10 100

10_99

10_101

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10 100 Quick Notes

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Knot presentations

Planar diagram presentation	X6271 X18,6,19,5 X20,13,1,14 X14,7,15,8 X10,3,11,4 X16,9,17,10 X4,11,5,12 X8,15,9,16 X12,19,13,20 X2,18,3,17
Gauss code	1, -10, 5, -7, 2, -1, 4, -8, 6, -5, 7, -9, 3, -4, 8, -6, 10, -2, 9, -3
Dowker-Thistlethwaite code	6 10 18 14 16 4 20 8 2 12
Conway Notation	[3:2:2]

Three dimensional invariants

Symmetry type Reversible Unknotting number [math]\displaystyle{ \{2,3\} }[/math] 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index 1 Maximal Thurston-Bennequin number [-12][0] Hyperbolic Volume 12.8109 A-Polynomial See Data:10 100/A-polynomial

[edit Notes for 10 100's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus [math]\displaystyle{ 2 }[/math] Topological 4 genus [math]\displaystyle{ 2 }[/math] Concordance genus [math]\displaystyle{ 4 }[/math] Rasmussen s-Invariant 4

[edit Notes for 10 100's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ t^4-4 t^3+9 t^2-12 t+13-12 t^{-1} +9 t^{-2} -4 t^{-3} + t^{-4} }[/math]
Conway polynomial	[math]\displaystyle{ z^8+4 z^6+5 z^4+4 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 65, -4 }
Jones polynomial	[math]\displaystyle{ -q+3-5 q^{-1} +8 q^{-2} -9 q^{-3} +11 q^{-4} -10 q^{-5} +8 q^{-6} -6 q^{-7} +3 q^{-8} - q^{-9} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ a^4 z^8-a^6 z^6+6 a^4 z^6-a^2 z^6-4 a^6 z^4+13 a^4 z^4-4 a^2 z^4-5 a^6 z^2+13 a^4 z^2-4 a^2 z^2-3 a^6+5 a^4-a^2 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ z^3 a^{11}+3 z^4 a^{10}+6 z^5 a^9-5 z^3 a^9+2 z a^9+8 z^6 a^8-11 z^4 a^8+4 z^2 a^8+8 z^7 a^7-14 z^5 a^7+5 z^3 a^7-2 z a^7+6 z^8 a^6-12 z^6 a^6+5 z^4 a^6-6 z^2 a^6+3 a^6+2 z^9 a^5+4 z^7 a^5-27 z^5 a^5+26 z^3 a^5-8 z a^5+9 z^8 a^4-33 z^6 a^4+36 z^4 a^4-17 z^2 a^4+5 a^4+2 z^9 a^3-3 z^7 a^3-11 z^5 a^3+20 z^3 a^3-6 z a^3+3 z^8 a^2-13 z^6 a^2+17 z^4 a^2-7 z^2 a^2+a^2+z^7 a-4 z^5 a+5 z^3 a-2 z a }[/math]
The A2 invariant	[math]\displaystyle{ -q^{26}+q^{24}-2 q^{22}-q^{18}-q^{16}+3 q^{14}-q^{12}+4 q^{10}+q^6+q^4-q^2+1- q^{-2} }[/math]
The G2 invariant	[math]\displaystyle{ q^{148}-2 q^{146}+3 q^{144}-4 q^{142}+3 q^{140}-2 q^{138}-q^{136}+8 q^{134}-12 q^{132}+16 q^{130}-17 q^{128}+11 q^{126}-4 q^{124}-9 q^{122}+24 q^{120}-33 q^{118}+35 q^{116}-28 q^{114}+15 q^{112}+3 q^{110}-20 q^{108}+39 q^{106}-51 q^{104}+48 q^{102}-36 q^{100}+7 q^{98}+24 q^{96}-52 q^{94}+65 q^{92}-53 q^{90}+17 q^{88}+22 q^{86}-59 q^{84}+59 q^{82}-30 q^{80}-23 q^{78}+66 q^{76}-82 q^{74}+58 q^{72}+2 q^{70}-67 q^{68}+112 q^{66}-116 q^{64}+77 q^{62}-10 q^{60}-57 q^{58}+107 q^{56}-111 q^{54}+88 q^{52}-35 q^{50}-20 q^{48}+66 q^{46}-81 q^{44}+66 q^{42}-23 q^{40}-26 q^{38}+64 q^{36}-68 q^{34}+41 q^{32}+16 q^{30}-68 q^{28}+98 q^{26}-86 q^{24}+34 q^{22}+32 q^{20}-84 q^{18}+103 q^{16}-81 q^{14}+36 q^{12}+12 q^{10}-49 q^8+59 q^6-47 q^4+24 q^2-2-12 q^{-2} +13 q^{-4} -11 q^{-6} +6 q^{-8} -2 q^{-10} + q^{-12} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 10_100.