8 18: Difference between revisions

Revision as of 20:05, 29 August 2005

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According to Mathematical Models by H. Martyn Cundy and A.P. Rollett, second edition, 1961 (Oxford University Press), p. 57, a flat ribbon or strip can be tightly folded into a heptagonal 8_18 knot (just as it can be tightly folded into a pentagonal trefoil knot).

This is the Carrick loop of practical knot tying. The Carrick bend of practical knot tying can be found at [math]\displaystyle{ 8^2_{7} }[/math].

Logo of the International Guild of Knot Tyers [1]	A charity logo in Porto [2]	A laser cut by Tom Longtin [3]	Knot in (pseudo-)Celtic decorative form
Less symmetrical	Within outer circle	Impossible figure	Mongolian ornament
Jump rope knot	Belt design	Bondage knot	Spheric depiction
A "Hungarian Knot", decorating French Military uniforms.	Carpet swatter.	Geodesic of the prolate ellipsoid.	Obtained with an epitrochoid.

Knot presentations

Planar diagram presentation	X₆₂₇₁ X₈₃₉₄ X_16,11,1,12 X_2,14,3,13 X_4,15,5,16 X_10,6,11,5 X_12,7,13,8 X_14,10,15,9
Gauss code	1, -4, 2, -5, 6, -1, 7, -2, 8, -6, 3, -7, 4, -8, 5, -3
Dowker-Thistlethwaite code	6 8 10 12 14 16 2 4
Conway Notation	[8*]

Minimum Braid Representative:

Length is 8, width is 3.

Braid index is 3.

A Morse Link Presentation:

Three dimensional invariants

Symmetry type	Fully amphicheiral
Unknotting number	2
3-genus	3
Bridge index	3
Super bridge index	4
Nakanishi index	2
Maximal Thurston-Bennequin number	[-5][-5]
Hyperbolic Volume	12.3509
A-Polynomial	See Data:8 18/A-polynomial

[edit Notes for 8 18's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus	[math]\displaystyle{ 1 }[/math]
Topological 4 genus	[math]\displaystyle{ 1 }[/math]
Concordance genus	[math]\displaystyle{ [1,3] }[/math]
Rasmussen s-Invariant	0

