8 7

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8 6.gif

8_6

8 8.gif

8_8

8 7.gif Visit 8 7's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 8 7's page at Knotilus!

Visit 8 7's page at the original Knot Atlas!

8 7 Quick Notes


8 7 Further Notes and Views

Knot presentations

Planar diagram presentation X1425 X3,10,4,11 X11,1,12,16 X5,13,6,12 X7,15,8,14 X13,7,14,6 X15,9,16,8 X9,2,10,3
Gauss code -1, 8, -2, 1, -4, 6, -5, 7, -8, 2, -3, 4, -6, 5, -7, 3
Dowker-Thistlethwaite code 4 10 12 14 2 16 6 8
Conway Notation [4112]

Three dimensional invariants

Symmetry type Reversible
Unknotting number 1
3-genus 3
Bridge index 2
Super bridge index [math]\displaystyle{ \{3,6\} }[/math]
Nakanishi index 1
Maximal Thurston-Bennequin number [-2][-8]
Hyperbolic Volume 7.0222
A-Polynomial See Data:8 7/A-polynomial

[edit Notes for 8 7'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{ 3 }[/math]
Rasmussen s-Invariant 2

[edit Notes for 8 7's four dimensional invariants]

Polynomial invariants

Alexander polynomial [math]\displaystyle{ t^3-3 t^2+5 t-5+5 t^{-1} -3 t^{-2} + t^{-3} }[/math]
Conway polynomial [math]\displaystyle{ z^6+3 z^4+2 z^2+1 }[/math]
2nd Alexander ideal (db, data sources) [math]\displaystyle{ \{1\} }[/math]
Determinant and Signature { 23, 2 }
Jones polynomial [math]\displaystyle{ -q^6+2 q^5-3 q^4+4 q^3-4 q^2+4 q-2+2 q^{-1} - q^{-2} }[/math]
HOMFLY-PT polynomial (db, data sources) [math]\displaystyle{ z^6 a^{-2} +5 z^4 a^{-2} -z^4 a^{-4} -z^4+8 z^2 a^{-2} -3 z^2 a^{-4} -3 z^2+4 a^{-2} -2 a^{-4} -1 }[/math]
Kauffman polynomial (db, data sources) [math]\displaystyle{ z^7 a^{-1} +z^7 a^{-3} +4 z^6 a^{-2} +2 z^6 a^{-4} +2 z^6+a z^5-z^5 a^{-1} +2 z^5 a^{-5} -12 z^4 a^{-2} -3 z^4 a^{-4} +2 z^4 a^{-6} -7 z^4-3 a z^3-3 z^3 a^{-1} -2 z^3 a^{-3} -z^3 a^{-5} +z^3 a^{-7} +12 z^2 a^{-2} +4 z^2 a^{-4} -2 z^2 a^{-6} +6 z^2+a z+2 z a^{-1} +2 z a^{-3} -z a^{-7} -4 a^{-2} -2 a^{-4} -1 }[/math]
The A2 invariant [math]\displaystyle{ -q^6+1+2 q^{-2} +2 q^{-6} + q^{-10} - q^{-14} - q^{-18} }[/math]
The G2 invariant [math]\displaystyle{ q^{32}-q^{30}+2 q^{28}-3 q^{26}+q^{24}-q^{22}-3 q^{20}+7 q^{18}-8 q^{16}+5 q^{14}-3 q^{12}-2 q^{10}+6 q^8-10 q^6+7 q^4-3 q^2+6 q^{-2} -6 q^{-4} +4 q^{-6} +3 q^{-8} - q^{-10} +4 q^{-12} -4 q^{-14} +2 q^{-16} +4 q^{-18} -5 q^{-20} +10 q^{-22} -7 q^{-24} +5 q^{-26} +3 q^{-28} -7 q^{-30} +10 q^{-32} -11 q^{-34} +8 q^{-36} -2 q^{-38} -3 q^{-40} +8 q^{-42} -8 q^{-44} +6 q^{-46} -3 q^{-50} +3 q^{-52} -3 q^{-54} - q^{-56} +4 q^{-58} -5 q^{-60} +5 q^{-62} -3 q^{-64} -2 q^{-66} +4 q^{-68} -7 q^{-70} +5 q^{-72} -5 q^{-74} +2 q^{-76} - q^{-78} -3 q^{-80} +4 q^{-82} -4 q^{-84} +4 q^{-86} -2 q^{-88} + q^{-90} -2 q^{-94} +2 q^{-96} - q^{-98} + q^{-100} }[/math]
Further Quantum Invariants
Further quantum knot invariants for 8_7.

