8 9

8_8

8_10

Visit 8 9's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 8 9's page at Knotilus! Visit 8 9's page at the original Knot Atlas! 8 9 Quick Notes

8 9 Further Notes and Views

Knot presentations

Planar diagram presentation	X6271 X14,8,15,7 X10,3,11,4 X2,13,3,14 X12,5,13,6 X4,11,5,12 X16,10,1,9 X8,16,9,15
Gauss code	1, -4, 3, -6, 5, -1, 2, -8, 7, -3, 6, -5, 4, -2, 8, -7
Dowker-Thistlethwaite code	6 10 12 14 16 4 2 8
Conway Notation	[3113]

Three dimensional invariants

Symmetry type Fully amphicheiral Unknotting number 1 3-genus 3 Bridge index 2 Super bridge index [math]\displaystyle{ \{3,6\} }[/math] Nakanishi index 1 Maximal Thurston-Bennequin number [-5][-5] Hyperbolic Volume 7.58818 A-Polynomial See Data:8 9/A-polynomial

[edit Notes for 8 9's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for 8 9's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ -t^3+3 t^2-5 t+7-5 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	{ 25, 0 }
Jones polynomial	[math]\displaystyle{ q^4-2 q^3+3 q^2-4 q+5-4 q^{-1} +3 q^{-2} -2 q^{-3} + q^{-4} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ -z^6+a^2 z^4+z^4 a^{-2} -5 z^4+3 a^2 z^2+3 z^2 a^{-2} -8 z^2+2 a^2+2 a^{-2} -3 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ a z^7+z^7 a^{-1} +2 a^2 z^6+2 z^6 a^{-2} +4 z^6+2 a^3 z^5+2 z^5 a^{-3} +a^4 z^4-4 a^2 z^4-4 z^4 a^{-2} +z^4 a^{-4} -10 z^4-4 a^3 z^3-a z^3-z^3 a^{-1} -4 z^3 a^{-3} -2 a^4 z^2+4 a^2 z^2+4 z^2 a^{-2} -2 z^2 a^{-4} +12 z^2+a^3 z+a z+z a^{-1} +z a^{-3} -2 a^2-2 a^{-2} -3 }[/math]
The A2 invariant	[math]\displaystyle{ q^{12}+q^8-q^4+q^2-1+ q^{-2} - q^{-4} + q^{-8} + q^{-12} }[/math]
The G2 invariant	[math]\displaystyle{ q^{66}-q^{64}+2 q^{62}-3 q^{60}+q^{58}-3 q^{54}+6 q^{52}-6 q^{50}+7 q^{48}-5 q^{46}+6 q^{42}-10 q^{40}+12 q^{38}-8 q^{36}+4 q^{34}+3 q^{32}-6 q^{30}+11 q^{28}-6 q^{26}+2 q^{24}+4 q^{22}-7 q^{20}+5 q^{18}-8 q^{14}+11 q^{12}-10 q^{10}+9 q^8-3 q^6-10 q^4+14 q^2-17+14 q^{-2} -10 q^{-4} -3 q^{-6} +9 q^{-8} -10 q^{-10} +11 q^{-12} -8 q^{-14} +5 q^{-18} -7 q^{-20} +4 q^{-22} +2 q^{-24} -6 q^{-26} +11 q^{-28} -6 q^{-30} +3 q^{-32} +4 q^{-34} -8 q^{-36} +12 q^{-38} -10 q^{-40} +6 q^{-42} -5 q^{-46} +7 q^{-48} -6 q^{-50} +6 q^{-52} -3 q^{-54} + q^{-58} -3 q^{-60} +2 q^{-62} - q^{-64} + q^{-66} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 8_9.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ q^9-q^7+q^5-q^3+q+ q^{-1} - q^{-3} + q^{-5} - q^{-7} + q^{-9} }[/math]
2	[math]\displaystyle{ q^{26}-q^{24}-q^{22}+3 q^{20}-q^{18}-3 q^{16}+4 q^{14}-5 q^{10}+3 q^8+q^6-3 q^4+2 q^2+3+2 q^{-2} -3 q^{-4} + q^{-6} +3 q^{-8} -5 q^{-10} +4 q^{-14} -3 q^{-16} - q^{-18} +3 q^{-20} - q^{-22} - q^{-24} + q^{-26} }[/math]