[edit Notes for 8 18's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ -t^3+5 t^2-10 t+13-10 t^{-1} +5 t^{-2} - t^{-3} }[/math]
Conway polynomial	[math]\displaystyle{ -z^6-z^4+z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \left\{t^2-t+1\right\} }[/math]
Determinant and Signature	{ 45, 0 }
Jones polynomial	[math]\displaystyle{ q^4-4 q^3+6 q^2-7 q+9-7 q^{-1} +6 q^{-2} -4 q^{-3} + q^{-4} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ -z^6+a^2 z^4+z^4 a^{-2} -3 z^4+a^2 z^2+z^2 a^{-2} -z^2-a^2- a^{-2} +3 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ 3 a z^7+3 z^7 a^{-1} +6 a^2 z^6+6 z^6 a^{-2} +12 z^6+4 a^3 z^5+3 a z^5+3 z^5 a^{-1} +4 z^5 a^{-3} +a^4 z^4-9 a^2 z^4-9 z^4 a^{-2} +z^4 a^{-4} -20 z^4-4 a^3 z^3-9 a z^3-9 z^3 a^{-1} -4 z^3 a^{-3} +3 a^2 z^2+3 z^2 a^{-2} +6 z^2+a z+z a^{-1} +a^2+ a^{-2} +3 }[/math]
The A2 invariant	[math]\displaystyle{ q^{12}-2 q^{10}-q^6-q^4+4 q^2+1+4 q^{-2} - q^{-4} - q^{-6} -2 q^{-10} + q^{-12} }[/math]
The G2 invariant	[math]\displaystyle{ q^{66}-3 q^{64}+6 q^{62}-10 q^{60}+8 q^{58}-4 q^{56}-5 q^{54}+23 q^{52}-36 q^{50}+48 q^{48}-38 q^{46}+7 q^{44}+28 q^{42}-67 q^{40}+84 q^{38}-71 q^{36}+29 q^{34}+17 q^{32}-58 q^{30}+77 q^{28}-56 q^{26}+8 q^{24}+34 q^{22}-59 q^{20}+45 q^{18}-6 q^{16}-45 q^{14}+81 q^{12}-81 q^{10}+64 q^8-11 q^6-48 q^4+97 q^2-111+97 q^{-2} -48 q^{-4} -11 q^{-6} +64 q^{-8} -81 q^{-10} +81 q^{-12} -45 q^{-14} -6 q^{-16} +45 q^{-18} -59 q^{-20} +34 q^{-22} +8 q^{-24} -56 q^{-26} +77 q^{-28} -58 q^{-30} +17 q^{-32} +29 q^{-34} -71 q^{-36} +84 q^{-38} -67 q^{-40} +28 q^{-42} +7 q^{-44} -38 q^{-46} +48 q^{-48} -36 q^{-50} +23 q^{-52} -5 q^{-54} -4 q^{-56} +8 q^{-58} -10 q^{-60} +6 q^{-62} -3 q^{-64} + q^{-66} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 8_18.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ q^9-3 q^7+2 q^5-q^3+2 q+2 q^{-1} - q^{-3} +2 q^{-5} -3 q^{-7} + q^{-9} }[/math]
2	[math]\displaystyle{ q^{26}-3 q^{24}-q^{22}+11 q^{20}-6 q^{18}-12 q^{16}+16 q^{14}-q^{12}-17 q^{10}+11 q^8+6 q^6-10 q^4+2 q^2+9+2 q^{-2} -10 q^{-4} +6 q^{-6} +11 q^{-8} -17 q^{-10} - q^{-12} +16 q^{-14} -12 q^{-16} -6 q^{-18} +11 q^{-20} - q^{-22} -3 q^{-24} + q^{-26} }[/math]
3	[math]\displaystyle{ q^{51}-3 q^{49}-q^{47}+8 q^{45}+6 q^{43}-14 q^{41}-26 q^{39}+20 q^{37}+48 q^{35}-7 q^{33}-70 q^{31}-17 q^{29}+86 q^{27}+44 q^{25}-82 q^{23}-71 q^{21}+65 q^{19}+81 q^{17}-42 q^{15}-87 q^{13}+21 q^{11}+73 q^9+6 q^7-58 q^5-21 q^3+42 q+42 q^{-1} -21 q^{-3} -58 q^{-5} +6 q^{-7} +73 q^{-9} +21 q^{-11} -87 q^{-13} -42 q^{-15} +81 q^{-17} +65 q^{-19} -71 q^{-21} -82 q^{-23} +44 q^{-25} +86 q^{-27} -17 q^{-29} -70 q^{-31} -7 q^{-33} +48 q^{-35} +20 q^{-37} -26 q^{-39} -14 q^{-41} +6 q^{-43} +8 q^{-45} - q^{-47} -3 q^{-49} + q^{-51} }[/math]
4	[math]\displaystyle{ q^{84}-3 q^{82}-q^{80}+8 q^{78}+3 q^{76}-2 q^{74}-28 q^{72}-16 q^{70}+41 q^{68}+56 q^{66}+40 q^{64}-103 q^{62}-153 q^{60}-2 q^{58}+172 q^{56}+269 q^{54}-41 q^{52}-349 q^{50}-286 q^{48}+88 q^{46}+541 q^{44}+295 q^{42}-291 q^{40}-582 q^{38}-260 q^{36}+507 q^{34}+589 q^{32}+29 q^{30}-565 q^{28}-529 q^{26}+216 q^{24}+566 q^{22}+282 q^{20}-309 q^{18}-511 q^{16}-45 q^{14}+349 q^{12}+338 q^{10}-61 q^8-348 q^6-197 q^4+133 q^2+323+133 q^{-2} -197 q^{-4} -348 q^{-6} -61 q^{-8} +338 q^{-10} +349 q^{-12} -45 q^{-14} -511 q^{-16} -309 q^{-18} +282 q^{-20} +566 q^{-22} +216 q^{-24} -529 q^{-26} -565 q^{-28} +29 q^{-30} +589 q^{-32} +507 q^{-34} -260 q^{-36} -582 q^{-38} -291 q^{-40} +295 q^{-42} +541 q^{-44} +88 q^{-46} -286 q^{-48} -349 q^{-50} -41 q^{-52} +269 q^{-54} +172 q^{-56} -2 q^{-58} -153 q^{-60} -103 q^{-62} +40 q^{-64} +56 q^{-66} +41 q^{-68} -16 q^{-70} -28 q^{-72} -2 q^{-74} +3 q^{-76} +8 q^{-78} - q^{-80} -3 q^{-82} + q^{-84} }[/math]