A1 Invariants.

Weight Invariant
1 [math]\displaystyle{ -q^5+q^3+2 q^{-1} + q^{-7} - q^{-9} + q^{-11} - q^{-13} }[/math]
2 [math]\displaystyle{ q^{16}-q^{14}-2 q^{12}+2 q^{10}-3 q^6+2 q^4+3 q^2-2+2 q^{-2} +3 q^{-4} -2 q^{-6} +2 q^{-10} -2 q^{-14} - q^{-16} +3 q^{-18} -2 q^{-20} -2 q^{-22} +4 q^{-24} - q^{-26} -2 q^{-28} +2 q^{-30} - q^{-32} - q^{-34} + q^{-36} }[/math]
3 [math]\displaystyle{ -q^{33}+q^{31}+2 q^{29}-3 q^{25}-2 q^{23}+4 q^{21}+3 q^{19}-4 q^{17}-6 q^{15}+q^{13}+6 q^{11}+2 q^9-7 q^7-2 q^5+6 q^3+7 q-3 q^{-1} -5 q^{-3} +3 q^{-5} +7 q^{-7} - q^{-9} -6 q^{-11} +2 q^{-13} +5 q^{-15} -6 q^{-19} -2 q^{-21} +3 q^{-23} +3 q^{-25} - q^{-27} -5 q^{-29} -2 q^{-31} +7 q^{-33} +4 q^{-35} -6 q^{-37} -6 q^{-39} +5 q^{-41} +6 q^{-43} -3 q^{-45} -5 q^{-47} +2 q^{-49} +3 q^{-51} - q^{-53} -2 q^{-55} + q^{-57} - q^{-61} + q^{-65} + q^{-67} - q^{-69} }[/math]
4 [math]\displaystyle{ q^{56}-q^{54}-2 q^{52}+q^{48}+5 q^{46}-4 q^{42}-4 q^{40}-3 q^{38}+10 q^{36}+7 q^{34}-2 q^{32}-8 q^{30}-13 q^{28}+4 q^{26}+10 q^{24}+9 q^{22}-18 q^{18}-8 q^{16}+2 q^{14}+15 q^{12}+14 q^{10}-8 q^8-13 q^6-12 q^4+10 q^2+23+5 q^{-2} -9 q^{-4} -20 q^{-6} +21 q^{-10} +10 q^{-12} -5 q^{-14} -19 q^{-16} -4 q^{-18} +15 q^{-20} +8 q^{-22} -4 q^{-24} -15 q^{-26} -4 q^{-28} +10 q^{-30} +10 q^{-32} -2 q^{-34} -10 q^{-36} -6 q^{-38} + q^{-40} +11 q^{-42} +7 q^{-44} + q^{-46} -13 q^{-48} -15 q^{-50} +8 q^{-52} +15 q^{-54} +14 q^{-56} -10 q^{-58} -25 q^{-60} -3 q^{-62} +12 q^{-64} +23 q^{-66} + q^{-68} -21 q^{-70} -9 q^{-72} +16 q^{-76} +7 q^{-78} -8 q^{-80} -4 q^{-82} -6 q^{-84} +5 q^{-86} +4 q^{-88} - q^{-90} +2 q^{-92} -5 q^{-94} + q^{-98} +3 q^{-102} - q^{-104} - q^{-108} - q^{-110} + q^{-112} }[/math]
5 [math]\displaystyle{ -q^{85}+q^{83}+2 q^{81}-q^{77}-3 q^{75}-3 q^{73}+6 q^{69}+6 q^{67}+q^{65}-5 q^{63}-10 q^{61}-8 q^{59}+3 q^{57}+16 q^{55}+15 q^{53}+3 q^{51}-11 q^{49}-22 q^{47}-15 q^{45}+4 q^{43}+22 q^{41}+23 q^{39}+9 q^{37}-12 q^{35}-30 q^{33}-24 q^{31}-3 q^{29}+22 q^{27}+33 q^{25}+21 q^{23}-9 q^{21}-34 q^{19}-38 q^{17}-14 q^{15}+28 q^{13}+51 q^{11}+31 q^9-8 q^7-48 q^5-50 q^3-5 q+48 q^{-1} +59 q^{-3} +23 q^{-5} -33 q^{-7} -63 q^{-9} -34 q^{-11} +25 q^{-13} +60 q^{-15} +39 q^{-17} -15 q^{-19} -52 q^{-21} -40 q^{-23} +7 q^{-25} +45 q^{-27} +34 q^{-29} -7 q^{-31} -36 q^{-33} -30 q^{-35} +4 q^{-37} +33 q^{-39} +24 q^{-41} -7 q^{-43} -26 q^{-45} -20 q^{-47} +2 q^{-49} +23 q^{-51} +20 q^{-53} +5 q^{-55} -15 q^{-57} -23 q^{-59} -15 q^{-61} +3 q^{-63} +25 q^{-65} +32 q^{-67} +14 q^{-69} -24 q^{-71} -47 q^{-73} -31 q^{-75} +12 