3	[math]\displaystyle{ q^{51}-q^{49}-q^{47}+q^{45}+2 q^{43}-q^{41}-4 q^{39}+q^{37}+6 q^{35}-8 q^{31}-3 q^{29}+9 q^{27}+7 q^{25}-9 q^{23}-9 q^{21}+8 q^{19}+11 q^{17}-6 q^{15}-11 q^{13}+4 q^{11}+9 q^9-q^7-8 q^5+5 q+5 q^{-1} -8 q^{-5} - q^{-7} +9 q^{-9} +4 q^{-11} -11 q^{-13} -6 q^{-15} +11 q^{-17} +8 q^{-19} -9 q^{-21} -9 q^{-23} +7 q^{-25} +9 q^{-27} -3 q^{-29} -8 q^{-31} +6 q^{-35} + q^{-37} -4 q^{-39} - q^{-41} +2 q^{-43} + q^{-45} - q^{-47} - q^{-49} + q^{-51} }[/math]
4	[math]\displaystyle{ q^{84}-q^{82}-q^{80}+q^{78}+2 q^{74}-3 q^{72}-2 q^{70}+3 q^{68}+q^{66}+6 q^{64}-6 q^{62}-9 q^{60}+q^{58}+5 q^{56}+17 q^{54}-3 q^{52}-17 q^{50}-13 q^{48}-2 q^{46}+31 q^{44}+16 q^{42}-12 q^{40}-29 q^{38}-21 q^{36}+31 q^{34}+32 q^{32}+4 q^{30}-31 q^{28}-36 q^{26}+18 q^{24}+31 q^{22}+14 q^{20}-19 q^{18}-33 q^{16}+5 q^{14}+20 q^{12}+16 q^{10}-6 q^8-20 q^6-4 q^4+8 q^2+15+8 q^{-2} -4 q^{-4} -20 q^{-6} -6 q^{-8} +16 q^{-10} +20 q^{-12} +5 q^{-14} -33 q^{-16} -19 q^{-18} +14 q^{-20} +31 q^{-22} +18 q^{-24} -36 q^{-26} -31 q^{-28} +4 q^{-30} +32 q^{-32} +31 q^{-34} -21 q^{-36} -29 q^{-38} -12 q^{-40} +16 q^{-42} +31 q^{-44} -2 q^{-46} -13 q^{-48} -17 q^{-50} -3 q^{-52} +17 q^{-54} +5 q^{-56} + q^{-58} -9 q^{-60} -6 q^{-62} +6 q^{-64} + q^{-66} +3 q^{-68} -2 q^{-70} -3 q^{-72} +2 q^{-74} + q^{-78} - q^{-80} - q^{-82} + q^{-84} }[/math]
5	[math]\displaystyle{ q^{125}-q^{123}-q^{121}+q^{119}-q^{111}-q^{109}+3 q^{107}+4 q^{105}-q^{103}-4 q^{101}-6 q^{99}-4 q^{97}+4 q^{95}+13 q^{93}+11 q^{91}-3 q^{89}-17 q^{87}-22 q^{85}-8 q^{83}+17 q^{81}+36 q^{79}+28 q^{77}-8 q^{75}-42 q^{73}-51 q^{71}-18 q^{69}+40 q^{67}+73 q^{65}+50 q^{63}-23 q^{61}-87 q^{59}-83 q^{57}-6 q^{55}+85 q^{53}+111 q^{51}+39 q^{49}-72 q^{47}-124 q^{45}-67 q^{43}+49 q^{41}+122 q^{39}+87 q^{37}-25 q^{35}-108 q^{33}-91 q^{31}+5 q^{29}+85 q^{27}+84 q^{25}+11 q^{23}-63 q^{21}-73 q^{19}-15 q^{17}+42 q^{15}+54 q^{13}+23 q^{11}-22 q^9-45 q^7-22 q^5+8 q^3+31 q+31 q^{-1} +8 q^{-3} -22 q^{-5} -45 q^{-7} -22 q^{-9} +23 q^{-11} +54 q^{-13} +42 q^{-15} -15 q^{-17} -73 q^{-19} -63 q^{-21} +11 q^{-23} +84 q^{-25} +85 q^{-27} +5 q^{-29} -91 q^{-31} -108 q^{-33} -25 q^{-35} +87 q^{-37} +122 q^{-39} +49 q^{-41} -67 q^{-43} -124 q^{-45} -72 q^{-47} +39 q^{-49} +111 q^{-51} +85 q^{-53} -6 q^{-55} -83 q^{-57} -87 q^{-59} -23 q^{-61} +50 q^{-63} +73 q^{-65} +40 q^{-67} -18 q^{-69} -51 q^{-71} -42 q^{-73} -8 q^{-75} +28 q^{-77} +36 q^{-79} +17 q^{-81} -8 q^{-83} -22 q^{-85} -17 q^{-87} -3 q^{-89} +11 q^{-91} +13 q^{-93} +4 q^{-95} -4 q^{-97} -6 q^{-99} -4 q^{-101} - q^{-103} +4 q^{-105} +3 q^{-107} - q^{-109} - q^{-111} + q^{-119} - q^{-121} - q^{-123} + q^{-125} }[/math]