5	[math]\displaystyle{ q^{125}-3 q^{123}-q^{121}+8 q^{119}+3 q^{117}-5 q^{115}-16 q^{113}-18 q^{111}+5 q^{109}+55 q^{107}+75 q^{105}+5 q^{103}-117 q^{101}-199 q^{99}-116 q^{97}+135 q^{95}+429 q^{93}+425 q^{91}-31 q^{89}-637 q^{87}-905 q^{85}-422 q^{83}+649 q^{81}+1500 q^{79}+1200 q^{77}-277 q^{75}-1879 q^{73}-2201 q^{71}-621 q^{69}+1832 q^{67}+3129 q^{65}+1860 q^{63}-1222 q^{61}-3614 q^{59}-3154 q^{57}+104 q^{55}+3538 q^{53}+4145 q^{51}+1191 q^{49}-2871 q^{47}-4563 q^{45}-2378 q^{43}+1862 q^{41}+4404 q^{39}+3127 q^{37}-751 q^{35}-3793 q^{33}-3415 q^{31}-164 q^{29}+2928 q^{27}+3251 q^{25}+827 q^{23}-2050 q^{21}-2874 q^{19}-1163 q^{17}+1281 q^{15}+2369 q^{13}+1365 q^{11}-676 q^9-1958 q^7-1474 q^5+218 q^3+1650 q+1650 q^{-1} +218 q^{-3} -1474 q^{-5} -1958 q^{-7} -676 q^{-9} +1365 q^{-11} +2369 q^{-13} +1281 q^{-15} -1163 q^{-17} -2874 q^{-19} -2050 q^{-21} +827 q^{-23} +3251 q^{-25} +2928 q^{-27} -164 q^{-29} -3415 q^{-31} -3793 q^{-33} -751 q^{-35} +3127 q^{-37} +4404 q^{-39} +1862 q^{-41} -2378 q^{-43} -4563 q^{-45} -2871 q^{-47} +1191 q^{-49} +4145 q^{-51} +3538 q^{-53} +104 q^{-55} -3154 q^{-57} -3614 q^{-59} -1222 q^{-61} +1860 q^{-63} +3129 q^{-65} +1832 q^{-67} -621 q^{-69} -2201 q^{-71} -1879 q^{-73} -277 q^{-75} +1200 q^{-77} +1500 q^{-79} +649 q^{-81} -422 q^{-83} -905 q^{-85} -637 q^{-87} -31 q^{-89} +425 q^{-91} +429 q^{-93} +135 q^{-95} -116 q^{-97} -199 q^{-99} -117 q^{-101} +5 q^{-103} +75 q^{-105} +55 q^{-107} +5 q^{-109} -18 q^{-111} -16 q^{-113} -5 q^{-115} +3 q^{-117} +8 q^{-119} - q^{-121} -3 q^{-123} + q^{-125} }[/math]
6	[math]\displaystyle{ q^{174}-3 q^{172}-q^{170}+8 q^{168}+3 q^{166}-5 q^{164}-19 q^{162}-6 q^{160}+3 q^{158}+19 q^{156}+74 q^{154}+46 q^{152}-37 q^{150}-165 q^{148}-191 q^{146}-117 q^{144}+105 q^{142}+493 q^{140}+615 q^{138}+334 q^{136}-418 q^{134}-1106 q^{132}-1471 q^{130}-907 q^{128}+794 q^{126}+2478 q^{124}+3133 q^{122}+1676 q^{120}-1190 q^{118}-4531 q^{116}-5858 q^{114}-3284 q^{112}+2081 q^{110}+7610 q^{108}+9275 q^{106}+5639 q^{104}-3099 q^{102}-11670 q^{100}-14037 q^{98}-7817 q^{96}+4735 q^{94}+16040 q^{92}+19384 q^{90}+9942 q^{88}-7128 q^{86}-21493 q^{84}-23749 q^{82}-10860 q^{80}+9930 q^{78}+26865 q^{76}+27036 q^{74}+10046 q^{72}-14232 q^{70}-30534 q^{68}-27888 q^{66}-7581 q^{64}+18756 q^{62}+32595 q^{60}+25896 q^{58}+2664 q^{56}-21877 q^{54}-31950 q^{52}-21520 q^{50}+2933 q^{48}+23687 q^{46}+28502 q^{44}+14738 q^{42}-7366 q^{40}-23259 q^{38}-23130 q^{36}-7741 q^{34}+10599 q^{32}+20653 q^{30}+16332 q^{28}+2223 q^{26}-11946 q^{24}-16899 q^{22}-10125 q^{20}+2028 q^{18}+11760 q^{16}+12663 q^{14}+5442 q^{12}-4872 q^{10}-11043 q^8-9364 q^6-1663 q^4+7039 q^2+10561+7039 q^{-2} -1663 q^{-4} -9364 q^{-6} -11043 q^{-8} -4872 q^{-10} +5442 q^{-12} +12663 q^{-14} +11760 q^{-16} +2028 q^{-18} -10125 q^{-20} -16899 q^{-22} -11946 q^{-24} +2223 q^{-26} +16332 q^{-28} +20653 q^{-30} +10599 q^{-32} -7741 q^{-34} -23130 q^{-36} -23259 q^{-38} -7366 q^{-40} +14738 q^{-42} +28502 q^{-44} +23687 q^{-46} +2933 q^{-48} -21520 q^{-50} -31950 q^{-52} -21877 q^{-54} +2664 q^{-56} +25896 q^{-58} +32595 q^{-60} +18756 q^{-62} -7581 q^{-64} -27888 q^{-66} -30534 q^{-68} -14232 q^{-70} +10046 q^{-72} +27036 q^{-74} +26865 q^{-76} +9930 q^{-78} -10860 q^{-80} -23749 q^{-82} -21493 q^{-84} -7128 q^{-86} +9942 q^{-88} +19384 q^{-90} +16040 q^{-92} +4735 q^{-94} -7817 q^{-96} -14037 q^{-98} -11670 q^{-100} -3099 q^{-102} +5639 q^{-104} +9275 q^{-106} +7610 q^{-108} +2081 q^{-110} -3284 q^{-112} -5858 q^{-114} -4531 q^{-116} -1190 q^{-118} +1676 q^{-120} +3133 q^{-122} +2478 q^{-124} +794 q^{-126} -907 q^{-128} -1471 q^{-130} -1106 q^{-132} -418 q^{-134} +334 q^{-136} +615 q^{-138} +493 q^{-140} +105 q^{-142} -117 q^{-144} -191 q^{-146} -165 q^{-148} -37 q^{-150} +46 q^{-152} +74 q^{-154} +19 q^{-156} +3 q^{-158} -6 q^{-160} -19 q^{-162} -5 q^{-164} +3 q^{-166} +8 q^{-168} - q^{-170} -3 q^{-172} + q^{-174} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ q^{12}-2 q^{10}-q^6-q^4+4 q^2+1+4 q^{-2} - q^{-4} - q^{-6} -2 q^{-10} + q^{-12} }[/math]
1,1	[math]\displaystyle{ q^{36}-6 q^{34}+18 q^{32}-38 q^{30}+71 q^{28}-124 q^{26}+188 q^{24}-246 q^{22}+300 q^{20}-322 q^{18}+298 q^{16}-236 q^{14}+111 q^{12}+36 q^{10}-204 q^8+360 q^6-482 q^4+578 q^2-598+578 q^{-2} -482 q^{-4} +360 q^{-6} -204 q^{-8} +36 q^{-10} +111 q^{-12} -236 q^{-14} +298 q^{-16} -322 q^{-18} +300 q^{-20} -246 q^{-22} +188 q^{-24} -124 q^{-26} +71 q^{-28} -38 q^{-30} +18 q^{-32} -6 q^{-34} + q^{-36} }[/math]
2,0	[math]\displaystyle{ q^{32}-2 q^{30}-2 q^{28}+5 q^{26}+2 q^{24}-3 q^{22}-2 q^{20}+5 q^{18}+2 q^{16}-11 q^{14}-2 q^{12}+3 q^{10}-6 q^8-4 q^6+7 q^4+8 q^2+4+8 q^{-2} +7 q^{-4} -4 q^{-6} -6 q^{-8} +3 q^{-10} -2 q^{-12} -11 q^{-14} +2 q^{-16} +5 q^{-18} -2 q^{-20} -3 q^{-22} +2 q^{-24} +5 q^{-26} -2 q^{-28} -2 q^{-30} + q^{-32} }[/math]