q^{-77} +54 q^{-79} +54 q^{-81} +2 q^{-83} -57 q^{-85} -69 q^{-87} -21 q^{-89} +46 q^{-91} +77 q^{-93} +39 q^{-95} -30 q^{-97} -71 q^{-99} -49 q^{-101} +13 q^{-103} +56 q^{-105} +49 q^{-107} +3 q^{-109} -37 q^{-111} -43 q^{-113} -12 q^{-115} +22 q^{-117} +29 q^{-119} +14 q^{-121} -7 q^{-123} -20 q^{-125} -14 q^{-127} +2 q^{-129} +11 q^{-131} +10 q^{-133} +3 q^{-135} -5 q^{-137} -8 q^{-139} -4 q^{-141} +3 q^{-143} +5 q^{-145} +2 q^{-147} + q^{-149} -2 q^{-151} -3 q^{-153} - q^{-155} + q^{-157} + q^{-161} + q^{-163} - q^{-165} }[/math]
6 [math]\displaystyle{ q^{120}-q^{118}-2 q^{116}+q^{112}+3 q^{110}+q^{108}+3 q^{106}-2 q^{104}-8 q^{102}-5 q^{100}-q^{98}+6 q^{96}+7 q^{94}+14 q^{92}+3 q^{90}-12 q^{88}-18 q^{86}-17 q^{84}-3 q^{82}+6 q^{80}+33 q^{78}+30 q^{76}+8 q^{74}-15 q^{72}-34 q^{70}-34 q^{68}-29 q^{66}+17 q^{64}+42 q^{62}+47 q^{60}+28 q^{58}-q^{56}-31 q^{54}-67 q^{52}-42 q^{50}-12 q^{48}+30 q^{46}+56 q^{44}+66 q^{42}+42 q^{40}-27 q^{38}-62 q^{36}-86 q^{34}-62 q^{32}-8 q^{30}+75 q^{28}+117 q^{26}+81 q^{24}+18 q^{22}-81 q^{20}-134 q^{18}-123 q^{16}-12 q^{14}+104 q^{12}+154 q^{10}+136 q^8+15 q^6-116 q^4-190 q^2-123+12 q^{-2} +137 q^{-4} +195 q^{-6} +116 q^{-8} -37 q^{-10} -177 q^{-12} -176 q^{-14} -74 q^{-16} +70 q^{-18} +179 q^{-20} +159 q^{-22} +32 q^{-24} -123 q^{-26} -164 q^{-28} -104 q^{-30} +17 q^{-32} +128 q^{-34} +140 q^{-36} +53 q^{-38} -75 q^{-40} -117 q^{-42} -85 q^{-44} +2 q^{-46} +83 q^{-48} +94 q^{-50} +35 q^{-52} -51 q^{-54} -74 q^{-56} -49 q^{-58} +13 q^{-60} +56 q^{-62} +58 q^{-64} +18 q^{-66} -36 q^{-68} -54 q^{-70} -42 q^{-72} +5 q^{-74} +37 q^{-76} +54 q^{-78} +40 q^{-80} +2 q^{-82} -46 q^{-84} -73 q^{-86} -52 q^{-88} -13 q^{-90} +59 q^{-92} +102 q^{-94} +94 q^{-96} +4 q^{-98} -96 q^{-100} -138 q^{-102} -118 q^{-104} +5 q^{-106} +136 q^{-108} +202 q^{-110} +116 q^{-112} -44 q^{-114} -177 q^{-116} -226 q^{-118} -111 q^{-120} +80 q^{-122} +236 q^{-124} +213 q^{-126} +65 q^{-128} -113 q^{-130} -237 q^{-132} -188 q^{-134} -24 q^{-136} +157 q^{-138} +200 q^{-140} +125 q^{-142} -8 q^{-144} -144 q^{-146} -162 q^{-148} -76 q^{-150} +53 q^{-152} +108 q^{-154} +97 q^{-156} +40 q^{-158} -50 q^{-160} -84 q^{-162} -59 q^{-164} +4 q^{-166} +35 q^{-168} +45 q^{-170} +37 q^{-172} -10 q^{-174} -33 q^{-176} -29 q^{-178} -4 q^{-180} +7 q^{-182} +16 q^{-184} +23 q^{-186} -12 q^{-190} -14 q^{-192} -4 q^{-194} - q^{-196} +5 q^{-198} +13 q^{-200} +2 q^{-202} -3 q^{-204} -6 q^{-206} -2 q^{-208} -3 q^{-210} +5 q^{-214} + q^{-216} + q^{-218} - q^{-220} - q^{-224} - q^{-226} + q^{-228} }[/math]