6	[math]\displaystyle{ q^{174}-q^{172}-q^{170}+q^{168}-2 q^{162}+2 q^{160}-q^{156}+5 q^{154}+2 q^{152}-2 q^{150}-9 q^{148}-2 q^{146}-3 q^{144}+15 q^{140}+14 q^{138}+5 q^{136}-17 q^{134}-14 q^{132}-22 q^{130}-16 q^{128}+21 q^{126}+41 q^{124}+41 q^{122}+4 q^{120}-14 q^{118}-58 q^{116}-75 q^{114}-28 q^{112}+36 q^{110}+91 q^{108}+86 q^{106}+70 q^{104}-36 q^{102}-135 q^{100}-149 q^{98}-82 q^{96}+48 q^{94}+153 q^{92}+232 q^{90}+123 q^{88}-71 q^{86}-232 q^{84}-271 q^{82}-145 q^{80}+72 q^{78}+322 q^{76}+329 q^{74}+140 q^{72}-149 q^{70}-359 q^{68}-355 q^{66}-134 q^{64}+243 q^{62}+408 q^{60}+333 q^{58}+39 q^{56}-274 q^{54}-412 q^{52}-292 q^{50}+73 q^{48}+316 q^{46}+360 q^{44}+164 q^{42}-116 q^{40}-311 q^{38}-296 q^{36}-43 q^{34}+162 q^{32}+254 q^{30}+165 q^{28}-6 q^{26}-163 q^{24}-198 q^{22}-67 q^{20}+51 q^{18}+132 q^{16}+109 q^{14}+35 q^{12}-60 q^{10}-105 q^8-64 q^6-5 q^4+61 q^2+79+61 q^{-2} -5 q^{-4} -64 q^{-6} -105 q^{-8} -60 q^{-10} +35 q^{-12} +109 q^{-14} +132 q^{-16} +51 q^{-18} -67 q^{-20} -198 q^{-22} -163 q^{-24} -6 q^{-26} +165 q^{-28} +254 q^{-30} +162 q^{-32} -43 q^{-34} -296 q^{-36} -311 q^{-38} -116 q^{-40} +164 q^{-42} +360 q^{-44} +316 q^{-46} +73 q^{-48} -292 q^{-50} -412 q^{-52} -274 q^{-54} +39 q^{-56} +333 q^{-58} +408 q^{-60} +243 q^{-62} -134 q^{-64} -355 q^{-66} -359 q^{-68} -149 q^{-70} +140 q^{-72} +329 q^{-74} +322 q^{-76} +72 q^{-78} -145 q^{-80} -271 q^{-82} -232 q^{-84} -71 q^{-86} +123 q^{-88} +232 q^{-90} +153 q^{-92} +48 q^{-94} -82 q^{-96} -149 q^{-98} -135 q^{-100} -36 q^{-102} +70 q^{-104} +86 q^{-106} +91 q^{-108} +36 q^{-110} -28 q^{-112} -75 q^{-114} -58 q^{-116} -14 q^{-118} +4 q^{-120} +41 q^{-122} +41 q^{-124} +21 q^{-126} -16 q^{-128} -22 q^{-130} -14 q^{-132} -17 q^{-134} +5 q^{-136} +14 q^{-138} +15 q^{-140} -3 q^{-144} -2 q^{-146} -9 q^{-148} -2 q^{-150} +2 q^{-152} +5 q^{-154} - q^{-156} +2 q^{-160} -2 q^{-162} + q^{-168} - q^{-170} - q^{-172} + q^{-174} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ q^{12}+q^8-q^4+q^2-1+ q^{-2} - q^{-4} + q^{-8} + q^{-12} }[/math]
1,1	[math]\displaystyle{ q^{36}-2 q^{34}+4 q^{32}-8 q^{30}+13 q^{28}-14 q^{26}+22 q^{24}-28 q^{22}+27 q^{20}-28 q^{18}+24 q^{16}-20 q^{14}+7 q^{12}+10 q^{10}-20 q^8+36 q^6-47 q^4+56 q^2-58+56 q^{-2} -47 q^{-4} +36 q^{-6} -20 q^{-8} +10 q^{-10} +7 q^{-12} -20 q^{-14} +24 q^{-16} -28 q^{-18} +27 q^{-20} -28 q^{-22} +22 q^{-24} -14 q^{-26} +13 q^{-28} -8 q^{-30} +4 q^{-32} -2 q^{-34} + q^{-36} }[/math]
2,0	[math]\displaystyle{ q^{32}+q^{26}+q^{24}-q^{22}-q^{20}+q^{18}-3 q^{14}-q^{12}+q^{10}-2 q^8-q^6+3 q^4+3 q^2+2+3 q^{-2} +3 q^{-4} - q^{-6} -2 q^{-8} + q^{-10} - q^{-12} -3 q^{-14} + q^{-18} - q^{-20} - q^{-22} + q^{-24} + q^{-26} + q^{-32} }[/math]
3,0	[math]\displaystyle{ q^{60}+q^{52}+q^{50}-2 q^{48}-2 q^{46}+4 q^{42}+q^{40}-5 q^{38}-6 q^{36}+q^{34}+7 q^{32}+2 q^{30}-7 q^{28}-5 q^{26}+5 q^{24}+11 q^{22}+3 q^{20}-6 q^{18}-q^{16}+5 q^{14}+5 q^{12}-5 q^{10}-7 q^8+q^6+4 q^4+q^2-4+ q^{-2} +4 q^{-4} + q^{-6} -7 q^{-8} -5 q^{-10} +5 q^{-12} +5 q^{-14} - q^{-16} -6 q^{-18} +3 q^{-20} +11 q^{-22} +5 q^{-24} -5 q^{-26} -7 q^{-28} +2 q^{-30} +7 q^{-32} + q^{-34} -6 q^{-36} -5 q^{-38} + q^{-40} +4 q^{-42} -2 q^{-46} -2 q^{-48} + q^{-50} + q^{-52} + q^{-60} }[/math]

