66} -26624 q^{-67} +6394 q^{-68} +33388 q^{-69} +34266 q^{-70} +13927 q^{-71} -12191 q^{-72} -25040 q^{-73} -21824 q^{-74} -6427 q^{-75} +11918 q^{-76} +18345 q^{-77} +12565 q^{-78} +130 q^{-79} -8631 q^{-80} -11256 q^{-81} -7032 q^{-82} +1861 q^{-83} +6599 q^{-84} +6378 q^{-85} +2139 q^{-86} -1540 q^{-87} -3953 q^{-88} -3710 q^{-89} -260 q^{-90} +1755 q^{-91} +2312 q^{-92} +1069 q^{-93} +20 q^{-94} -1161 q^{-95} -1533 q^{-96} -185 q^{-97} +463 q^{-98} +820 q^{-99} +386 q^{-100} +163 q^{-101} -364 q^{-102} -661 q^{-103} -98 q^{-104} +96 q^{-105} +311 q^{-106} +156 q^{-107} +153 q^{-108} -84 q^{-109} -261 q^{-110} -60 q^{-111} -24 q^{-112} +87 q^{-113} +43 q^{-114} +87 q^{-115} +2 q^{-116} -72 q^{-117} -15 q^{-118} -23 q^{-119} +16 q^{-120} +25 q^{-122} +5 q^{-123} -15 q^{-124} +2 q^{-125} -6 q^{-126} +3 q^{-127} - q^{-128} +3 q^{-129} + q^{-130} -3 q^{-131} + q^{-132} }
7	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{105}-3 q^{104}+q^{103}+5 q^{102}-3 q^{101}-3 q^{100}-6 q^{99}+6 q^{98}+12 q^{97}-8 q^{96}+6 q^{95}+10 q^{94}-16 q^{93}-26 q^{92}-41 q^{91}+6 q^{90}+76 q^{89}+46 q^{88}+67 q^{87}+39 q^{86}-79 q^{85}-150 q^{84}-274 q^{83}-157 q^{82}+146 q^{81}+311 q^{80}+515 q^{79}+476 q^{78}+82 q^{77}-345 q^{76}-1034 q^{75}-1244 q^{74}-641 q^{73}+155 q^{72}+1419 q^{71}+2168 q^{70}+1927 q^{69}+1023 q^{68}-1195 q^{67}-3255 q^{66}-3837 q^{65}-3138 q^{64}-240 q^{63}+3190 q^{62}+5443 q^{61}+6370 q^{60}+3622 q^{59}-1238 q^{58}-5822 q^{57}-9224 q^{56}-7979 q^{55}-3200 q^{54}+2987 q^{53}+9746 q^{52}+11975 q^{51}+9178 q^{50}+2980 q^{49}-6316 q^{48}-12313 q^{47}-13642 q^{46}-10644 q^{45}-1690 q^{44}+6900 q^{43}+13181 q^{42}+15963 q^{41}+11257 q^{40}+4073 q^{39}-4503 q^{38}-13756 q^{37}-17258 q^{36}-17364 q^{35}-11832 q^{34}+905 q^{33}+13252 q^{32}+25589 q^{31}+31084 q^{30}+22737 q^{29}+5227 q^{28}-21326 q^{27}-45132 q^{26}-51161 q^{25}-37738 q^{24}-1252 q^{23}+44366 q^{22}+74957 q^{21}+78577 q^{20}+42293 q^{19}-22043 q^{18}-83392 q^{17}-117236 q^{16}-96077 q^{15}-23117 q^{14}+68266 q^{13}+141795 q^{12}+152136 q^{11}+86431 q^{10}-26730 q^9-143331 q^8-198453 q^7-157725 q^6-37398 q^5+117268 q^4+224549 q^3+225545 q^2+116048 q-65594-225596 q^{-1} -279447 q^{-2} -197716 q^{-3} -4731 q^{-4} +200891 q^{-5} +312574 q^{-6} +273053 q^{-7} +84760 q^{-8} -155868 q^{-9} -323974 q^{-10} -334103 q^{-11} -164262 q^{-12} +97583 q^{-13} +315163 q^{-14} +377966 q^{-15} +236813 q^{-16} -34453 q^{-17} -292507 q^{-18} -404840 q^{-19} -296738 q^{-20} -26454 q^{-21} +261167 q^{-22} +417350 q^{-23} +343536 q^{-24} +80554 q^{-25} -228062 q^{-26} -419971 q^{-27} -377010 q^{-28} -124735 q^{-29} +196753 q^{-30} +416587 q^{-31} +400284 q^{-32} +158918 q^{-33} -170943 q^{-34} -411425 q^{-35} -415711 q^{-36} -183943 q^{-37} +151132 q^{-38} +407191 q^{-39} +427414 q^{-40} +202430 q^{-41} -137432 q^{-42} -405703 q^{-43} -438133 q^{-44} -218015 q^{-45} +127069 q^{-46} +407318 q^{-47} +451159 q^{-48} +234645 q^{-49} -116843 q^{-50} -410023 q^{-51} -467104 q^{-52} -256389 q^{-53} +101019 q^{-54} +410025 q^{-55} +485514 q^{-56} +285538 q^{-57} -75241 q^{-58} -401454 q^{-59} -501391 q^{-60} -321524 q^{-61} +35128 q^{-62} +377573 q^{-63} +508816 q^{-64} +360309 q^{-65} +18722 q^{-66} -333930 q^{-67} -499139 q^{-68} -393445 q^{-69} -82426 q^{-70} +268990 q^{-71} +466223 q^{-72} +411992 q^{-73} +146770 q^{-74} -187803 q^{-75} -407913 q^{-76} -407073 q^{-77} -200167 q^{-78} +99580 q^{-79} +327782 q^{-80} +375102 q^{-81} +232986 q^{-82} -17228 q^{-83} -236211 q^{-84} -318697 q^{-85} -238645 q^{-86} -47240 q^{-87} +145329 q^{-88} +246292 q^{-89} +218615 q^{-90} +86908 q^{-91} -68242 q^{-92} -170403 q^{-93} -179246 q^{-94} -100329 q^{-95} +12623 q^{-96} +102195 q^{-97} +131276 q^{-98} +92925 q^{-99} +19427 q^{-100} -49942 q^{-101} -84989 q^{-102} -73050 q^{-103} -31339 q^{-104} +16175 q^{-105} +47526 q^{-106} +49468 q^{-107} +29851 q^{-108} +1405 q^{-109} -21905 q^{-110} -28819 q^{-111} -22099 q^{-112} -7330 q^{-113} +7487 q^{-114} +13906 q^{-115} +13180 q^{-116} +7051 q^{-117} -883 q^{-118} -5172 q^{-119} -6478 q^{-120} -4481 q^{-121} -748 q^{-122} +1140 q^{-123} +2197 q^{-124} +1916 q^{-125} +595 q^{-126} +229 q^{-127} -347 q^{-128} -433 q^{-129} +110 q^{-130} -192 q^{-131} -185 q^{-132} -307 q^{-133} -579 q^{-134} -9 q^{-135} +196 q^{-136} +329 q^{-137} +583 q^{-138} +249 q^{-139} +70 q^{-140} -204 q^{-141} -551 q^{-142} -273 q^{-143} -96 q^{-144} +53 q^{-145} +286 q^{-146} +186 q^{-147} +184 q^{-148} +57 q^{-149} -182 q^{-150} -131 q^{-151} -99 q^{-152} -51 q^{-153} +58 q^{-154} +39 q^{-155} +71 q^{-156} +60 q^{-157} -33 q^{-158} -18 q^{-159} -29 q^{-160} -24 q^{-161} +10 q^{-162} - q^{-163} +13 q^{-164} +14 q^{-165} -7 q^{-166} -2 q^{-168} -6 q^{-169} +3 q^{-170} - q^{-171} +3 q^{-172} + q^{-173} -3 q^{-174} + q^{-175} }

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 29, 2005, 15:27:48)...