A2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ -q^6+1+2 q^{-2} +2 q^{-6} + q^{-10} - q^{-14} - q^{-18} }[/math]
1,1 [math]\displaystyle{ q^{20}-2 q^{18}+4 q^{16}-8 q^{14}+13 q^{12}-18 q^{10}+18 q^8-24 q^6+20 q^4-16 q^2+10+4 q^{-2} -4 q^{-4} +22 q^{-6} -22 q^{-8} +34 q^{-10} -34 q^{-12} +34 q^{-14} -34 q^{-16} +26 q^{-18} -22 q^{-20} +12 q^{-22} -4 q^{-24} -4 q^{-26} +7 q^{-28} -12 q^{-30} +14 q^{-32} -14 q^{-34} +13 q^{-36} -12 q^{-38} +12 q^{-40} -10 q^{-42} +7 q^{-44} -6 q^{-46} +4 q^{-48} -2 q^{-50} + q^{-52} }[/math]
2,0 [math]\displaystyle{ q^{18}-q^{14}-q^{12}-q^8-3 q^6+2 q^2+2+ q^{-2} +4 q^{-4} +2 q^{-6} + q^{-8} +2 q^{-10} +2 q^{-12} - q^{-16} + q^{-18} - q^{-20} -3 q^{-22} + q^{-26} - q^{-28} - q^{-30} + q^{-32} - q^{-36} - q^{-38} + q^{-46} }[/math]
3,0 [math]\displaystyle{ -q^{36}+q^{32}+2 q^{30}+q^{28}-2 q^{26}-q^{24}+q^{22}+4 q^{20}-6 q^{16}-7 q^{14}-q^{12}+3 q^{10}-6 q^6-4 q^4+5 q^2+10+6 q^{-2} -2 q^{-4} + q^{-6} +5 q^{-8} +8 q^{-10} + q^{-12} +2 q^{-16} +4 q^{-18} + q^{-20} -4 q^{-22} - q^{-24} -3 q^{-28} -7 q^{-30} -5 q^{-32} +3 q^{-34} +3 q^{-36} -3 q^{-38} -6 q^{-40} + q^{-42} +8 q^{-44} +5 q^{-46} -3 q^{-48} -6 q^{-50} + q^{-52} +6 q^{-54} +2 q^{-56} -4 q^{-58} -5 q^{-60} +3 q^{-64} - q^{-68} -2 q^{-70} + q^{-72} + q^{-74} + q^{-76} + q^{-78} - q^{-84} }[/math]

A3 Invariants.