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ -q^{19}+2 q^{17}-3 q^{15}+2 q^{13}-2 q^{11}+q^9+2 q^7-q^5+3 q^3-2 q+2 q^{-1} - q^{-3} }[/math]
2	[math]\displaystyle{ q^{52}-2 q^{50}+q^{48}+4 q^{46}-7 q^{44}+3 q^{42}+5 q^{40}-13 q^{38}+8 q^{36}+9 q^{34}-16 q^{32}+q^{30}+13 q^{28}-7 q^{26}-9 q^{24}+9 q^{22}+5 q^{20}-11 q^{18}-q^{16}+14 q^{14}-7 q^{12}-9 q^{10}+17 q^8-15 q^4+11 q^2+7-12 q^{-2} + q^{-4} +7 q^{-6} -3 q^{-8} -2 q^{-10} + q^{-12} }[/math]
3	[math]\displaystyle{ -q^{99}+2 q^{97}-q^{95}-2 q^{93}+q^{91}+3 q^{89}-4 q^{85}+3 q^{83}-q^{81}-7 q^{79}+11 q^{77}+16 q^{75}-18 q^{73}-37 q^{71}+24 q^{69}+58 q^{67}-16 q^{65}-76 q^{63}-9 q^{61}+80 q^{59}+36 q^{57}-59 q^{55}-62 q^{53}+28 q^{51}+72 q^{49}+12 q^{47}-72 q^{45}-38 q^{43}+55 q^{41}+59 q^{39}-43 q^{37}-68 q^{35}+26 q^{33}+70 q^{31}-13 q^{29}-73 q^{27}-2 q^{25}+73 q^{23}+26 q^{21}-70 q^{19}-46 q^{17}+57 q^{15}+69 q^{13}-31 q^{11}-78 q^9-q^7+78 q^5+30 q^3-54 q-50 q^{-1} +25 q^{-3} +51 q^{-5} + q^{-7} -37 q^{-9} -16 q^{-11} +17 q^{-13} +18 q^{-15} -4 q^{-17} -10 q^{-19} -2 q^{-21} +3 q^{-23} +2 q^{-25} - q^{-27} }[/math]
4	[math]\displaystyle{ q^{160}-2 q^{158}+q^{156}+2 q^{154}-3 q^{152}+3 q^{150}-6 q^{148}+2 q^{146}+5 q^{144}-4 q^{142}+14 q^{140}-19 q^{138}-14 q^{136}+4 q^{134}+19 q^{132}+64 q^{130}-35 q^{128}-93 q^{126}-57 q^{124}+75 q^{122}+225 q^{120}+21 q^{118}-245 q^{116}-278 q^{114}+43 q^{112}+474 q^{110}+296 q^{108}-253 q^{106}-586 q^{104}-271 q^{102}+477 q^{100}+631 q^{98}+116 q^{96}-542 q^{94}-632 q^{92}+28 q^{90}+547 q^{88}+518 q^{86}-43 q^{84}-546 q^{82}-413 q^{80}+58 q^{78}+497 q^{76}+395 q^{74}-145 q^{72}-472 q^{70}-300 q^{68}+242 q^{66}+469 q^{64}+109 q^{62}-362 q^{60}-367 q^{58}+112 q^{56}+432 q^{54}+180 q^{52}-339 q^{50}-399 q^{48}+57 q^{46}+463 q^{44}+313 q^{42}-290 q^{40}-512 q^{38}-152 q^{36}+407 q^{34}+536 q^{32}-12 q^{30}-480 q^{28}-467 q^{26}+67 q^{24}+561 q^{22}+378 q^{20}-114 q^{18}-527 q^{16}-368 q^{14}+190 q^{12}+450 q^{10}+330 q^8-161 q^6-435 q^4-232 q^2+102+369 q^{-2} +206 q^{-4} -105 q^{-6} -245 q^{-8} -191 q^{-10} +76 q^{-12} +181 q^{-14} +116 q^{-16} -16 q^{-18} -132 q^{-20} -71 q^{-22} +7 q^{-24} +60 q^{-26} +55 q^{-28} -9 q^{-30} -24 q^{-32} -23 q^{-34} -3 q^{-36} +13 q^{-38} +5 q^{-40} +2 q^{-42} -3 q^{-44} -2 q^{-46} + q^{-48} }[/math]