In[2]:=	PD[Knot[10, 82]]
Out[2]=	PD[X[1, 6, 2, 7], X[7, 16, 8, 17], X[3, 9, 4, 8], X[15, 3, 16, 2], X[5, 15, 6, 14], X[9, 5, 10, 4], X[11, 18, 12, 19], X[13, 20, 14, 1], X[17, 10, 18, 11], X[19, 12, 20, 13]]
In[3]:=	GaussCode[Knot[10, 82]]
Out[3]=	GaussCode[-1, 4, -3, 6, -5, 1, -2, 3, -6, 9, -7, 10, -8, 5, -4, 2, -9, 7, -10, 8]
In[4]:=	DTCode[Knot[10, 82]]
Out[4]=	DTCode[6, 8, 14, 16, 4, 18, 20, 2, 10, 12]
In[5]:=	br = BR[Knot[10, 82]]
Out[5]=	BR[3, {-1, -1, -1, -1, 2, -1, 2, -1, 2, 2}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{3, 10}
In[7]:=	BraidIndex[Knot[10, 82]]
Out[7]=	3
In[8]:=	Show[DrawMorseLink[Knot[10, 82]]]

`Out[8]=`	`-Graphics-`
In[9]:=	(#[Knot[10, 82]]&) /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Chiral, 1, 4, 3, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 82]][t]
Out[10]=	-4 4 8 12 2 3 4 -13 - t + -- - -- + -- + 12 t - 8 t + 4 t - t 3 2 t t t
In[11]:=	Conway[Knot[10, 82]][z]
Out[11]=	4 6 8 1 - 4 z - 4 z - z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 82]}
In[13]:=	{KnotDet[Knot[10, 82]], KnotSignature[Knot[10, 82]]}
Out[13]=	{63, -2}
In[14]:=	Jones[Knot[10, 82]][q]
Out[14]=	-7 3 5 8 10 10 10 2 3 -7 + q - -- + -- - -- + -- - -- + -- + 5 q - 3 q + q 6 5 4 3 2 q q q q q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] \|\| (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[10, 82]}
In[16]:=	A2Invariant[Knot[10, 82]][q]
Out[16]=	-20 -18 -16 2 -12 -10 -8 4 -4 2 2 q - q + q - --- - q + q - q + -- - q + -- - q + 14 6 2 q q q 4 6 8 q - q + q
In[17]:=	HOMFLYPT[Knot[10, 82]][a, z]
Out[17]=	2 2 2 4 2 4 2 4 4 4 6 1 + 4 z - 8 a z + 4 a z + 4 z - 12 a z + 4 a z + z - 2 6 4 6 2 8 6 a z + a z - a z
In[18]:=	Kauffman[Knot[10, 82]][a, z]
Out[18]=	2 z 5 7 2 z 2 2 4 2 1 - - - 2 a z + 2 a z + a z - 6 z + -- - 13 a z - 5 a z - a 2 a 3 4 8 2 7 z 3 3 3 5 3 7 3 4 3 z a z + ---- + 10 a z + 5 a z - 2 a z - 4 a z + 14 z - ---- + a 2 a 5 2 4 4 4 6 4 8 4 10 z 5 3 5 32 a z + 10 a z - 4 a z + a z - ----- - 8 a z - 4 a z - a 6 5 5 7 5 6 z 2 6 4 6 6 6 3 a z + 3 a z - 14 z + -- - 27 a z - 8 a z + 4 a z + 2 a 7 3 z 7 3 7 5 7 8 2 8 4 8 9 ---- - 2 a z - a z + 4 a z + 4 z + 8 a z + 4 a z + 2 a z + a 3 9 2 a z
In[19]:=	{Vassiliev[2][Knot[10, 82]], Vassiliev[3][Knot[10, 82]]}
Out[19]=	{0, 0}
In[20]:=	Kh[Knot[10, 82]][q, t]
Out[20]=	5 6 1 2 1 3 2 5 3 -- + - + ------ + ------ + ------ + ------ + ----- + ----- + ----- + 3 q 15 6 13 5 11 5 11 4 9 4 9 3 7 3 q q t q t q t q t q t q t q t 5 5 5 5 3 t 2 3 2 ----- + ----- + ---- + ---- + --- + 4 q t + 2 q t + 3 q t + 7 2 5 2 5 3 q q t q t q t q t 3 3 5 3 7 4 q t + 2 q t + q t
In[21]:=	ColouredJones[Knot[10, 82], 2][q]
Out[21]=	-20 3 -18 7 12 4 17 31 11 36 -20 + q - --- + q + --- - --- + --- + --- - --- + --- + --- - 19 17 16 15 14 13 12 11 q q q q q q q q 58 13 60 75 5 74 70 9 71 49 2 --- + -- + -- - -- + -- + -- - -- - -- + -- - -- + 54 q - 23 q - 10 9 8 7 6 5 4 3 2 q q q q q q q q q q 3 4 5 6 7 8 9 10 23 q + 30 q - 4 q - 14 q + 9 q + q - 3 q + q

See/edit the Rolfsen_Splice_Template.