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^{28}-3 q^{26}+7 q^{22}-8 q^{20}+2 q^{18}+11 q^{16}-14 q^{14}-q^{12}+7 q^{10}-12 q^8-2 q^6+9 q^4+4 q^2+4+4 q^{-2} +9 q^{-4} -2 q^{-6} -12 q^{-8} +7 q^{-10} - q^{-12} -14 q^{-14} +11 q^{-16} +2 q^{-18} -8 q^{-20} +7 q^{-22} -3 q^{-26} + q^{-28} }[/math]
1,0,0	[math]\displaystyle{ q^{15}-2 q^{13}+q^{11}-3 q^9-q^5+3 q^3+3 q+3 q^{-1} +3 q^{-3} - q^{-5} -3 q^{-9} + q^{-11} -2 q^{-13} + q^{-15} }[/math]
1,0,1	[math]\displaystyle{ q^{46}-6 q^{44}+15 q^{42}-17 q^{40}-3 q^{38}+48 q^{36}-95 q^{34}+100 q^{32}-28 q^{30}-107 q^{28}+237 q^{26}-277 q^{24}+195 q^{22}+11 q^{20}-243 q^{18}+384 q^{16}-393 q^{14}+209 q^{12}+14 q^{10}-210 q^8+257 q^6-153 q^4+47 q^2+43+47 q^{-2} -153 q^{-4} +257 q^{-6} -210 q^{-8} +14 q^{-10} +209 q^{-12} -393 q^{-14} +384 q^{-16} -243 q^{-18} +11 q^{-20} +195 q^{-22} -277 q^{-24} +237 q^{-26} -107 q^{-28} -28 q^{-30} +100 q^{-32} -95 q^{-34} +48 q^{-36} -3 q^{-38} -17 q^{-40} +15 q^{-42} -6 q^{-44} + q^{-46} }[/math]