Weight Invariant
0,1,0 [math]\displaystyle{ q^{14}-q^{12}-3 q^6+q^4-2+2 q^{-2} +3 q^{-4} - q^{-6} +4 q^{-8} +5 q^{-10} +2 q^{-12} +2 q^{-14} + q^{-16} -3 q^{-20} -3 q^{-22} + q^{-24} -3 q^{-26} -2 q^{-28} +3 q^{-30} - q^{-32} -2 q^{-34} +2 q^{-36} - q^{-40} + q^{-42} }[/math]
1,0,0 [math]\displaystyle{ -q^7-q^3+q+2 q^{-3} + q^{-5} +2 q^{-7} +2 q^{-9} + q^{-11} + q^{-13} - q^{-15} -2 q^{-19} - q^{-23} }[/math]
1,0,1 [math]\displaystyle{ q^{24}-2 q^{22}+3 q^{20}-3 q^{18}+q^{16}+5 q^{14}-10 q^{12}+11 q^{10}-12 q^8+q^6+q^4-16 q^2+16-18 q^{-2} +15 q^{-4} +6 q^{-6} +22 q^{-10} +2 q^{-12} +11 q^{-14} +4 q^{-16} +4 q^{-18} -8 q^{-20} +7 q^{-22} -16 q^{-24} -18 q^{-30} +16 q^{-32} -15 q^{-34} +2 q^{-36} +9 q^{-38} -13 q^{-40} +10 q^{-42} - q^{-44} -6 q^{-46} +11 q^{-48} -7 q^{-50} +2 q^{-52} +4 q^{-54} -8 q^{-56} +6 q^{-58} -3 q^{-60} - q^{-62} +3 q^{-64} -2 q^{-66} + q^{-68} }[/math]

A4 Invariants.

Weight Invariant
0,1,0,0 [math]\displaystyle{ q^{16}-q^8-2 q^6-q^4-2 q^2-3+ q^{-4} + q^{-6} + q^{-8} +7 q^{-10} +7 q^{-12} +5 q^{-14} +6 q^{-16} +8 q^{-18} + q^{-20} - q^{-22} -4 q^{-26} -6 q^{-28} -3 q^{-30} -2 q^{-32} -4 q^{-34} -2 q^{-36} + q^{-38} -2 q^{-42} + q^{-44} +2 q^{-46} + q^{-52} }[/math]
1,0,0,0 [math]\displaystyle{ -q^8-q^4+2 q^{-4} + q^{-6} +3 q^{-8} +2 q^{-10} +3 q^{-12} + q^{-14} + q^{-16} - q^{-18} - q^{-20} - q^{-22} -2 q^{-24} - q^{-28} }[/math]

B2 Invariants.

Weight Invariant
0,1 [math]\displaystyle{ -q^{14}+q^{12}-2 q^{10}+2 q^8-3 q^6+3 q^4-2 q^2+2+ q^{-4} +3 q^{-6} -2 q^{-8} +5 q^{-10} -4 q^{-12} +6 q^{-14} -5 q^{-16} +4 q^{-18} -3 q^{-20} + q^{-22} - q^{-24} - q^{-26} +2 q^{-28} -3 q^{-30} +3 q^{-32} -2 q^{-34} +2 q^{-36} -2 q^{-38} + q^{-40} - q^{-42} }[/math]
1,0 [math]\displaystyle{ q^{24}-q^{20}-q^{18}+q^{16}+q^{14}-2 q^{12}-3 q^{10}+3 q^6+q^4-2 q^2-2+2 q^{-2} +3 q^{-4} +2 q^{-6} -2 q^{-8} +2 q^{-12} +3 q^{-14} +2 q^{-20} +3 q^{-22} -2 q^{-26} +2 q^{-30} -3 q^{-34} -2 q^{-36} + q^{-38} + q^{-40} -2 q^{-42} -3 q^{-44} +3 q^{-48} + q^{-50} -2 q^{-52} -2 q^{-54} +2 q^{-58} + q^{-60} - q^{-62} - q^{-64} + q^{-68} }[/math]

D4 Invariants.