@@ Line 16: / Line 16: @@
 {{Knot Presentations}}
+<center><table border=1 cellpadding=10><tr align=center valign=top>
+<td>
+[[Braid Representatives|Minimum Braid Representative]]:
+<table cellspacing=0 cellpadding=0 border=0>
+<tr><td>[[Image:BraidPart3.gif]][[Image:BraidPart3.gif]][[Image:BraidPart3.gif]][[Image:BraidPart3.gif]][[Image:BraidPart0.gif]][[Image:BraidPart3.gif]][[Image:BraidPart0.gif]][[Image:BraidPart3.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]]</td></tr>
+<tr><td>[[Image:BraidPart4.gif]][[Image:BraidPart4.gif]][[Image:BraidPart4.gif]][[Image:BraidPart4.gif]][[Image:BraidPart1.gif]][[Image:BraidPart4.gif]][[Image:BraidPart1.gif]][[Image:BraidPart4.gif]][[Image:BraidPart1.gif]][[Image:BraidPart1.gif]]</td></tr>
+<tr><td>[[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart0.gif]][[Image:BraidPart2.gif]][[Image:BraidPart0.gif]][[Image:BraidPart2.gif]][[Image:BraidPart0.gif]][[Image:BraidPart2.gif]][[Image:BraidPart2.gif]]</td></tr>
+</table>
+[[Invariants from Braid Theory|Length]] is 10, width is 3.
+[[Invariants from Braid Theory|Braid index]] is 3.
+</td>
+<td>
+[[Lightly Documented Features|A Morse Link Presentation]]:
+[[Image:{{PAGENAME}}_ML.gif]]
+</td>
+</tr></table></center>
 {{3D Invariants}}
 {{4D Invariants}}
 {{Polynomial Invariants}}
+=== "Similar" Knots (within the Atlas) ===
+Same [[The Alexander-Conway Polynomial|Alexander/Conway Polynomial]]:
+{...}
+Same [[The Jones Polynomial|Jones Polynomial]] (up to mirroring, <math>q\leftrightarrow q^{-1}</math>):
+{...}
 {{Vassiliev Invariants}}
@@ Line 42: / Line 72: @@
 <tr align=center><td>-15</td><td bgcolor=yellow>1</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>1</td></tr>
 </table>}}
+{{Display Coloured Jones|J2=<math>q^{10}-3 q^9+q^8+9 q^7-14 q^6-4 q^5+30 q^4-23 q^3-23 q^2+54 q-20-49 q^{-1} +71 q^{-2} -9 q^{-3} -70 q^{-4} +74 q^{-5} +5 q^{-6} -75 q^{-7} +60 q^{-8} +13 q^{-9} -58 q^{-10} +36 q^{-11} +11 q^{-12} -31 q^{-13} +17 q^{-14} +4 q^{-15} -12 q^{-16} +7 q^{-17} + q^{-18} -3 q^{-19} + q^{-20} </math>|J3=<math>q^{21}-3 q^{20}+q^{19}+5 q^{18}+q^{17}-14 q^{16}-6 q^{15}+28 q^{14}+19 q^{13}-41 q^{12}-44 q^{11}+46 q^{10}+81 q^9-39 q^8-119 q^7+13 q^6+149 q^5+30 q^4-165 q^3-83 q^2+166 q+130-140 q^{-1} -183 q^{-2} +119 q^{-3} +211 q^{-4} -72 q^{-5} -250 q^{-6} +45 q^{-7} +259 q^{-8} +2 q^{-9} -276 q^{-10} -31 q^{-11} +263 q^{-12} +71 q^{-13} -247 q^{-14} -90 q^{-15} +203 q^{-16} +105 q^{-17} -153 q^{-18} -103 q^{-19} +103 q^{-20} +80 q^{-21} -54 q^{-22} -56 q^{-23} +28 q^{-24} +25 q^{-25} -11 q^{-26} -7 q^{-27} +10 q^{-28} -6 q^{-29} -8 q^{-30} +6 q^{-31} +10 q^{-32} -5 q^{-33} -8 q^{-34} +3 q^{-35} +3 q^{-36} + q^{-37} -3 q^{-38} + q^{-39} </math>|J4=<math>q^{36}-3 q^{35}+q^{34}+5 q^{33}-3 q^{32}+q^{31}-17 q^{30}+6 q^{29}+31 q^{28}+2 q^{26}-80 q^{25}-20 q^{24}+92 q^{23}+63 q^{22}+70 q^{21}-192 q^{20}-160 q^{19}+76 q^{18}+156 q^{17}+317 q^{16}-183 q^{15}-346 q^{14}-142 q^{13}+54 q^{12}+610 q^{11}+75 q^{10}-282 q^9-387 q^8-362 q^7+613 q^6+361 q^5+142 q^4-315 q^3-838 q^2+230 q+347+671 q^{-1} +132 q^{-2} -1066 q^{-3} -287 q^{-4} +8 q^{-5} +1021 q^{-6} +702 q^{-7} -1021 q^{-8} -699 q^{-9} -426 q^{-10} +1169 q^{-11} +1170 q^{-12} -859 q^{-13} -969 q^{-14} -796 q^{-15} +1194 q^{-16} +1505 q^{-17} -624 q^{-18} -1125 q^{-19} -1119 q^{-20} +1053 q^{-21} +1697 q^{-22} -245 q^{-23} -1049 q^{-24} -1369 q^{-25} +629 q^{-26} +1593 q^{-27} +218 q^{-28} -631 q^{-29} -1346 q^{-30} +61 q^{-31} +1095 q^{-32} +466 q^{-33} -62 q^{-34} -942 q^{-35} -289 q^{-36} +457 q^{-37} +353 q^{-38} +270 q^{-39} -420 q^{-40} -273 q^{-41} +58 q^{-42} +102 q^{-43} +265 q^{-44} -99 q^{-45} -113 q^{-46} -41 q^{-47} -33 q^{-48} +132 q^{-49} -3 q^{-50} -13 q^{-51} -21 q^{-52} -44 q^{-53} +40 q^{-54} +3 q^{-55} +8 q^{-56} - q^{-57} -17 q^{-58} +7 q^{-59} - q^{-60} +3 q^{-61} + q^{-62} -3 q^{-63} + q^{-64} </math>|J5=<math>q^{55}-3 q^{54}+q^{53}+5 q^{52}-3 q^{51}-3 q^{50}-2 q^{49}-5 q^{48}+8 q^{47}+26 q^{46}+4 q^{45}-31 q^{44}-41 q^{43}-32 q^{42}+31 q^{41}+108 q^{40}+106 q^{39}-27 q^{38}-179 q^{37}-220 q^{36}-83 q^{35}+215 q^{34}+417 q^{33}+287 q^{32}-154 q^{31}-571 q^{30}-595 q^{29}-108 q^{28}+598 q^{27}+929 q^{26}+536 q^{25}-383 q^{24}-1102 q^{23}-1021 q^{22}-135 q^{21}+962 q^{20}+1393 q^{19}+795 q^{18}-423 q^{17}-1376 q^{16}-1401 q^{15}-459 q^{14}+859 q^{13}+1665 q^{12}+1436 q^{11}+186 q^{10}-1382 q^9-2220 q^8-1566 q^7+444 q^6+2563 q^5+3042 q^4+999 q^3-2321 q^2-4246 q-2819+1451 q^{-1} +5150 q^{-2} +4654 q^{-3} -199 q^{-4} -5449 q^{-5} -6364 q^{-6} -1414 q^{-7} +5459 q^{-8} +7770 q^{-9} +2920 q^{-10} -5011 q^{-11} -8830 q^{-12} -4455 q^{-13} +4544 q^{-14} +9608 