A4 Invariants.

Weight	Invariant
0,1,0,0	[math]\displaystyle{ q^{34}-2 q^{32}-2 q^{30}+5 q^{28}-2 q^{26}-4 q^{24}+8 q^{22}+7 q^{20}-9 q^{18}-5 q^{16}+4 q^{14}-7 q^{12}-16 q^{10}+12 q^6-4 q^4+7 q^2+24+7 q^{-2} -4 q^{-4} +12 q^{-6} -16 q^{-10} -7 q^{-12} +4 q^{-14} -5 q^{-16} -9 q^{-18} +7 q^{-20} +8 q^{-22} -4 q^{-24} -2 q^{-26} +5 q^{-28} -2 q^{-30} -2 q^{-32} + q^{-34} }[/math]
1,0,0,0	[math]\displaystyle{ q^{18}-2 q^{16}+q^{14}-2 q^{12}-2 q^{10}-q^6+3 q^4+2 q^2+5+2 q^{-2} +3 q^{-4} - q^{-6} -2 q^{-10} -2 q^{-12} + q^{-14} -2 q^{-16} + q^{-18} }[/math]

B2 Invariants.

Weight

Invariant

0,1

[math]\displaystyle{ q^{28}-3 q^{26}+6 q^{24}-9 q^{22}+12 q^{20}-14 q^{18}+11 q^{16}-10 q^{14}+5 q^{12}-q^{10}-6 q^8+14 q^6-17 q^4+24 q^2-22+24 q^{-2} -17 q^{-4} +14 q^{-6} -6 q^{-8} - q^{-10} +5 q^{-12} -10 q^{-14} +11 q^{-16} -14 q^{-18} +12 q^{-20} -9 q^{-22} +6 q^{-24} -3 q^{-26} + q^{-28} }[/math]

1,0

[math]\displaystyle{ q^{46}-3 q^{42}-3 q^{40}+3 q^{38}+8 q^{36}+q^{34}-10 q^{32}-6 q^{30}+11 q^{28}+11 q^{26}-5 q^{24}-15 q^{22}-3 q^{20}+11 q^{18}+4 q^{16}-11 q^{14}-9 q^{12}+6 q^{10}+8 q^8-q^6-7 q^4+5 q^2+13+5 q^{-2} -7 q^{-4} - q^{-6} +8 q^{-8} +6 q^{-10} -9 q^{-12} -11 q^{-14} +4 q^{-16} +11 q^{-18} -3 q^{-20} -15 q^{-22} -5 q^{-24} +11 q^{-26} +11 q^{-28} -6 q^{-30} -10 q^{-32} + q^{-34} +8 q^{-36} +3 q^{-38} -3 q^{-40} -3 q^{-42} + q^{-46} }[/math]

D4 Invariants.

Weight	Invariant
1,0,0,0	[math]\displaystyle{ q^{38}-3 q^{36}+3 q^{34}-4 q^{32}+8 q^{30}-10 q^{28}+10 q^{26}-9 q^{24}+11 q^{22}-9 q^{20}+3 q^{18}-6 q^{16}+2 q^{12}-11 q^{10}+9 q^8-10 q^6+21 q^4-13 q^2+22-13 q^{-2} +21 q^{-4} -10 q^{-6} +9 q^{-8} -11 q^{-10} +2 q^{-12} -6 q^{-16} +3 q^{-18} -9 q^{-20} +11 q^{-22} -9 q^{-24} +10 q^{-26} -10 q^{-28} +8 q^{-30} -4 q^{-32} +3 q^{-34} -3 q^{-36} + q^{-38} }[/math]

G2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ q^{66}-3 q^{64}+6 q^{62}-10 q^{60}+8 q^{58}-4 q^{56}-5 q^{54}+23 q^{52}-36 q^{50}+48 q^{48}-38 q^{46}+7 q^{44}+28 q^{42}-67 q^{40}+84 q^{38}-71 q^{36}+29 q^{34}+17 q^{32}-58 q^{30}+77 q^{28}-56 q^{26}+8 q^{24}+34 q^{22}-59 q^{20}+45 q^{18}-6 q^{16}-45 q^{14}+81 q^{12}-81 q^{10}+64 q^8-11 q^6-48 q^4+97 q^2-111+97 q^{-2} -48 q^{-4} -11 q^{-6} +64 q^{-8} -81 q^{-10} +81 q^{-12} -45 q^{-14} -6 q^{-16} +45 q^{-18} -59 q^{-20} +34 q^{-22} +8 q^{-24} -56 q^{-26} +77 q^{-28} -58 q^{-30} +17 q^{-32} +29 q^{-34} -71 q^{-36} +84 q^{-38} -67 q^{-40} +28 q^{-42} +7 q^{-44} -38 q^{-46} +48 q^{-48} -36 q^{-50} +23 q^{-52} -5 q^{-54} -4 q^{-56} +8 q^{-58} -10 q^{-60} +6 q^{-62} -3 q^{-64} + q^{-66} }[/math]

.