Weight Invariant
1,0,0,0 [math]\displaystyle{ q^{18}-q^{16}+q^{14}-2 q^{12}+2 q^{10}-3 q^8+q^6-3 q^4+q^2-2+ q^{-6} +3 q^{-8} + q^{-10} +7 q^{-12} +2 q^{-14} +8 q^{-16} - q^{-18} +6 q^{-20} -3 q^{-22} +3 q^{-24} -5 q^{-26} - q^{-28} -4 q^{-30} - q^{-32} - q^{-34} -2 q^{-36} -2 q^{-40} +3 q^{-42} -2 q^{-44} + q^{-46} -2 q^{-48} +2 q^{-50} - q^{-52} + q^{-54} - q^{-56} + q^{-58} }[/math]

G2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ q^{32}-q^{30}+2 q^{28}-3 q^{26}+q^{24}-q^{22}-3 q^{20}+7 q^{18}-8 q^{16}+5 q^{14}-3 q^{12}-2 q^{10}+6 q^8-10 q^6+7 q^4-3 q^2+6 q^{-2} -6 q^{-4} +4 q^{-6} +3 q^{-8} - q^{-10} +4 q^{-12} -4 q^{-14} +2 q^{-16} +4 q^{-18} -5 q^{-20} +10 q^{-22} -7 q^{-24} +5 q^{-26} +3 q^{-28} -7 q^{-30} +10 q^{-32} -11 q^{-34} +8 q^{-36} -2 q^{-38} -3 q^{-40} +8 q^{-42} -8 q^{-44} +6 q^{-46} -3 q^{-50} +3 q^{-52} -3 q^{-54} - q^{-56} +4 q^{-58} -5 q^{-60} +5 q^{-62} -3 q^{-64} -2 q^{-66} +4 q^{-68} -7 q^{-70} +5 q^{-72} -5 q^{-74} +2 q^{-76} - q^{-78} -3 q^{-80} +4 q^{-82} -4 q^{-84} +4 q^{-86} -2 q^{-88} + q^{-90} -2 q^{-94} +2 q^{-96} - q^{-98} + q^{-100} }[/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.

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

In[3]:= K = Knot["8 7"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ t^3-3 t^2+5 t-5+5 t^{-1} -3 t^{-2} + t^{-3} }[/math]
In[5]:= Conway[K][z]
Out[5]= [math]\displaystyle{ z^6+3 z^4+2 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]= { 23, 2 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ -q^6+2 q^5-3 q^4+4 q^3-4 q^2+4 q-2+2 q^{-1} - q^{-2} }[/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} +5 z^4 a^{-2} -z^4 a^{-4} -z^4+8 z^2 a^{-2} -3 z^2 a^{-4} -3 z^2+4 a^{-2} -2 a^{-4} -1 }[/math]
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= [math]\displaystyle{ z^7 a^{-1} +z^7 a^{-3} +4 z^6 a^{-2} +2 z^6 a^{-4} +2 z^6+a z^5-z^5 a^{-1} +2 z^5 a^{-5} -12 z^4 a^{-2} -3 z^4 a^{-4} +2 z^4 a^{-6} -7 z^4-3 a z^3-3 z^3 a^{-1} -2 z^3 a^{-3} -z^3 a^{-5} +z^3 a^{-7} +12 z^2 a^{-2} +4 z^2 a^{-4} -2 z^2 a^{-6} +6 z^2+a z+2 z a^{-1} +2 z a^{-3} -z a^{-7} -4 a^{-2} -2 a^{-4} -1 }[/math]

Vassiliev invariants

V2 and V3: (2, 2)
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{ 8 }[/math] [math]\displaystyle{ 16 }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ \frac{124}{3} }[/math] [math]\displaystyle{ -\frac{28}{3} }[/math] [math]\displaystyle{ 128 }[/math] [math]\displaystyle{ \frac{448}{3} }[/math] [math]\displaystyle{ \frac{64}{3} }[/math] [math]\displaystyle{ -16 }[/math] [math]\displaystyle{ \frac{256}{3} }[/math] [math]\displaystyle{ 128 }[/math] [math]\displaystyle{ \frac{992}{3} }[/math] [math]\displaystyle{ -\frac{224}{3} }[/math] [math]\displaystyle{ \frac{7951}{15} }[/math] [math]\displaystyle{ \frac{1052}{5} }[/math] [math]\displaystyle{ -\frac{6596}{45} }[/math] [math]\displaystyle{ \frac{65}{9} }[/math] [math]\displaystyle{ -\frac{929}{15} }[/math]