q^{-15} +5625 q^{-16} -3920 q^{-17} -10143 q^{-18} -6722 q^{-19} +3459 q^{-20} +10556 q^{-21} +7526 q^{-22} -2918 q^{-23} -10877 q^{-24} -8367 q^{-25} +2456 q^{-26} +11079 q^{-27} +9088 q^{-28} -1726 q^{-29} -11082 q^{-30} -9890 q^{-31} +827 q^{-32} +10747 q^{-33} +10478 q^{-34} +438 q^{-35} -9888 q^{-36} -10864 q^{-37} -1855 q^{-38} +8497 q^{-39} +10718 q^{-40} +3252 q^{-41} -6585 q^{-42} -9951 q^{-43} -4402 q^{-44} +4437 q^{-45} +8603 q^{-46} +4982 q^{-47} -2356 q^{-48} -6754 q^{-49} -4992 q^{-50} +622 q^{-51} +4837 q^{-52} +4407 q^{-53} +532 q^{-54} -3037 q^{-55} -3508 q^{-56} -1115 q^{-57} +1653 q^{-58} +2498 q^{-59} +1232 q^{-60} -696 q^{-61} -1640 q^{-62} -1061 q^{-63} +178 q^{-64} +944 q^{-65} +802 q^{-66} +90 q^{-67} -520 q^{-68} -546 q^{-69} -145 q^{-70} +237 q^{-71} +332 q^{-72} +155 q^{-73} -88 q^{-74} -198 q^{-75} -120 q^{-76} +30 q^{-77} +93 q^{-78} +73 q^{-79} +16 q^{-80} -43 q^{-81} -53 q^{-82} -6 q^{-83} +17 q^{-84} +12 q^{-85} +16 q^{-86} -3 q^{-87} -13 q^{-88} -2 q^{-89} +3 q^{-90} - q^{-91} +3 q^{-92} + q^{-93} -3 q^{-94} + q^{-95} </math>|J6=<math>q^{78}-3 q^{77}+q^{76}+5 q^{75}-3 q^{74}-3 q^{73}-6 q^{72}+10 q^{71}-3 q^{70}+3 q^{69}+29 q^{68}-15 q^{67}-31 q^{66}-50 q^{65}+16 q^{64}+18 q^{63}+57 q^{62}+149 q^{61}+13 q^{60}-108 q^{59}-269 q^{58}-138 q^{57}-83 q^{56}+165 q^{55}+588 q^{54}+435 q^{53}+110 q^{52}-592 q^{51}-726 q^{50}-907 q^{49}-356 q^{48}+952 q^{47}+1486 q^{46}+1487 q^{45}+150 q^{44}-801 q^{43}-2351 q^{42}-2430 q^{41}-528 q^{40}+1390 q^{39}+3160 q^{38}+2606 q^{37}+1802 q^{36}-1595 q^{35}-3927 q^{34}-3673 q^{33}-2004 q^{32}+1294 q^{31}+3033 q^{30}+5316 q^{29}+2875 q^{28}-323 q^{27}-2954 q^{26}-4788 q^{25}-4200 q^{24}-3084 q^{23}+2559 q^{22}+4694 q^{21}+6288 q^{20}+5394 q^{19}+1266 q^{18}-4553 q^{17}-10970 q^{16}-8683 q^{15}-4599 q^{14}+5123 q^{13}+13899 q^{12}+16169 q^{11}+8088 q^{10}-8616 q^9-18535 q^8-22543 q^7-10353 q^6+10285 q^5+28258 q^4+28942 q^3+8648 q^2-15292 q-36555-33436 q^{-1} -8032 q^{-2} +27043 q^{-3} +45373 q^{-4} +32754 q^{-5} +1665 q^{-6} -37824 q^{-7} -52042 q^{-8} -32205 q^{-9} +13503 q^{-10} +50279 q^{-11} +52582 q^{-12} +23344 q^{-13} -28472 q^{-14} -60533 q^{-15} -52249 q^{-16} -3779 q^{-17} +46271 q^{-18} +63647 q^{-19} +41071 q^{-20} -16261 q^{-21} -61493 q^{-22} -64549 q^{-23} -17582 q^{-24} +39891 q^{-25} +68311 q^{-26} +52193 q^{-27} -6878 q^{-28} -60252 q^{-29} -71364 q^{-30} -26411 q^{-31} +35246 q^{-32} +70996 q^{-33} +59376 q^{-34} -459 q^{-35} -59542 q^{-36} -76726 q^{-37} -33819 q^{-38} +31002 q^{-39} +73370 q^{-40} +66752 q^{-41} +7641 q^{-42} -56589 q^{-43} -81421 q^{-44} -44089 q^{-45} +21451 q^{-46} +71209 q^{-47} +74062 q^{-48} +21515 q^{-49} -44801 q^{-50} -79670 q^{-51} -55378 q^{-52} +3232 q^{-53} +57328 q^{-54} +73753 q^{-55} +37124 q^{-56} -22204 q^{-57} -64074 q^{-58} -58169 q^{-59} -16982 q^{-60} +31733 q^{-61} +58619 q^{-62} +43188 q^{-63} +1708 q^{-64} -37044 q^{-65} -45851 q^{-66} -26624 q^{-67} +6394 q^{-68} +33388 q^{-69} +34266 q^{-70} +13927 q^{-71} -12191 q^{-72} -25040 q^{-73} -21824 q^{-74} -6427 q^{-75} +11918 q^{-76} +18345 q^{-77} +12565 q^{-78} +130 q^{-79} -8631 q^{-80} -11256 q^{-81} -7032 q^{-82} +1861 q^{-83} +6599 q^{-84} +6378 q^{-85} +2139 q^{-86} -1540 q^{-87} -3953 q^{-88} -3710 q^{-89} -260 q^{-90} +1755 q^{-91} +2312 q^{-92} +1069 q^{-93} +20 q^{-94} -1161 q^{-95} -1533 q^{-96} -185 q^{-97} +463 q^{-98} +820 q^{-99} +386 q^{-100} +163 q^{-101} -364 q^{-102} -661 q^{-103} -98 q^{-104} +96 q^{-105} +311 q^{-106} +156 q^{-107} +153 q^{-108} -84 q^{-109} -261 q^{-110} -60 q^{-111} -24 q^{-112} +87 q^{-113} +43 q^{-114} +87 q^{-115} +2 q^{-116} -72 q^{-117} -15 q^{-118} -23 q^{-119} +16 q^{-120} +25 q^{-122} +5 q^{-123} -15 q^{-124} +2 q^{-125} -6 q^{-126} +3 q^{-127} - q^{-128} +3 q^{-129} + q^{-130} -3 q^{-131} + q^{-132} </math>|J7=<math>q^{105}-3 q^{104}+q^{103}+5 q^{102}-3 q^{101}-3 q^{100}-6 q^{99}+6 q^{98}+12 q^{97}-8 q^{96}+6 q^{95}+10 q^{94}-16 q^{93}-26 q^{92}-41 q^{91}+6 q^{90}+76 q^{89}+46 q^{88}+67 q^{87}+39 q^{86}-79 q^{85}-150 q^{84}-274 q^{83}-157 q^{82}+146 q^{81}+311 q^{80}+515 q^{79}+476 q^{78}+82 q^{77}-345 q^{76}-1034 q^{75}-1244 q^{74}-641 q^{73}+155 q^{72}+1419 q^{71}+2168 q^{70}+1927 q^{69}+1023 q^{68}-1195 q^{67}-3255 q^{66}-3837 q^{65}-3138 q^{64}-240 q^{63}+3190 q^{62}+5443 q^{61}+6370 q^{60}+3622 q^{59}-1238 q^{58}-5822 q^{57}-9224 q^{56}-7979 q^{55}-3200 q^{54}+2987 q^{53}+9746 q^{52}+11975 q^{51}+9178 q^{50}+2980 q^{49}-6316 q^{48}-12313 q^{47}-13642 q^{46}-10644 q^{45}-1690 q^{44}+6900 q^{43}+13181 q^{42}+15963 q^{41}+11257 q^{40}+4073 q^{39}-4503 q^{38}-13756 q^{37}-17258 q^{36}-17364 q^{35}-11832 