5	[math]\displaystyle{ -q^{235}+2 q^{233}-q^{231}-2 q^{229}+3 q^{227}-q^{225}+4 q^{221}-3 q^{219}-7 q^{217}+6 q^{213}+11 q^{211}+13 q^{209}-12 q^{207}-43 q^{205}-37 q^{203}+33 q^{201}+102 q^{199}+83 q^{197}-46 q^{195}-219 q^{193}-212 q^{191}+75 q^{189}+430 q^{187}+437 q^{185}-71 q^{183}-733 q^{181}-856 q^{179}-57 q^{177}+1139 q^{175}+1534 q^{173}+394 q^{171}-1517 q^{169}-2419 q^{167}-1129 q^{165}+1636 q^{163}+3441 q^{161}+2280 q^{159}-1275 q^{157}-4213 q^{155}-3700 q^{153}+200 q^{151}+4353 q^{149}+5050 q^{147}+1432 q^{145}-3587 q^{143}-5764 q^{141}-3220 q^{139}+1909 q^{137}+5472 q^{135}+4635 q^{133}+211 q^{131}-4147 q^{129}-5115 q^{127}-2212 q^{125}+2127 q^{123}+4602 q^{121}+3517 q^{119}-39 q^{117}-3323 q^{115}-3943 q^{113}-1567 q^{111}+1816 q^{109}+3594 q^{107}+2458 q^{105}-569 q^{103}-2919 q^{101}-2637 q^{99}-116 q^{97}+2274 q^{95}+2430 q^{93}+299 q^{91}-2001 q^{89}-2180 q^{87}-114 q^{85}+2073 q^{83}+2148 q^{81}-68 q^{79}-2417 q^{77}-2472 q^{75}+35 q^{73}+2786 q^{71}+3059 q^{69}+395 q^{67}-2896 q^{65}-3777 q^{63}-1250 q^{61}+2583 q^{59}+4333 q^{57}+2370 q^{55}-1696 q^{53}-4460 q^{51}-3550 q^{49}+316 q^{47}+3955 q^{45}+4401 q^{43}+1337 q^{41}-2732 q^{39}-4609 q^{37}-2884 q^{35}+972 q^{33}+3967 q^{31}+3895 q^{29}+934 q^{27}-2536 q^{25}-3984 q^{23}-2509 q^{21}+662 q^{19}+3157 q^{17}+3260 q^{15}+1089 q^{13}-1621 q^{11}-3008 q^9-2237 q^7-29 q^5+1969 q^3+2440 q+1255 q^{-1} -592 q^{-3} -1834 q^{-5} -1729 q^{-7} -531 q^{-9} +828 q^{-11} +1455 q^{-13} +1060 q^{-15} +89 q^{-17} -774 q^{-19} -1005 q^{-21} -572 q^{-23} +124 q^{-25} +589 q^{-27} +595 q^{-29} +259 q^{-31} -160 q^{-33} -380 q^{-35} -312 q^{-37} -80 q^{-39} +123 q^{-41} +196 q^{-43} +141 q^{-45} +13 q^{-47} -75 q^{-49} -84 q^{-51} -43 q^{-53} +2 q^{-55} +30 q^{-57} +31 q^{-59} +8 q^{-61} -6 q^{-63} -8 q^{-65} -5 q^{-67} -2 q^{-69} +3 q^{-71} +2 q^{-73} - q^{-75} }[/math]
6	[math]\displaystyle{ q^{324}-2 q^{322}+q^{320}+2 q^{318}-3 q^{316}+q^{314}-2 q^{312}+2 q^{310}-3 q^{308}+5 q^{306}+11 q^{304}-17 q^{302}-7 q^{300}-4 q^{298}+6 q^{296}+13 q^{294}+35 q^{292}+29 q^{290}-71 q^{288}-80 q^{286}-42 q^{284}+48 q^{282}+142 q^{280}+196 q^{278}+58 q^{276}-312 q^{274}-428 q^{272}-228 q^{270}+268 q^{268}+756 q^{266}+834 q^{264}+102 q^{262}-1223 q^{260}-1797 q^{258}-1011 q^{256}+978 q^{254}+2952 q^{252}+3195 q^{250}+648 q^{248}-3626 q^{246}-6114 q^{244}-4334 q^{242}+1537 q^{240}+8114 q^{238}+10269 q^{236}+4754 q^{234}-6257 q^{232}-15068 q^{230}-14515 q^{228}-3008 q^{226}+13371 q^{224}+23406 q^{222}+18261 q^{220}-1169 q^{218}-22733 q^{216}-31591 q^{214}-19984 q^{212}+6895 q^{210}+32245 q^{208}+38539 q^{206}+19620 q^{204}-13403 q^{202}-39833 q^{200}-42397 q^{198}-18056 q^{196}+18506 q^{194}+44418 q^{192}+43076 q^{190}+15682 q^{188}-20956 q^{186}-44742 q^{184}-41414 q^{182}-13850 q^{180}+20803 q^{178}+41655 q^{176}+37876 q^{174}+12847 q^{172}-18062 q^{170}-36839 q^{168}-33796 q^{166}-12064 