V2,1 through V6,9 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]2 is the signature of 8 7. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\ r
  \  
j \
 -3-2-1012345χ
13        1-1
11       1 1
9      21 -1
7     21  1
5    22   0
3   22    0
1  13     2
-1 11      0
-3 1       1
-51        -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[8, 7]]
Out[2]=  
8
In[3]:=
PD[Knot[8, 7]]
Out[3]=  
PD[X[1, 4, 2, 5], X[3, 10, 4, 11], X[11, 1, 12, 16], X[5, 13, 6, 12], 
  X[7, 15, 8, 14], X[13, 7, 14, 6], X[15, 9, 16, 8], X[9, 2, 10, 3]]
In[4]:=
GaussCode[Knot[8, 7]]
Out[4]=  
GaussCode[-1, 8, -2, 1, -4, 6, -5, 7, -8, 2, -3, 4, -6, 5, -7, 3]
In[5]:=
BR[Knot[8, 7]]
Out[5]=  
BR[3, {1, 1, 1, 1, -2, 1, -2, -2}]
In[6]:=
alex = Alexander[Knot[8, 7]][t]
Out[6]=  
      -3   3    5            2    3

-5 + t   - -- + - + 5 t - 3 t  + t



           2   t


           t
In[7]:=
Conway[Knot[8, 7]][z]
Out[7]=  
       2      4    6
1 + 2 z  + 3 z  + z
In[8]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=  
{Knot[8, 7], Knot[11, NonAlternating, 24]}
In[9]:=
{KnotDet[Knot[8, 7]], KnotSignature[Knot[8, 7]]}
Out[9]=  
{23, 2}
In[10]:=
J=Jones[Knot[8, 7]][q]
Out[10]=  
      -2   2            2      3      4      5    6

-2 - q   + - + 4 q - 4 q  + 4 q  - 3 q  + 2 q  - q


           q
In[11]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=  
{Knot[8, 7]}
In[12]:=
A2Invariant[Knot[8, 7]][q]
Out[12]=  
     -6      2      6    10    14    18
1 - q   + 2 q  + 2 q  + q   - q   - q
In[13]:=
Kauffman[Knot[8, 7]][a, z]
Out[13]=  
                                                2      2       2    3

    2    4    z    2 z   2 z            2   2 z    4 z    12 z    z



-1 - -- - -- - -- + --- + --- + a z + 6 z  - ---- + ---- + ----- + -- - 



     4    2    7    3     a                   6      4      2      7
    a    a    a    a                         a      a      a      a

  3      3      3                      4      4       4      5    5
 z    2 z    3 z         3      4   2 z    3 z    12 z    2 z    z
 -- - ---- - ---- - 3 a z  - 7 z  + ---- - ---- - ----- + ---- - -- + 
  5     3     a                       6      4      2       5    a
 a     a                             a      a      a       a

                  6      6    7    7
    5      6   2 z    4 z    z    z
 a z  + 2 z  + ---- + ---- + -- + --
                 4      2     3   a


                 a      a     a
In[14]:=
{Vassiliev[2][Knot[8, 7]], Vassiliev[3][Knot[8, 7]]}
Out[14]=  
{0, 2}
In[15]:=
Kh[Knot[8, 7]][q, t]
Out[15]=  
         3     1       1      1      1    q      3        5

3 q + 2 q  + ----- + ----- + ---- + --- + - + 2 q  t + 2 q  t + 



             5  3    3  2      2   q t   t
            q  t    q  t    q t

    5  2      7  2    7  3      9  3    9  4    11  4    13  5


  2 q  t  + 2 q  t  + q  t  + 2 q  t  + q  t  + q   t  + q   t
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