q^{34}+905 q^{33}+13252 q^{32}+25589 q^{31}+31084 q^{30}+22737 q^{29}+5227 q^{28}-21326 q^{27}-45132 q^{26}-51161 q^{25}-37738 q^{24}-1252 q^{23}+44366 q^{22}+74957 q^{21}+78577 q^{20}+42293 q^{19}-22043 q^{18}-83392 q^{17}-117236 q^{16}-96077 q^{15}-23117 q^{14}+68266 q^{13}+141795 q^{12}+152136 q^{11}+86431 q^{10}-26730 q^9-143331 q^8-198453 q^7-157725 q^6-37398 q^5+117268 q^4+224549 q^3+225545 q^2+116048 q-65594-225596 q^{-1} -279447 q^{-2} -197716 q^{-3} -4731 q^{-4} +200891 q^{-5} +312574 q^{-6} +273053 q^{-7} +84760 q^{-8} -155868 q^{-9} -323974 q^{-10} -334103 q^{-11} -164262 q^{-12} +97583 q^{-13} +315163 q^{-14} +377966 q^{-15} +236813 q^{-16} -34453 q^{-17} -292507 q^{-18} -404840 q^{-19} -296738 q^{-20} -26454 q^{-21} +261167 q^{-22} +417350 q^{-23} +343536 q^{-24} +80554 q^{-25} -228062 q^{-26} -419971 q^{-27} -377010 q^{-28} -124735 q^{-29} +196753 q^{-30} +416587 q^{-31} +400284 q^{-32} +158918 q^{-33} -170943 q^{-34} -411425 q^{-35} -415711 q^{-36} -183943 q^{-37} +151132 q^{-38} +407191 q^{-39} +427414 q^{-40} +202430 q^{-41} -137432 q^{-42} -405703 q^{-43} -438133 q^{-44} -218015 q^{-45} +127069 q^{-46} +407318 q^{-47} +451159 q^{-48} +234645 q^{-49} -116843 q^{-50} -410023 q^{-51} -467104 q^{-52} -256389 q^{-53} +101019 q^{-54} +410025 q^{-55} +485514 q^{-56} +285538 q^{-57} -75241 q^{-58} -401454 q^{-59} -501391 q^{-60} -321524 q^{-61} +35128 q^{-62} +377573 q^{-63} +508816 q^{-64} +360309 q^{-65} +18722 q^{-66} -333930 q^{-67} -499139 q^{-68} -393445 q^{-69} -82426 q^{-70} +268990 q^{-71} +466223 q^{-72} +411992 q^{-73} +146770 q^{-74} -187803 q^{-75} -407913 q^{-76} -407073 q^{-77} -200167 q^{-78} +99580 q^{-79} +327782 q^{-80} +375102 q^{-81} +232986 q^{-82} -17228 q^{-83} -236211 q^{-84} -318697 q^{-85} -238645 q^{-86} -47240 q^{-87} +145329 q^{-88} +246292 q^{-89} +218615 q^{-90} +86908 q^{-91} -68242 q^{-92} -170403 q^{-93} -179246 q^{-94} -100329 q^{-95} +12623 q^{-96} +102195 q^{-97} +131276 q^{-98} +92925 q^{-99} +19427 q^{-100} -49942 q^{-101} -84989 q^{-102} -73050 q^{-103} -31339 q^{-104} +16175 q^{-105} +47526 q^{-106} +49468 q^{-107} +29851 q^{-108} +1405 q^{-109} -21905 q^{-110} -28819 q^{-111} -22099 q^{-112} -7330 q^{-113} +7487 q^{-114} +13906 q^{-115} +13180 q^{-116} +7051 q^{-117} -883 q^{-118} -5172 q^{-119} -6478 q^{-120} -4481 q^{-121} -748 q^{-122} +1140 q^{-123} +2197 q^{-124} +1916 q^{-125} +595 q^{-126} +229 q^{-127} -347 q^{-128} -433 q^{-129} +110 q^{-130} -192 q^{-131} -185 q^{-132} -307 q^{-133} -579 q^{-134} -9 q^{-135} +196 q^{-136} +329 q^{-137} +583 q^{-138} +249 q^{-139} +70 q^{-140} -204 q^{-141} -551 q^{-142} -273 q^{-143} -96 q^{-144} +53 q^{-145} +286 q^{-146} +186 q^{-147} +184 q^{-148} +57 q^{-149} -182 q^{-150} -131 q^{-151} -99 q^{-152} -51 q^{-153} +58 q^{-154} +39 q^{-155} +71 q^{-156} +60 q^{-157} -33 q^{-158} -18 q^{-159} -29 q^{-160} -24 q^{-161} +10 q^{-162} - q^{-163} +13 q^{-164} +14 q^{-165} -7 q^{-166} -2 q^{-168} -6 q^{-169} +3 q^{-170} - q^{-171} +3 q^{-172} + q^{-173} -3 q^{-174} + q^{-175} </math>}}
 {{Computer Talk Header}}
@@ Line 49: / Line 82: @@
 <td align=left><pre style="color: red; border: 0px; padding: 0em">&lt;&lt; KnotTheory`</pre></td>
 </tr>
-<tr valign=top><td colspan=2><pre style="border: 0px; padding: 0em">Loading KnotTheory` (version of August 17, 2005, 14:44:34)...</pre></td></tr>
+<tr valign=top><td colspan=2><pre style="border: 0px; padding: 0em">Loading KnotTheory` (version of August 29, 2005, 15:27:48)...</pre></td></tr>
-<tr valign=top><td><pre style="color:  blue; border: 0px; padding: 0em"><nowiki>In[2]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>Crossings[Knot[10, 82]]</nowiki></pre></td></tr>
-<tr valign=top><td><pre  style="color: blue; border: 0px; padding: 0em"><nowiki>Out[2]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>10</nowiki></pre></td></tr>
+<tr valign=top><td><pre style="color:  blue; border: 0px; padding: 0em"><nowiki>In[2]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>PD[Knot[10, 82]]</nowiki></pre></td></tr>
-<tr valign=top><td><pre style="color:  blue; border: 0px; padding: 0em"><nowiki>In[3]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>PD[Knot[10, 82]]</nowiki></pre></td></tr>
+<tr valign=top><td><pre  style="color: blue; border: 0px; padding: 0em"><nowiki>Out[2]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>PD[X[1, 6, 2, 7], X[7, 16, 8, 17], X[3, 9, 4, 8], X[15, 3, 16, 2],
-<tr valign=top><td><pre  style="color: blue; border: 0px; padding: 0em"><nowiki>Out[3]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>PD[X[1, 6, 2, 7], X[7, 16, 8, 17], X[3, 9, 4, 8], X[15, 3, 16, 2],