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^{28}-q^{26}+q^{22}-3 q^{20}+q^{18}+4 q^{16}-2 q^{14}+2 q^{12}+5 q^{10}-2 q^8-q^6+q^4-2 q^2-2-2 q^{-2} + q^{-4} - q^{-6} -2 q^{-8} +5 q^{-10} +2 q^{-12} -2 q^{-14} +4 q^{-16} + q^{-18} -3 q^{-20} + q^{-22} - q^{-26} + q^{-28} }[/math]
1,0,0	[math]\displaystyle{ q^{15}+2 q^{11}+q^7-q^5-q- q^{-1} - q^{-5} + q^{-7} +2 q^{-11} + q^{-15} }[/math]
1,0,1	[math]\displaystyle{ q^{46}-2 q^{44}+3 q^{42}-3 q^{40}-q^{38}+8 q^{36}-9 q^{34}+13 q^{32}-3 q^{30}-9 q^{28}+18 q^{26}-27 q^{24}+19 q^{22}-2 q^{20}-20 q^{18}+36 q^{16}-34 q^{14}+16 q^{12}+5 q^{10}-22 q^8+21 q^6-9 q^4+4 q^2+9+4 q^{-2} -9 q^{-4} +21 q^{-6} -22 q^{-8} +5 q^{-10} +16 q^{-12} -34 q^{-14} +36 q^{-16} -20 q^{-18} -2 q^{-20} +19 q^{-22} -27 q^{-24} +18 q^{-26} -9 q^{-28} -3 q^{-30} +13 q^{-32} -9 q^{-34} +8 q^{-36} - q^{-38} -3 q^{-40} +3 q^{-42} -2 q^{-44} + q^{-46} }[/math]