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["8 18"];

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

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

Out[4]= [math]\displaystyle{ -t^3+5 t^2-10 t+13-10 t^{-1} +5 t^{-2} - t^{-3} }[/math]

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

Out[5]= [math]\displaystyle{ -z^6-z^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{ \left\{t^2-t+1\right\} }[/math]

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

Out[7]= { 45, 0 }

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

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

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

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

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

Out[9]= [math]\displaystyle{ -z^6+a^2 z^4+z^4 a^{-2} -3 z^4+a^2 z^2+z^2 a^{-2} -z^2-a^2- a^{-2} +3 }[/math]

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

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

Out[10]= [math]\displaystyle{ 3 a z^7+3 z^7 a^{-1} +6 a^2 z^6+6 z^6 a^{-2} +12 z^6+4 a^3 z^5+3 a z^5+3 z^5 a^{-1} +4 z^5 a^{-3} +a^4 z^4-9 a^2 z^4-9 z^4 a^{-2} +z^4 a^{-4} -20 z^4-4 a^3 z^3-9 a z^3-9 z^3 a^{-1} -4 z^3 a^{-3} +3 a^2 z^2+3 z^2 a^{-2} +6 z^2+a z+z a^{-1} +a^2+ a^{-2} +3 }[/math]

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {9_24, K11n85, K11n164, ...}

Same Jones Polynomial (up to mirroring, [math]\displaystyle{ q\leftrightarrow q^{-1} }[/math]): {...}

Vassiliev invariants

V₂ and V₃:

(1, 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

[math]\displaystyle{ 4 }[/math]

[math]\displaystyle{ 0 }[/math]

[math]\displaystyle{ 8 }[/math]

[math]\displaystyle{ \frac{62}{3} }[/math]

[math]\displaystyle{ \frac{34}{3} }[/math]

[math]\displaystyle{ 0 }[/math]

[math]\displaystyle{ \frac{32}{3} }[/math]

[math]\displaystyle{ 0 }[/math]

[math]\displaystyle{ \frac{248}{3} }[/math]

[math]\displaystyle{ \frac{136}{3} }[/math]

[math]\displaystyle{ \frac{2671}{30} }[/math]

[math]\displaystyle{ -\frac{382}{15} }[/math]

[math]\displaystyle{ \frac{2942}{45} }[/math]

[math]\displaystyle{ \frac{401}{18} }[/math]

[math]\displaystyle{ \frac{751}{30} }[/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]0 is the signature of 8 18. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\		r
	\
j		\

-4

-3

-2

-1

0

1

2

3

4

χ

9

1

7

3

-3

5

3

1

2

3

4

3

-1

1

5

3

2

-1

3

5

2

-3

3

4

-1

-5

1

3

2

-7

3

-3

-9

1

Integral Khovanov Homology

(db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math]	[math]\displaystyle{ i=-1 }[/math]	[math]\displaystyle{ i=1 }[/math]
[math]\displaystyle{ r=-4 }[/math]	[math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-3 }[/math]	[math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math]	[math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-2 }[/math]	[math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math]	[math]\displaystyle{ {\mathbb Z}^{3} }[/math]
[math]\displaystyle{ r=-1 }[/math]	[math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} }[/math]	[math]\displaystyle{ {\mathbb Z}^{3} }[/math]
[math]\displaystyle{ r=0 }[/math]	[math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} }[/math]	[math]\displaystyle{ {\mathbb Z}^{5} }[/math]
[math]\displaystyle{ r=1 }[/math]	[math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} }[/math]	[math]\displaystyle{ {\mathbb Z}^{4} }[/math]
[math]\displaystyle{ r=2 }[/math]	[math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math]	[math]\displaystyle{ {\mathbb Z}^{3} }[/math]
[math]\displaystyle{ r=3 }[/math]	[math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math]	[math]\displaystyle{ {\mathbb Z}^{3} }[/math]
[math]\displaystyle{ r=4 }[/math]	[math]\displaystyle{ {\mathbb Z}_2 }[/math]	[math]\displaystyle{ {\mathbb Z} }[/math]

The Coloured Jones Polynomials

The Coloured Jones Polynomials (in the [math]\displaystyle{ n+1 }[/math]-dimensional representation of [math]\displaystyle{ sl(2) }[/math])