   X[5, 15, 6, 14], X[9, 5, 10, 4], X[11, 18, 12, 19], X[13, 20, 14, 1],
   X[17, 10, 18, 11], X[19, 12, 20, 13]]</nowiki></pre></td></tr>
-<tr valign=top><td><pre style="color:  blue; border: 0px; padding: 0em"><nowiki>In[4]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>GaussCode[Knot[10, 82]]</nowiki></pre></td></tr>
-<tr valign=top><td><pre  style="color: blue; border: 0px; padding: 0em"><nowiki>Out[4]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>GaussCode[-1, 4, -3, 6, -5, 1, -2, 3, -6, 9, -7, 10, -8, 5, -4, 2, -9,
+<tr valign=top><td><pre style="color:  blue; border: 0px; padding: 0em"><nowiki>In[3]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>GaussCode[Knot[10, 82]]</nowiki></pre></td></tr>
+<tr valign=top><td><pre  style="color: blue; border: 0px; padding: 0em"><nowiki>Out[3]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>GaussCode[-1, 4, -3, 6, -5, 1, -2, 3, -6, 9, -7, 10, -8, 5, -4, 2, -9,
 , -10, 8]</nowiki></pre></td></tr>
-<tr valign=top><td><pre style="color:  blue; border: 0px; padding: 0em"><nowiki>In[5]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>BR[Knot[10, 82]]</nowiki></pre></td></tr>
+<tr valign=top><td><pre style="color:  blue; border: 0px; padding: 0em"><nowiki>In[4]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>DTCode[Knot[10, 82]]</nowiki></pre></td></tr>
+<tr valign=top><td><pre  style="color: blue; border: 0px; padding: 0em"><nowiki>Out[4]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>DTCode[6, 8, 14, 16, 4, 18, 20, 2, 10, 12]</nowiki></pre></td></tr>
+<tr valign=top><td><pre style="color:  blue; border: 0px; padding: 0em"><nowiki>In[5]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>br = BR[Knot[10, 82]]</nowiki></pre></td></tr>
 <tr valign=top><td><pre  style="color: blue; border: 0px; padding: 0em"><nowiki>Out[5]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>BR[3, {-1, -1, -1, -1, 2, -1, 2, -1, 2, 2}]</nowiki></pre></td></tr>
-<tr valign=top><td><pre style="color:  blue; border: 0px; padding: 0em"><nowiki>In[6]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>alex = Alexander[Knot[10, 82]][t]</nowiki></pre></td></tr>
-<tr valign=top><td><pre  style="color: blue; border: 0px; padding: 0em"><nowiki>Out[6]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>       -4   4    8    12             2      3    4
+<tr valign=top><td><pre style="color:  blue; border: 0px; padding: 0em"><nowiki>In[6]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>{First[br], Crossings[br]}</nowiki></pre></td></tr>
+<tr valign=top><td><pre  style="color: blue; border: 0px; padding: 0em"><nowiki>Out[6]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>{3, 10}</nowiki></pre></td></tr>
+<tr valign=top><td><pre style="color:  blue; border: 0px; padding: 0em"><nowiki>In[7]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>BraidIndex[Knot[10, 82]]</nowiki></pre></td></tr>
+<tr valign=top><td><pre  style="color: blue; border: 0px; padding: 0em"><nowiki>Out[7]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>3</nowiki></pre></td></tr>
+<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[8]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>Show[DrawMorseLink[Knot[10, 82]]]</nowiki></pre></td></tr><tr><td></td><td align=left>[[Image:10_82_ML.gif]]</td></tr><tr valign=top><td><tt><font color=blue>Out[8]=</font></tt><td><tt><font color=black>-Graphics-</font></tt></td></tr>
+<tr valign=top><td><pre style="color:  blue; border: 0px; padding: 0em"><nowiki>In[9]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>(#[Knot[10, 82]]&) /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}</nowiki></pre></td></tr>
+<tr valign=top><td><pre  style="color: blue; border: 0px; padding: 0em"><nowiki>Out[9]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>{Chiral, 1, 4, 3, NotAvailable, 1}</nowiki></pre></td></tr>
+<tr valign=top><td><pre style="color:  blue; border: 0px; padding: 0em"><nowiki>In[10]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>alex = Alexander[Knot[10, 82]][t]</nowiki></pre></td></tr>
+<tr valign=top><td><pre  style="color: blue; border: 0px; padding: 0em"><nowiki>Out[10]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>       -4   4    8    12             2      3    4
 -13 - t   + -- - -- + -- + 12 t - 8 t  + 4 t  - t
 2   t
             t    t</nowiki></pre></td></tr>
-<tr valign=top><td><pre style="color:  blue; border: 0px; padding: 0em"><nowiki>In[7]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>Conway[Knot[10, 82]][z]</nowiki></pre></td></tr>