A4 Invariants.

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

B2 Invariants.

Weight	Invariant
0,1	[math]\displaystyle{ q^{28}-q^{26}+2 q^{24}-3 q^{22}+3 q^{20}-3 q^{18}+4 q^{16}-2 q^{14}+2 q^{12}+q^{10}-2 q^8+3 q^6-5 q^4+6 q^2-8+6 q^{-2} -5 q^{-4} +3 q^{-6} -2 q^{-8} + q^{-10} +2 q^{-12} -2 q^{-14} +4 q^{-16} -3 q^{-18} +3 q^{-20} -3 q^{-22} +2 q^{-24} - q^{-26} + q^{-28} }[/math]
1,0	[math]\displaystyle{ q^{46}-q^{42}-q^{40}+q^{38}+2 q^{36}-q^{34}-3 q^{32}-q^{30}+3 q^{28}+4 q^{26}-q^{24}-3 q^{22}+4 q^{18}+2 q^{16}-2 q^{14}-2 q^{12}+q^{10}+2 q^8-q^6-3 q^4+3-3 q^{-4} - q^{-6} +2 q^{-8} + q^{-10} -2 q^{-12} -2 q^{-14} +2 q^{-16} +4 q^{-18} -3 q^{-22} - q^{-24} +4 q^{-26} +3 q^{-28} - q^{-30} -3 q^{-32} - q^{-34} +2 q^{-36} + q^{-38} - q^{-40} - q^{-42} + q^{-46} }[/math]

D4 Invariants.

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

G2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ q^{66}-q^{64}+2 q^{62}-3 q^{60}+q^{58}-3 q^{54}+6 q^{52}-6 q^{50}+7 q^{48}-5 q^{46}+6 q^{42}-10 q^{40}+12 q^{38}-8 q^{36}+4 q^{34}+3 q^{32}-6 q^{30}+11 q^{28}-6 q^{26}+2 q^{24}+4 q^{22}-7 q^{20}+5 q^{18}-8 q^{14}+11 q^{12}-10 q^{10}+9 q^8-3 q^6-10 q^4+14 q^2-17+14 q^{-2} -10 q^{-4} -3 q^{-6} +9 q^{-8} -10 q^{-10} +11 q^{-12} -8 q^{-14} +5 q^{-18} -7 q^{-20} +4 q^{-22} +2 q^{-24} -6 q^{-26} +11 q^{-28} -6 q^{-30} +3 q^{-32} +4 q^{-34} -8 q^{-36} +12 q^{-38} -10 q^{-40} +6 q^{-42} -5 q^{-46} +7 q^{-48} -6 q^{-50} +6 q^{-52} -3 q^{-54} + q^{-58} -3 q^{-60} +2 q^{-62} - 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. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["8 9"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ -t^3+3 t^2-5 t+7-5 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]=