q^{164}+14534 q^{162}+31365 q^{160}+29468 q^{158}+11216 q^{156}-11970 q^{154}-26525 q^{152}-24577 q^{150}-8839 q^{148}+10877 q^{146}+22143 q^{144}+18940 q^{142}+4305 q^{140}-11305 q^{138}-17551 q^{136}-11529 q^{134}+1751 q^{132}+11894 q^{130}+12056 q^{128}+2561 q^{126}-8290 q^{124}-11518 q^{122}-4645 q^{120}+7030 q^{118}+13532 q^{116}+9179 q^{114}-4082 q^{112}-16057 q^{110}-16761 q^{108}-4271 q^{106}+13370 q^{104}+23163 q^{102}+17274 q^{100}-2151 q^{98}-22058 q^{96}-28251 q^{94}-15167 q^{92}+9324 q^{90}+29120 q^{88}+30840 q^{86}+11821 q^{84}-15989 q^{82}-34611 q^{80}-31303 q^{78}-7901 q^{76}+21178 q^{74}+37823 q^{72}+30855 q^{70}+4582 q^{68}-24862 q^{66}-38978 q^{64}-29693 q^{62}-2558 q^{60}+26144 q^{58}+38872 q^{56}+28552 q^{54}+1666 q^{52}-25236 q^{50}-37234 q^{48}-27629 q^{46}-2573 q^{44}+22889 q^{42}+34537 q^{40}+26467 q^{38}+4665 q^{36}-18893 q^{34}-30874 q^{32}-25532 q^{30}-6956 q^{28}+14042 q^{26}+26109 q^{24}+24115 q^{22}+9452 q^{20}-8826 q^{18}-21131 q^{16}-21622 q^{14}-11272 q^{12}+3679 q^{10}+15769 q^8+18588 q^6+12113 q^4+286 q^2-10275-14854 q^{-2} -11857 q^{-4} -3231 q^{-6} +5658 q^{-8} +10894 q^{-10} +10205 q^{-12} +5009 q^{-14} -1998 q^{-16} -7148 q^{-18} -8053 q^{-20} -5262 q^{-22} -392 q^{-24} +3769 q^{-26} +5760 q^{-28} +4633 q^{-30} +1598 q^{-32} -1512 q^{-34} -3457 q^{-36} -3445 q^{-38} -2010 q^{-40} +212 q^{-42} +1754 q^{-44} +2204 q^{-46} +1631 q^{-48} +461 q^{-50} -634 q^{-52} -1264 q^{-54} -1064 q^{-56} -539 q^{-58} +96 q^{-60} +519 q^{-62} +617 q^{-64} +418 q^{-66} +51 q^{-68} -156 q^{-70} -268 q^{-72} -221 q^{-74} -101 q^{-76} +37 q^{-78} +103 q^{-80} +80 q^{-82} +56 q^{-84} +5 q^{-86} -24 q^{-88} -37 q^{-90} -16 q^{-92} + q^{-94} + q^{-96} +8 q^{-98} +5 q^{-100} +2 q^{-102} -3 q^{-104} -2 q^{-106} + q^{-108} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ -q^{26}+q^{24}-2 q^{22}-q^{18}-q^{16}+3 q^{14}-q^{12}+4 q^{10}+q^6+q^4-q^2+1- q^{-2} }[/math]
1,1	[math]\displaystyle{ q^{76}-4 q^{74}+8 q^{72}-12 q^{70}+22 q^{68}-36 q^{66}+48 q^{64}-60 q^{62}+81 q^{60}-102 q^{58}+110 q^{56}-116 q^{54}+138 q^{52}-154 q^{50}+156 q^{48}-174 q^{46}+180 q^{44}-170 q^{42}+128 q^{40}-52 q^{38}-45 q^{36}+176 q^{34}-308 q^{32}+416 q^{30}-512 q^{28}+552 q^{26}-548 q^{24}+494 q^{22}-403 q^{20}+292 q^{18}-134 q^{16}-10 q^{14}+147 q^{12}-248 q^{10}+326 q^8-348 q^6+318 q^4-266 q^2+202-132 q^{-2} +74 q^{-4} -38 q^{-6} +14 q^{-8} -4 q^{-10} + q^{-12} }[/math]
2,0	[math]\displaystyle{ q^{66}-q^{64}+q^{62}+2 q^{60}-q^{58}+2 q^{56}-q^{54}-2 q^{52}-q^{50}+q^{48}-q^{46}-8 q^{44}+5 q^{40}-2 q^{38}-6 q^{36}+7 q^{34}+4 q^{32}-5 q^{30}+2 q^{26}-4 q^{22}+5 q^{20}+3 q^{18}-3 q^{16}+4 q^{14}+7 q^{12}-3 q^{10}-3 q^8+5 q^6+3 q^4-4 q^2-2+3 q^{-2} + q^{-4} -3 q^{-6} - q^{-8} + q^{-10} }[/math]