[math]\displaystyle{ n }[/math]	[math]\displaystyle{ J_n }[/math]
2	[math]\displaystyle{ q^{12}-4 q^{11}+2 q^{10}+13 q^9-21 q^8-4 q^7+41 q^6-38 q^5-20 q^4+69 q^3-43 q^2-36 q+81-36 q^{-1} -43 q^{-2} +69 q^{-3} -20 q^{-4} -38 q^{-5} +41 q^{-6} -4 q^{-7} -21 q^{-8} +13 q^{-9} +2 q^{-10} -4 q^{-11} + q^{-12} }[/math]
3	[math]\displaystyle{ q^{24}-4 q^{23}+2 q^{22}+9 q^{21}-q^{20}-24 q^{19}-10 q^{18}+55 q^{17}+27 q^{16}-79 q^{15}-73 q^{14}+108 q^{13}+130 q^{12}-121 q^{11}-199 q^{10}+119 q^9+266 q^8-105 q^7-322 q^6+74 q^5+374 q^4-53 q^3-389 q^2+10 q+411+10 q^{-1} -389 q^{-2} -53 q^{-3} +374 q^{-4} +74 q^{-5} -322 q^{-6} -105 q^{-7} +266 q^{-8} +119 q^{-9} -199 q^{-10} -121 q^{-11} +130 q^{-12} +108 q^{-13} -73 q^{-14} -79 q^{-15} +27 q^{-16} +55 q^{-17} -10 q^{-18} -24 q^{-19} - q^{-20} +9 q^{-21} +2 q^{-22} -4 q^{-23} + q^{-24} }[/math]
4	[math]\displaystyle{ q^{40}-4 q^{39}+2 q^{38}+9 q^{37}-5 q^{36}-4 q^{35}-30 q^{34}+14 q^{33}+66 q^{32}+10 q^{31}-20 q^{30}-173 q^{29}-36 q^{28}+217 q^{27}+184 q^{26}+77 q^{25}-483 q^{24}-344 q^{23}+280 q^{22}+558 q^{21}+530 q^{20}-729 q^{19}-930 q^{18}-11 q^{17}+880 q^{16}+1297 q^{15}-647 q^{14}-1490 q^{13}-605 q^{12}+916 q^{11}+2042 q^{10}-297 q^9-1774 q^8-1196 q^7+714 q^6+2508 q^5+97 q^4-1785 q^3-1595 q^2+427 q+2659+427 q^{-1} -1595 q^{-2} -1785 q^{-3} +97 q^{-4} +2508 q^{-5} +714 q^{-6} -1196 q^{-7} -1774 q^{-8} -297 q^{-9} +2042 q^{-10} +916 q^{-11} -605 q^{-12} -1490 q^{-13} -647 q^{-14} +1297 q^{-15} +880 q^{-16} -11 q^{-17} -930 q^{-18} -729 q^{-19} +530 q^{-20} +558 q^{-21} +280 q^{-22} -344 q^{-23} -483 q^{-24} +77 q^{-25} +184 q^{-26} +217 q^{-27} -36 q^{-28} -173 q^{-29} -20 q^{-30} +10 q^{-31} +66 q^{-32} +14 q^{-33} -30 q^{-34} -4 q^{-35} -5 q^{-36} +9 q^{-37} +2 q^{-38} -4 q^{-39} + q^{-40} }[/math]
5	[math]\displaystyle{ q^{60}-4 q^{59}+2 q^{58}+9 q^{57}-5 q^{56}-8 q^{55}-10 q^{54}-6 q^{53}+25 q^{52}+59 q^{51}+15 q^{50}-78 q^{49}-132 q^{48}-88 q^{47}+108 q^{46}+310 q^{45}+309 q^{44}-82 q^{43}-588 q^{42}-694 q^{41}-160 q^{40}+793 q^{39}+1380 q^{38}+769 q^{37}-888 q^{36}-2171 q^{35}-1762 q^{34}+471 q^{33}+2960 q^{32}+3222 q^{31}+409 q^{30}-3440 q^{29}-4844 q^{28}-1921 q^{27}+3420 q^{26}+6480 q^{25}+3843 q^{24}-2833 q^{23}-7798 q^{22}-5983 q^{21}+1728 q^{20}+8665 q^{19}+8083 q^{18}-291 q^{17}-9075 q^{16}-9861 q^{15}-1314 q^{14}+9043 q^{13}+11334 q^{12}+2801 q^{11}-8752 q^{10}-12285 q^9-4191 q^8+8219 q^7+13045 q^6+5245 q^5-7664 q^4-13289 q^3-6232 q^2+6937 q+13529+6937 q^{-1} -6232 q^{-2} -13289 q^{-3} -7664 q^{-4} +5245 q^{-5} +13045 q^{-6} +8219 q^{-7} -4191 q^{-8} -12285 q^{-9} -8752 q^{-10} +2801 q^{-11} +11334 q^{-12} +9043 q^{-13} -1314 q^{-14} -9861 q^{-15} -9075 q^{-16} -291 q^{-17} +8083 q^{-18} +8665 q^{-19} +1728 q^{-20} -5983 q^{-21} -7798 q^{-22} -2833 q^{-23} +3843 q^{-24} +6480 q^{-25} +3420 q^{-26} -1921 q^{-27} -4844 q^{-28} -3440 q^{-29} +409 q^{-30} +3222 q^{-31} +2960 q^{-32} +471 q^{-33} -1762 q^{-34} -2171 q^{-35} -888 q^{-36} +769 q^{-37} +1380 q^{-38} +793 q^{-39} -160 q^{-40} -694 q^{-41} -588 q^{-42} -82 q^{-43} +309 q^{-44} +310 q^{-45} +108 q^{-46} -88 q^{-47} -132 q^{-48} -78 q^{-49} +15 q^{-50} +59 q^{-51} +25 q^{-52} -6 q^{-53} -10 q^{-54} -8 q^{-55} -5 q^{-56} +9 q^{-57} +2 q^{-58} -4 q^{-59} + q^{-60} }[/math]
6	[math]\displaystyle{ q^{84}-4 q^{83}+2 q^{82}+9 q^{81}-5 q^{80}-8 q^{79}-14 q^{78}+14 q^{77}+5 q^{76}+18 q^{75}+64 q^{74}-33 q^{73}-91 q^{72}-142 q^{71}-12 q^{70}+79 q^{69}+240 q^{68}+452 q^{67}+89 q^{66}-372 q^{65}-894 q^{64}-700 q^{63}-286 q^{62}+804 q^{61}+2153 q^{60}+1773 q^{59}+283 q^{58}-2351 q^{57}-3566 q^{56}-3627 q^{55}-523 q^{54}+4727 q^{53}+7138 q^{52}+5812 q^{51}-686 q^{50}-7202 q^{49}-12365 q^{48}-9094 q^{47}+2360 q^{46}+13358 q^{45}+18364 q^{44}+10619 q^{43}-3858 q^{42}-21807 q^{41}-26164 q^{40}-12005 q^{39}+11102 q^{38}+31253 q^{37}+31409 q^{36}+13077 q^{35}-21636 q^{34}-43154 q^{33}-36283 q^{32}-5200 q^{31}+33899 q^{30}+51716 q^{29}+39414 q^{28}-7797 q^{27}-49853 q^{26}-59515 q^{25}-29741 q^{24}+23826 q^{23}+62146 q^{22}+63867 q^{21}+12957 q^{20}-45038 q^{19}-73279 q^{18}-51845 q^{17}+7933 q^{16}+62275 q^{15}+79256 q^{14}+31297 q^{13}-34984 q^{12}-77600 q^{11}-65954 q^{10}-6236 q^9+57322 q^8+86030 q^7+43450 q^6-25252 q^5-76697 q^4-73175 q^3-16550 q^2+51151 q+87709+51151 q^{-1} -16550 q^{-2} -73175 q^{-3} -76697 q^{-4} -25252 q^{-5} +43450 q^{-6} +86030 q^{-7} +57322 q^{-8} -6236 q^{-9} -65954 q^{-10} -77600 q^{-11} -34984 q^{-12} +31297 q^{-13} +79256 q^{-14} +62275 q^{-15} +7933 q^{-16} -51845 q^{-17} -73279 q^{-18} -45038 q^{-19} +12957 q^{-20} +63867 q^{-21} +62146 q^{-22} +23826 q^{-23} -29741 q^{-24} -59515 q^{-25} -49853 q^{-26} -7797 q^{-27} +39414 q^{-28} +51716 q^{-29} +33899 q^{-30} -5200 q^{-31} -36283 q^{-32} -43154 q^{-33} -21636 q^{-34} +13077 q^{-35} +31409 q^{-36} +31253 q^{-37} +11102 q^{-38} -12005 q^{-39} -26164 q^{-40} -21807 q^{-41} -3858 q^{-42} +10619 q^{-43} +18364 q^{-44} +13358 q^{-45} +2360 q^{-46} -9094 q^{-47} -12365 q^{-48} -7202 q^{-49} -686 q^{-50} +5812 q^{-51} +7138 q^{-52} +4727 q^{-53} -523 q^{-54} -3627 q^{-55} -3566 q^{-56} -2351 q^{-57} +283 q^{-58} +1773 q^{-59} +2153 q^{-60} +804 q^{-61} -286 q^{-62} -700 q^{-63} -894 q^{-64} -372 q^{-65} +89 q^{-66} +452 q^{-67} +240 q^{-68} +79 q^{-69} -12 q^{-70} -142 q^{-71} -91 q^{-72} -33 q^{-73} +64 q^{-74} +18 q^{-75} +5 q^{-76} +14 q^{-77} -14 q^{-78} -8 q^{-79} -5 q^{-80} +9 q^{-81} +2 q^{-82} -4 q^{-83} + q^{-84} }[/math]
7	[math]\displaystyle{ q^{112}-4 q^{111}+2 q^{110}+9 q^{109}-5 q^{108}-8 q^{107}-14 q^{106}+10 q^{105}+25 q^{104}-2 q^{103}+23 q^{102}+16 q^{101}-46 q^{100}-91 q^{99}-120 q^{98}+q^{97}+194 q^{96}+228 q^{95}+305 q^{94}+159 q^{93}-265 q^{92}-667 q^{91}-1062 q^{90}-701 q^{89}+318 q^{88}+1373 q^{87}+2402 q^{86}+2319 q^{85}+673 q^{84}-1836 q^{83}-4899 q^{82}-5994 q^{81}-3738 q^{80}+996 q^{79}+7628 q^{78}+11884 q^{77}+10820 q^{76}+3971 q^{75}-8456 q^{74}-19772 q^{73}-23149 q^{72}-15565 q^{71}+3896 q^{70}+26043 q^{69}+39541 q^{68}+36591 q^{67}+11339 q^{66}-25804 q^{65}-57204 q^{64}-66358 q^{63}-39973 q^{62}+12511 q^{61}+68336 q^{60}+100926 q^{59}+83792 q^{58}+19179 q^{57}-66151 q^{56}-133172 q^{55}-137679 q^{54}-70621 q^{53}+42889 q^{52}+153359 q^{51}+194934 q^{50}+139172 q^{49}+3442 q^{48}-154136 q^{47}-245138 q^{46}-216584 q^{45}-71245 q^{44}+130992 q^{43}+279711 q^{42}+293462 q^{41}+153405 q^{40}-85147 q^{39}-293239 q^{38}-360206 q^{37}-240305 q^{36}+21900 q^{35}+284890 q^{34}+410240 q^{33}+322851 q^{32}+50497 q^{31}-258503 q^{30}-441071 q^{29}-393503 q^{28}-123370 q^{27}+219717 q^{26}+454109 q^{25}+448852 q^{24}+189820 q^{23}-176053 q^{22}-453071 q^{21}-487746 q^{20}-245868 q^{19}+132559 q^{18}+443069 q^{17}+513256 q^{16}+289741 q^{15}-94142 q^{14}-428270 q^{13}-527350 q^{12}-322834 q^{11}+60878 q^{10}+412390 q^9+535248 q^8+347117 q^7-34348 q^6-396718 q^5-537790 q^4-365850 q^3+10559 q^2+381625 q+539297+381625 q^{-1} +10559 q^{-2} -365850 q^{-3} -537790 q^{-4} -396718 q^{-5} -34348 q^{-6} +347117 q^{-7} +535248 q^{-8} +412390 q^{-9} +60878 q^{-10} -322834 q^{-11} -527350 q^{-12} -428270 q^{-13} -94142 q^{-14} +289741 q^{-15} +513256 q^{-16} +443069 q^{-17} +132559 q^{-18} -245868 q^{-19} -487746 q^{-20} -453071 q^{-21} -176053 q^{-22} +189820 q^{-23} +448852 q^{-24} +454109 q^{-25} +219717 q^{-26} -123370 q^{-27} -393503 q^{-28} -441071 q^{-29} -258503 q^{-30} +50497 q^{-31} +322851 q^{-32} +410240 q^{-33} +284890 q^{-34} +21900 q^{-35} -240305 q^{-36} -360206 q^{-37} -293239 q^{-38} -85147 q^{-39} +153405 q^{-40} +293462 q^{-41} +279711 q^{-42} +130992 q^{-43} -71245 q^{-44} -216584 q^{-45} -245138 q^{-46} -154136 q^{-47} +3442 q^{-48} +139172 q^{-49} +194934 q^{-50} +153359 q^{-51} +42889 q^{-52} -70621 q^{-53} -137679 q^{-54} -133172 q^{-55} -66151 q^{-56} +19179 q^{-57} +83792 q^{-58} +100926 q^{-59} +68336 q^{-60} +12511 q^{-61} -39973 q^{-62} -66358 q^{-63} -57204 q^{-64} -25804 q^{-65} +11339 q^{-66} +36591 q^{-67} +39541 q^{-68} +26043 q^{-69} +3896 q^{-70} -15565 q^{-71} -23149 q^{-72} -19772 q^{-73} -8456 q^{-74} +3971 q^{-75} +10820 q^{-76} +11884 q^{-77} +7628 q^{-78} +996 q^{-79} -3738 q^{-80} -5994 q^{-81} -4899 q^{-82} -1836 q^{-83} +673 q^{-84} +2319 q^{-85} +2402 q^{-86} +1373 q^{-87} +318 q^{-88} -701 q^{-89} -1062 q^{-90} -667 q^{-91} -265 q^{-92} +159 q^{-93} +305 q^{-94} +228 q^{-95} +194 q^{-96} + q^{-97} -120 q^{-98} -91 q^{-99} -46 q^{-100} +16 q^{-101} +23 q^{-102} -2 q^{-103} +25 q^{-104} +10 q^{-105} -14 q^{-106} -8 q^{-107} -5 q^{-108} +9 q^{-109} +2 q^{-110} -4 q^{-111} + q^{-112} }[/math]