{ 25, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ q^4-2 q^3+3 q^2-4 q+5-4 q^{-1} +3 q^{-2} -2 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} -5 z^4+3 a^2 z^2+3 z^2 a^{-2} -8 z^2+2 a^2+2 a^{-2} -3 }[/math]

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

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

Vassiliev invariants

V2 and V3:

(-2, 0)

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{ 0 }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ \frac{212}{3} }[/math] [math]\displaystyle{ \frac{124}{3} }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ -\frac{256}{3} }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ -\frac{1696}{3} }[/math] [math]\displaystyle{ -\frac{992}{3} }[/math] [math]\displaystyle{ -\frac{8071}{15} }[/math] [math]\displaystyle{ \frac{1668}{5} }[/math] [math]\displaystyle{ -\frac{37804}{45} }[/math] [math]\displaystyle{ \frac{1063}{9} }[/math] [math]\displaystyle{ -\frac{3271}{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]0 is the signature of 8 9. 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 1 7               1   -1 5             2 1   1 3           2 1     -1 1         3 2       1 -1       2 3         1 -3     1 2           -1 -5   1 2             1 -7   1               -1 -9 1                 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}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]

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, 9]]
Out[2]=	8
In[3]:=	PD[Knot[8, 9]]
Out[3]=	PD[X[6, 2, 7, 1], X[14, 8, 15, 7], X[10, 3, 11, 4], X[2, 13, 3, 14], X[12, 5, 13, 6], X[4, 11, 5, 12], X[16, 10, 1, 9], X[8, 16, 9, 15]]
In[4]:=	GaussCode[Knot[8, 9]]
Out[4]=	GaussCode[1, -4, 3, -6, 5, -1, 2, -8, 7, -3, 6, -5, 4, -2, 8, -7]
In[5]:=	BR[Knot[8, 9]]
Out[5]=	BR[3, {-1, -1, -1, 2, -1, 2, 2, 2}]
In[6]:=	alex = Alexander[Knot[8, 9]][t]
Out[6]=	-3 3 5 2 3 7 - t + -- - - - 5 t + 3 t - t 2 t t
In[7]:=	Conway[Knot[8, 9]][z]
Out[7]=	2 4 6 1 - 2 z - 3 z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[8, 9], Knot[10, 155], Knot[11, NonAlternating, 37]}
In[9]:=	{KnotDet[Knot[8, 9]], KnotSignature[Knot[8, 9]]}
Out[9]=	{25, 0}
In[10]:=	J=Jones[Knot[8, 9]][q]
Out[10]=	-4 2 3 4 2 3 4 5 + q - -- + -- - - - 4 q + 3 q - 2 q + q 3 2 q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[8, 9]}
In[12]:=	A2Invariant[Knot[8, 9]][q]
Out[12]=	-12 -8 -4 -2 2 4 8 12 -1 + q + q - q + q + q - q + q + q
In[13]:=	Kauffman[Knot[8, 9]][a, z]
Out[13]=	2 2 2 2 z z 3 2 2 z 4 z 2 2 -3 - -- - 2 a + -- + - + a z + a z + 12 z - ---- + ---- + 4 a z - 2 3 a 4 2 a a a a 3 3 4 4 4 2 4 z z 3 3 3 4 z 4 z 2 4 2 a z - ---- - -- - a z - 4 a z - 10 z + -- - ---- - 4 a z + 3 a 4 2 a a a 5 6 7 4 4 2 z 3 5 6 2 z 2 6 z 7 a z + ---- + 2 a z + 4 z + ---- + 2 a z + -- + a z 3 2 a a a
In[14]:=	{Vassiliev[2][Knot[8, 9]], Vassiliev[3][Knot[8, 9]]}
Out[14]=	{0, 0}
In[15]:=	Kh[Knot[8, 9]][q, t]
Out[15]=	3 1 1 1 2 1 2 2 - + 3 q + ----- + ----- + ----- + ----- + ----- + ---- + --- + 2 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 2 q t + q t + 2 q t + q t + q t + q t