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^{62}-2 q^{60}-q^{58}+6 q^{56}-3 q^{54}-6 q^{52}+10 q^{50}-q^{48}-10 q^{46}+10 q^{44}+3 q^{42}-12 q^{40}+7 q^{38}+2 q^{36}-12 q^{34}-2 q^{32}-q^{30}-q^{28}-6 q^{26}+4 q^{24}+13 q^{22}-3 q^{20}+3 q^{18}+14 q^{16}-5 q^{14}-2 q^{12}+9 q^{10}-8 q^8-3 q^6+6 q^4-6 q^2+1+2 q^{-2} -2 q^{-4} + q^{-6} }[/math]
1,0,0	[math]\displaystyle{ -q^{33}+q^{31}-3 q^{29}+q^{27}-4 q^{25}+q^{23}-q^{21}+2 q^{19}+2 q^{17}+2 q^{15}+3 q^{13}+3 q^9-2 q^7+2 q^5-2 q^3+q- q^{-1} }[/math]
1,0,1	[math]\displaystyle{ q^{100}-4 q^{98}+6 q^{96}-2 q^{94}-7 q^{92}+18 q^{90}-21 q^{88}+6 q^{86}+20 q^{84}-36 q^{82}+34 q^{80}-7 q^{78}-28 q^{76}+43 q^{74}-32 q^{72}+9 q^{70}+22 q^{68}-41 q^{66}+28 q^{64}+11 q^{62}-68 q^{60}+107 q^{58}-97 q^{56}+21 q^{54}+83 q^{52}-169 q^{50}+204 q^{48}-177 q^{46}+101 q^{44}-18 q^{42}-80 q^{40}+121 q^{38}-155 q^{36}+139 q^{34}-126 q^{32}+123 q^{30}-104 q^{28}+106 q^{26}-34 q^{24}-35 q^{22}+145 q^{20}-203 q^{18}+226 q^{16}-176 q^{14}+82 q^{12}+15 q^{10}-96 q^8+131 q^6-115 q^4+69 q^2-21-17 q^{-2} +26 q^{-4} -22 q^{-6} +12 q^{-8} -4 q^{-10} + q^{-12} }[/math]