-<tr valign=top><td><pre  style="color: blue; border: 0px; padding: 0em"><nowiki>Out[7]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>       4      6    8
+<tr valign=top><td><pre style="color:  blue; border: 0px; padding: 0em"><nowiki>In[11]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>Conway[Knot[10, 82]][z]</nowiki></pre></td></tr>
+<tr valign=top><td><pre  style="color: blue; border: 0px; padding: 0em"><nowiki>Out[11]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>       4      6    8
 - 4 z  - 4 z  - z</nowiki></pre></td></tr>
-<tr valign=top><td><pre style="color:  blue; border: 0px; padding: 0em"><nowiki>In[8]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>Select[AllKnots[], (alex === Alexander[#][t])&]</nowiki></pre></td></tr>
-<tr valign=top><td><pre  style="color: blue; border: 0px; padding: 0em"><nowiki>Out[8]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>{Knot[10, 82]}</nowiki></pre></td></tr>
+<tr valign=top><td><pre style="color:  blue; border: 0px; padding: 0em"><nowiki>In[12]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>Select[AllKnots[], (alex === Alexander[#][t])&]</nowiki></pre></td></tr>
-<tr valign=top><td><pre style="color:  blue; border: 0px; padding: 0em"><nowiki>In[9]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>{KnotDet[Knot[10, 82]], KnotSignature[Knot[10, 82]]}</nowiki></pre></td></tr>
+<tr valign=top><td><pre  style="color: blue; border: 0px; padding: 0em"><nowiki>Out[12]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>{Knot[10, 82]}</nowiki></pre></td></tr>
-<tr valign=top><td><pre  style="color: blue; border: 0px; padding: 0em"><nowiki>Out[9]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>{63, -2}</nowiki></pre></td></tr>
-<tr valign=top><td><pre style="color:  blue; border: 0px; padding: 0em"><nowiki>In[10]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>J=Jones[Knot[10, 82]][q]</nowiki></pre></td></tr>
+<tr valign=top><td><pre style="color:  blue; border: 0px; padding: 0em"><nowiki>In[13]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>{KnotDet[Knot[10, 82]], KnotSignature[Knot[10, 82]]}</nowiki></pre></td></tr>
-<tr valign=top><td><pre  style="color: blue; border: 0px; padding: 0em"><nowiki>Out[10]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>      -7   3    5    8    10   10   10            2    3
+<tr valign=top><td><pre  style="color: blue; border: 0px; padding: 0em"><nowiki>Out[13]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>{63, -2}</nowiki></pre></td></tr>
+<tr valign=top><td><pre style="color:  blue; border: 0px; padding: 0em"><nowiki>In[14]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>Jones[Knot[10, 82]][q]</nowiki></pre></td></tr>
+<tr valign=top><td><pre  style="color: blue; border: 0px; padding: 0em"><nowiki>Out[14]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>      -7   3    5    8    10   10   10            2    3
 -7 + q   - -- + -- - -- + -- - -- + -- + 5 q - 3 q  + q
 5    4    3    2   q
            q    q    q    q    q</nowiki></pre></td></tr>
-<tr valign=top><td><pre style="color:  blue; border: 0px; padding: 0em"><nowiki>In[11]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]</nowiki></pre></td></tr>
-<tr valign=top><td><pre  style="color: blue; border: 0px; padding: 0em"><nowiki>Out[11]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>{Knot[10, 82]}</nowiki></pre></td></tr>
+<tr valign=top><td><pre style="color:  blue; border: 0px; padding: 0em"><nowiki>In[15]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]</nowiki></pre></td></tr>
+<tr valign=top><td><pre  style="color: blue; border: 0px; padding: 0em"><nowiki>Out[15]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>{Knot[10, 82]}</nowiki></pre></td></tr>
-<math>\textrm{Include}(\textrm{ColouredJonesM.mhtml})</math>
-<tr valign=top><td><pre style="color:  blue; border: 0px; padding: 0em"><nowiki>In[12]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>A2Invariant[Knot[10, 82]][q]</nowiki></pre></td></tr>
-<tr valign=top><td><pre  style="color: blue; border: 0px; padding: 0em"><nowiki>Out[12]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> -20    -18    -16    2     -12    -10    -8   4     -4   2     2
+<tr valign=top><td><pre style="color:  blue; border: 0px; padding: 0em"><nowiki>In[16]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>A2Invariant[Knot[10, 82]][q]</nowiki></pre></td></tr>
+<tr valign=top><td><pre  style="color: blue; border: 0px; padding: 0em"><nowiki>Out[16]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> -20    -18    -16    2     -12    -10    -8   4     -4   2     2
 q    - q    + q    - --- - q    + q    - q   + -- - q   + -- - q  +