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[8, 18]]
Out[2]=	PD[X[6, 2, 7, 1], X[8, 3, 9, 4], X[16, 11, 1, 12], X[2, 14, 3, 13], X[4, 15, 5, 16], X[10, 6, 11, 5], X[12, 7, 13, 8], X[14, 10, 15, 9]]
In[3]:=	GaussCode[Knot[8, 18]]
Out[3]=	GaussCode[1, -4, 2, -5, 6, -1, 7, -2, 8, -6, 3, -7, 4, -8, 5, -3]
In[4]:=	DTCode[Knot[8, 18]]
Out[4]=	DTCode[6, 8, 10, 12, 14, 16, 2, 4]
In[5]:=	br = BR[Knot[8, 18]]
Out[5]=	BR[3, {-1, 2, -1, 2, -1, 2, -1, 2}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{3, 8}
In[7]:=	BraidIndex[Knot[8, 18]]
Out[7]=	3
In[8]:=	Show[DrawMorseLink[Knot[8, 18]]]

`Out[8]=`	`-Graphics-`
In[9]:=	(#[Knot[8, 18]]&) /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{FullyAmphicheiral, 2, 3, 3, 4, 2}
In[10]:=	alex = Alexander[Knot[8, 18]][t]
Out[10]=	-3 5 10 2 3 13 - t + -- - -- - 10 t + 5 t - t 2 t t
In[11]:=	Conway[Knot[8, 18]][z]
Out[11]=	2 4 6 1 + z - z - z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[8, 18], Knot[9, 24], Knot[11, NonAlternating, 85], Knot[11, NonAlternating, 164]}
In[13]:=	{KnotDet[Knot[8, 18]], KnotSignature[Knot[8, 18]]}
Out[13]=	{45, 0}
In[14]:=	Jones[Knot[8, 18]][q]
Out[14]=	-4 4 6 7 2 3 4 9 + q - -- + -- - - - 7 q + 6 q - 4 q + q 3 2 q q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] \|\| (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[8, 18]}
In[16]:=	A2Invariant[Knot[8, 18]][q]
Out[16]=	-12 2 -6 -4 4 2 4 6 10 12 1 + q - --- - q - q + -- + 4 q - q - q - 2 q + q 10 2 q q
In[17]:=	HOMFLYPT[Knot[8, 18]][a, z]
Out[17]=	2 4 -2 2 2 z 2 2 4 z 2 4 6 3 - a - a - z + -- + a z - 3 z + -- + a z - z 2 2 a a
In[18]:=	Kauffman[Knot[8, 18]][a, z]
Out[18]=	2 3 3 -2 2 z 2 3 z 2 2 4 z 9 z 3 3 + a + a + - + a z + 6 z + ---- + 3 a z - ---- - ---- - 9 a z - a 2 3 a a a 4 4 5 5 3 3 4 z 9 z 2 4 4 4 4 z 3 z 4 a z - 20 z + -- - ---- - 9 a z + a z + ---- + ---- + 4 2 3 a a a a 6 7 5 3 5 6 6 z 2 6 3 z 7 3 a z + 4 a z + 12 z + ---- + 6 a z + ---- + 3 a z 2 a a
In[19]:=	{Vassiliev[2][Knot[8, 18]], Vassiliev[3][Knot[8, 18]]}
Out[19]=	{1, 0}
In[20]:=	Kh[Knot[8, 18]][q, t]
Out[20]=	5 1 3 1 3 3 4 3 - + 5 q + ----- + ----- + ----- + ----- + ----- + ---- + --- + 3 q t + q 9 4 7 3 5 3 5 2 3 2 3 q t q t q t q t q t q t q t 3 3 2 5 2 5 3 7 3 9 4 4 q t + 3 q t + 3 q t + q t + 3 q t + q t
In[21]:=	ColouredJones[Knot[8, 18], 2][q]
Out[21]=	-12 4 2 13 21 4 41 38 20 69 43 36 81 + q - --- + --- + -- - -- - -- + -- - -- - -- + -- - -- - -- - 11 10 9 8 7 6 5 4 3 2 q q q q q q q q q q q 2 3 4 5 6 7 8 9 36 q - 43 q + 69 q - 20 q - 38 q + 41 q - 4 q - 21 q + 13 q + 10 11 12 2 q - 4 q + q

See/edit the Rolfsen_Splice_Template.

Back to the top.

8_17

8_19

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+{| width=100%
-See/edit the [[Rolfsen_Splice_Template]].
+|align=left|See/edit the [[Rolfsen_Splice_Template]].
+Back to the [[#top|top]].
+|align=right|{{Knot Navigation Links|ext=gif}}
+|}
  [[Category:Knot Page]]

8 18: Difference between revisions

Revision as of 20:05, 29 August 2005

Contents

Knot presentations

Three dimensional invariants

Four dimensional invariants

Polynomial invariants

"Similar" Knots (within the Atlas)

Vassiliev invariants

Khovanov Homology

The Coloured Jones Polynomials

Computer Talk

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