A4 Invariants.

Weight	Invariant
0,1,0,0	[math]\displaystyle{ q^{76}-q^{74}-2 q^{72}+4 q^{70}+2 q^{68}-7 q^{66}+2 q^{64}+10 q^{62}-4 q^{60}-6 q^{58}+10 q^{56}+4 q^{54}-8 q^{52}-2 q^{50}+7 q^{48}-12 q^{46}-17 q^{44}-8 q^{40}-20 q^{38}+3 q^{36}+15 q^{34}-4 q^{32}+10 q^{30}+23 q^{28}+13 q^{26}+q^{24}+8 q^{22}+7 q^{20}-5 q^{18}-7 q^{16}-4 q^{12}-6 q^{10}+2 q^8-2 q^4+2 q^2+1- q^{-2} + q^{-4} }[/math]
1,0,0,0	[math]\displaystyle{ -q^{40}+q^{38}-3 q^{36}-3 q^{32}-2 q^{30}-q^{26}+3 q^{24}+q^{22}+5 q^{20}+q^{18}+4 q^{16}+2 q^{12}-q^8+q^6-2 q^4+q^2-1 }[/math]

B2 Invariants.

Weight	Invariant
0,1	[math]\displaystyle{ -q^{62}+2 q^{60}-3 q^{58}+6 q^{56}-9 q^{54}+10 q^{52}-14 q^{50}+15 q^{48}-16 q^{46}+16 q^{44}-13 q^{42}+8 q^{40}-q^{38}-8 q^{36}+16 q^{34}-24 q^{32}+29 q^{30}-33 q^{28}+34 q^{26}-32 q^{24}+27 q^{22}-17 q^{20}+11 q^{18}-5 q^{14}+14 q^{12}-17 q^{10}+18 q^8-17 q^6+16 q^4-12 q^2+9-6 q^{-2} +2 q^{-4} - q^{-6} }[/math]
1,0	[math]\displaystyle{ q^{100}-2 q^{96}-2 q^{94}+q^{92}+6 q^{90}+3 q^{88}-6 q^{86}-8 q^{84}+12 q^{80}+7 q^{78}-8 q^{76}-13 q^{74}+15 q^{70}+8 q^{68}-11 q^{66}-14 q^{64}+5 q^{62}+16 q^{60}-q^{58}-17 q^{56}-7 q^{54}+11 q^{52}+7 q^{50}-10 q^{48}-11 q^{46}+5 q^{44}+10 q^{42}-4 q^{40}-9 q^{38}+5 q^{36}+15 q^{34}+2 q^{32}-13 q^{30}-6 q^{28}+16 q^{26}+16 q^{24}-5 q^{22}-19 q^{20}-q^{18}+18 q^{16}+10 q^{14}-12 q^{12}-15 q^{10}+4 q^8+13 q^6+q^4-9 q^2-5+5 q^{-2} +4 q^{-4} -2 q^{-6} -2 q^{-8} + q^{-12} }[/math]

D4 Invariants.

Weight

Invariant

1,0,0,0

[math]\displaystyle{ q^{86}-2 q^{84}+q^{82}-2 q^{80}+6 q^{78}-6 q^{76}+5 q^{74}-8 q^{72}+12 q^{70}-10 q^{68}+10 q^{66}-13 q^{64}+13 q^{62}-10 q^{60}+10 q^{58}-8 q^{56}+4 q^{54}-7 q^{50}+8 q^{48}-20 q^{46}+14 q^{44}-27 q^{42}+21 q^{40}-29 q^{38}+27 q^{36}-22 q^{34}+27 q^{32}-13 q^{30}+22 q^{28}-2 q^{26}+10 q^{24}+2 q^{22}-3 q^{20}+10 q^{18}-12 q^{16}+11 q^{14}-17 q^{12}+14 q^{10}-15 q^8+11 q^6-11 q^4+9 q^2-6+4 q^{-2} -2 q^{-4} + q^{-6} }[/math]