 6          2
@@ Line 92: / Line 146: @@
 6    8
   q  - q  + q</nowiki></pre></td></tr>
-<tr valign=top><td><pre style="color:  blue; border: 0px; padding: 0em"><nowiki>In[13]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>Kauffman[Knot[10, 82]][a, z]</nowiki></pre></td></tr>
-<tr valign=top><td><pre  style="color: blue; border: 0px; padding: 0em"><nowiki>Out[13]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>                                        2
+<tr valign=top><td><pre style="color:  blue; border: 0px; padding: 0em"><nowiki>In[17]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>HOMFLYPT[Knot[10, 82]][a, z]</nowiki></pre></td></tr>
+<tr valign=top><td><pre  style="color: blue; border: 0px; padding: 0em"><nowiki>Out[17]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>       2      2  2      4  2      4       2  4      4  4    6
++ 4 z  - 8 a  z  + 4 a  z  + 4 z  - 12 a  z  + 4 a  z  + z  -
+6    4  6    2  8
+a  z  + a  z  - a  z</nowiki></pre></td></tr>
+<tr valign=top><td><pre style="color:  blue; border: 0px; padding: 0em"><nowiki>In[18]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>Kauffman[Knot[10, 82]][a, z]</nowiki></pre></td></tr>
+<tr valign=top><td><pre  style="color: blue; border: 0px; padding: 0em"><nowiki>Out[18]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>                                        2
     z              5      7        2   z        2  2      4  2
 - - - 2 a z + 2 a  z + a  z - 6 z  + -- - 13 a  z  - 5 a  z  -
@@ Line 123: / Line 185: @@
 9
 a  z</nowiki></pre></td></tr>
-<tr valign=top><td><pre style="color:  blue; border: 0px; padding: 0em"><nowiki>In[14]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>{Vassiliev[2][Knot[10, 82]], Vassiliev[3][Knot[10, 82]]}</nowiki></pre></td></tr>
-<tr valign=top><td><pre  style="color: blue; border: 0px; padding: 0em"><nowiki>Out[14]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>{0, 0}</nowiki></pre></td></tr>
+<tr valign=top><td><pre style="color:  blue; border: 0px; padding: 0em"><nowiki>In[19]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>{Vassiliev[2][Knot[10, 82]], Vassiliev[3][Knot[10, 82]]}</nowiki></pre></td></tr>
-<tr valign=top><td><pre style="color:  blue; border: 0px; padding: 0em"><nowiki>In[15]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>Kh[Knot[10, 82]][q, t]</nowiki></pre></td></tr>
+<tr valign=top><td><pre  style="color: blue; border: 0px; padding: 0em"><nowiki>Out[19]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>{0, 0}</nowiki></pre></td></tr>
-<tr valign=top><td><pre  style="color: blue; border: 0px; padding: 0em"><nowiki>Out[15]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>5    6     1        2        1        3        2       5       3
+<tr valign=top><td><pre style="color:  blue; border: 0px; padding: 0em"><nowiki>In[20]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>Kh[Knot[10, 82]][q, t]</nowiki></pre></td></tr>
+<tr valign=top><td><pre  style="color: blue; border: 0px; padding: 0em"><nowiki>Out[20]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>5    6     1        2        1        3        2       5       3
 -- + - + ------ + ------ + ------ + ------ + ----- + ----- + ----- +
 q    15  6    13  5    11  5    11  4    9  4    9  3    7  3
@@ Line 138: / Line 202: @@
 3      5  3    7  4
   q  t  + 2 q  t  + q  t</nowiki></pre></td></tr>
+<tr valign=top><td><pre style="color:  blue; border: 0px; padding: 0em"><nowiki>In[21]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>ColouredJones[Knot[10, 82], 2][q]</nowiki></pre></td></tr>
+<tr valign=top><td><pre  style="color: blue; border: 0px; padding: 0em"><nowiki>Out[21]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>       -20    3     -18    7    12     4    17    31    11    36
+-20 + q    - --- + q    + --- - --- + --- + --- - --- + --- + --- -
+17    16    15    14    13    12    11
+             q            q     q     q     q     q     q     q
+13   60   75   5    74   70   9    71   49              2
+  --- + -- + -- - -- + -- + -- - -- - -- + -- - -- + 54 q - 23 q  -
+9    8    7    6    5    4    3    2   q
+  q     q    q    q    q    q    q    q    q
+4      5       6      7    8      9    10
+q  + 30 q  - 4 q  - 14 q  + 9 q  + q  - 3 q  + q</nowiki></pre></td></tr>
 </table>
+See/edit the [[Rolfsen_Splice_Template]].
  [[Category:Knot Page]]

10 82: Difference between revisions

Revision as of 17:20, 29 August 2005

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Four dimensional invariants

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