G2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ q^{148}-2 q^{146}+3 q^{144}-4 q^{142}+3 q^{140}-2 q^{138}-q^{136}+8 q^{134}-12 q^{132}+16 q^{130}-17 q^{128}+11 q^{126}-4 q^{124}-9 q^{122}+24 q^{120}-33 q^{118}+35 q^{116}-28 q^{114}+15 q^{112}+3 q^{110}-20 q^{108}+39 q^{106}-51 q^{104}+48 q^{102}-36 q^{100}+7 q^{98}+24 q^{96}-52 q^{94}+65 q^{92}-53 q^{90}+17 q^{88}+22 q^{86}-59 q^{84}+59 q^{82}-30 q^{80}-23 q^{78}+66 q^{76}-82 q^{74}+58 q^{72}+2 q^{70}-67 q^{68}+112 q^{66}-116 q^{64}+77 q^{62}-10 q^{60}-57 q^{58}+107 q^{56}-111 q^{54}+88 q^{52}-35 q^{50}-20 q^{48}+66 q^{46}-81 q^{44}+66 q^{42}-23 q^{40}-26 q^{38}+64 q^{36}-68 q^{34}+41 q^{32}+16 q^{30}-68 q^{28}+98 q^{26}-86 q^{24}+34 q^{22}+32 q^{20}-84 q^{18}+103 q^{16}-81 q^{14}+36 q^{12}+12 q^{10}-49 q^8+59 q^6-47 q^4+24 q^2-2-12 q^{-2} +13 q^{-4} -11 q^{-6} +6 q^{-8} -2 q^{-10} + q^{-12} }[/math]

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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. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["10 100"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ t^4-4 t^3+9 t^2-12 t+13-12 t^{-1} +9 t^{-2} -4 t^{-3} + t^{-4} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ z^8+4 z^6+5 z^4+4 z^2+1 }[/math]

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]=

[math]\displaystyle{ \{1\} }[/math]

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 65, -4 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ -q+3-5 q^{-1} +8 q^{-2} -9 q^{-3} +11 q^{-4} -10 q^{-5} +8 q^{-6} -6 q^{-7} +3 q^{-8} - q^{-9} }[/math]

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

[math]\displaystyle{ a^4 z^8-a^6 z^6+6 a^4 z^6-a^2 z^6-4 a^6 z^4+13 a^4 z^4-4 a^2 z^4-5 a^6 z^2+13 a^4 z^2-4 a^2 z^2-3 a^6+5 a^4-a^2 }[/math]

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ z^3 a^{11}+3 z^4 a^{10}+6 z^5 a^9-5 z^3 a^9+2 z a^9+8 z^6 a^8-11 z^4 a^8+4 z^2 a^8+8 z^7 a^7-14 z^5 a^7+5 z^3 a^7-2 z a^7+6 z^8 a^6-12 z^6 a^6+5 z^4 a^6-6 z^2 a^6+3 a^6+2 z^9 a^5+4 z^7 a^5-27 z^5 a^5+26 z^3 a^5-8 z a^5+9 z^8 a^4-33 z^6 a^4+36 z^4 a^4-17 z^2 a^4+5 a^4+2 z^9 a^3-3 z^7 a^3-11 z^5 a^3+20 z^3 a^3-6 z a^3+3 z^8 a^2-13 z^6 a^2+17 z^4 a^2-7 z^2 a^2+a^2+z^7 a-4 z^5 a+5 z^3 a-2 z a }[/math]

Vassiliev invariants

V2 and V3:

(4, -7)

V2,1 through V6,9:

V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9 [math]\displaystyle{ 16 }[/math] [math]\displaystyle{ -56 }[/math] [math]\displaystyle{ 128 }[/math] [math]\displaystyle{ \frac{776}{3} }[/math] [math]\displaystyle{ \frac{64}{3} }[/math] [math]\displaystyle{ -896 }[/math] [math]\displaystyle{ -\frac{4304}{3} }[/math] [math]\displaystyle{ -\frac{512}{3} }[/math] [math]\displaystyle{ -216 }[/math] [math]\displaystyle{ \frac{2048}{3} }[/math] [math]\displaystyle{ 1568 }[/math] [math]\displaystyle{ \frac{12416}{3} }[/math] [math]\displaystyle{ \frac{1024}{3} }[/math] [math]\displaystyle{ \frac{118262}{15} }[/math] [math]\displaystyle{ \frac{224}{5} }[/math] [math]\displaystyle{ \frac{128048}{45} }[/math] [math]\displaystyle{ \frac{1306}{9} }[/math] [math]\displaystyle{ \frac{4502}{15} }[/math]

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 [math]\displaystyle{ t^rq^j }[/math] are shown, along with their alternating sums [math]\displaystyle{ \chi }[/math] (fixed [math]\displaystyle{ j }[/math], alternation over [math]\displaystyle{ r }[/math]). The squares with yellow highlighting are those on the "critical diagonals", where [math]\displaystyle{ j-2r=s+1 }[/math] or [math]\displaystyle{ j-2r=s+1 }[/math], where [math]\displaystyle{ s= }[/math]-4 is the signature of 10 100. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \	-7	-6	-5	-4	-3	-2	-1	0	1	2	3	χ
3											1	-1
1										2		2
-1									3	1		-2
-3								5	2			3
-5							5	4				-1
-7						6	4					2
-9					4	5						1
-11				4	6							-2
-13			2	4								2
-15		1	4									-3
-17		2										2
-19	1											-1

Computer Talk

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

[math]\displaystyle{ \textrm{Include}(\textrm{ColouredJonesM.mhtml}) }[/math]

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 17, 2005, 14:44:34)...
In[2]:=	Crossings[Knot[10, 100]]
Out[2]=	10
In[3]:=	PD[Knot[10, 100]]
Out[3]=	PD[X[6, 2, 7, 1], X[18, 6, 19, 5], X[20, 13, 1, 14], X[14, 7, 15, 8], X[10, 3, 11, 4], X[16, 9, 17, 10], X[4, 11, 5, 12], X[8, 15, 9, 16], X[12, 19, 13, 20], X[2, 18, 3, 17]]
In[4]:=	GaussCode[Knot[10, 100]]
Out[4]=	GaussCode[1, -10, 5, -7, 2, -1, 4, -8, 6, -5, 7, -9, 3, -4, 8, -6, 10, -2, 9, -3]
In[5]:=	BR[Knot[10, 100]]
Out[5]=	BR[3, {-1, -1, -1, 2, -1, -1, 2, -1, -1, 2}]
In[6]:=	alex = Alexander[Knot[10, 100]][t]
Out[6]=	-4 4 9 12 2 3 4 13 + t - -- + -- - -- - 12 t + 9 t - 4 t + t 3 2 t t t
In[7]:=	Conway[Knot[10, 100]][z]
Out[7]=	2 4 6 8 1 + 4 z + 5 z + 4 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 100]}
In[9]:=	{KnotDet[Knot[10, 100]], KnotSignature[Knot[10, 100]]}
Out[9]=	{65, -4}
In[10]:=	J=Jones[Knot[10, 100]][q]
Out[10]=	-9 3 6 8 10 11 9 8 5 3 - q + -- - -- + -- - -- + -- - -- + -- - - - q 8 7 6 5 4 3 2 q q q q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[10, 100]}
In[12]:=	A2Invariant[Knot[10, 100]][q]
Out[12]=	-26 -24 2 -18 -16 3 -12 4 -6 -4 1 - q + q - --- - q - q + --- - q + --- + q + q - 22 14 10 q q q -2 2 q - q
In[13]:=	Kauffman[Knot[10, 100]][a, z]
Out[13]=	2 4 6 3 5 7 9 a + 5 a + 3 a - 2 a z - 6 a z - 8 a z - 2 a z + 2 a z - 2 2 4 2 6 2 8 2 3 3 3 7 a z - 17 a z - 6 a z + 4 a z + 5 a z + 20 a z + 5 3 7 3 9 3 11 3 2 4 4 4 26 a z + 5 a z - 5 a z + a z + 17 a z + 36 a z + 6 4 8 4 10 4 5 3 5 5 5 5 a z - 11 a z + 3 a z - 4 a z - 11 a z - 27 a z - 7 5 9 5 2 6 4 6 6 6 8 6 14 a z + 6 a z - 13 a z - 33 a z - 12 a z + 8 a z + 7 3 7 5 7 7 7 2 8 4 8 6 8 a z - 3 a z + 4 a z + 8 a z + 3 a z + 9 a z + 6 a z + 3 9 5 9 2 a z + 2 a z
In[14]:=	{Vassiliev[2][Knot[10, 100]], Vassiliev[3][Knot[10, 100]]}
Out[14]=	{0, -7}
In[15]:=	Kh[Knot[10, 100]][q, t]
Out[15]=	4 5 1 2 1 4 2 4 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + 5 3 19 7 17 6 15 6 15 5 13 5 13 4 q q q t q t q t q t q t q t 4 6 4 5 6 4 5 2 t 3 t ------ + ------ + ----- + ----- + ----- + ---- + ---- + --- + --- + 11 4 11 3 9 3 9 2 7 2 7 5 3 q q t q t q t q t q t q t q t q 2 t 2 3 3 -- + 2 q t + q t q