9 26

9_25

9_27

Visit 9 26's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 26's page at Knotilus! Visit 9 26's page at the original Knot Atlas! 9 26 Quick Notes

9 26 Further Notes and Views

Knot presentations

Planar diagram presentation	X1425 X5,12,6,13 X3,11,4,10 X11,3,12,2 X13,18,14,1 X7,15,8,14 X17,7,18,6 X9,17,10,16 X15,9,16,8
Gauss code	-1, 4, -3, 1, -2, 7, -6, 9, -8, 3, -4, 2, -5, 6, -9, 8, -7, 5
Dowker-Thistlethwaite code	4 10 12 14 16 2 18 8 6
Conway Notation	[311112]

Three dimensional invariants

Symmetry type Reversible Unknotting number 1 3-genus 3 Bridge index 2 Super bridge index [math]\displaystyle{ \{4,6\} }[/math] Nakanishi index 1 Maximal Thurston-Bennequin number [-2][-9] Hyperbolic Volume 10.5958 A-Polynomial See Data:9 26/A-polynomial

[edit Notes for 9 26'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 9 26's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ t^3-5 t^2+11 t-13+11 t^{-1} -5 t^{-2} + t^{-3} }[/math]
Conway polynomial	[math]\displaystyle{ z^6+z^4+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 47, 2 }
Jones polynomial	[math]\displaystyle{ q^7-3 q^6+5 q^5-7 q^4+8 q^3-8 q^2+7 q-4+3 q^{-1} - q^{-2} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ z^6 a^{-2} +4 z^4 a^{-2} -2 z^4 a^{-4} -z^4+6 z^2 a^{-2} -5 z^2 a^{-4} +z^2 a^{-6} -2 z^2+3 a^{-2} -3 a^{-4} + a^{-6} }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ z^8 a^{-2} +z^8 a^{-4} +3 z^7 a^{-1} +6 z^7 a^{-3} +3 z^7 a^{-5} +5 z^6 a^{-2} +6 z^6 a^{-4} +4 z^6 a^{-6} +3 z^6+a z^5-6 z^5 a^{-1} -11 z^5 a^{-3} -z^5 a^{-5} +3 z^5 a^{-7} -16 z^4 a^{-2} -14 z^4 a^{-4} -5 z^4 a^{-6} +z^4 a^{-8} -8 z^4-2 a z^3+3 z^3 a^{-1} +7 z^3 a^{-3} -2 z^3 a^{-5} -4 z^3 a^{-7} +13 z^2 a^{-2} +11 z^2 a^{-4} +2 z^2 a^{-6} -z^2 a^{-8} +5 z^2-z a^{-1} -z a^{-3} +z a^{-5} +z a^{-7} -3 a^{-2} -3 a^{-4} - a^{-6} }[/math]
The A2 invariant	[math]\displaystyle{ -q^6+q^4+1+3 q^{-2} - q^{-4} +2 q^{-6} - q^{-8} -2 q^{-14} + q^{-16} - q^{-18} + q^{-22} }[/math]
The G2 invariant	[math]\displaystyle{ q^{32}-2 q^{30}+4 q^{28}-7 q^{26}+5 q^{24}-4 q^{22}-4 q^{20}+16 q^{18}-23 q^{16}+28 q^{14}-23 q^{12}+8 q^{10}+15 q^8-39 q^6+53 q^4-49 q^2+30+2 q^{-2} -31 q^{-4} +51 q^{-6} -49 q^{-8} +35 q^{-10} -5 q^{-12} -23 q^{-14} +35 q^{-16} -28 q^{-18} +6 q^{-20} +25 q^{-22} -40 q^{-24} +44 q^{-26} -23 q^{-28} -10 q^{-30} +46 q^{-32} -73 q^{-34} +76 q^{-36} -53 q^{-38} +12 q^{-40} +33 q^{-42} -67 q^{-44} +78 q^{-46} -62 q^{-48} +28 q^{-50} +5 q^{-52} -38 q^{-54} +44 q^{-56} -31 q^{-58} +4 q^{-60} +21 q^{-62} -33 q^{-64} +26 q^{-66} -4 q^{-68} -25 q^{-70} +44 q^{-72} -50 q^{-74} +40 q^{-76} -15 q^{-78} -15 q^{-80} +38 q^{-82} -46 q^{-84} +44 q^{-86} -27 q^{-88} +9 q^{-90} +8 q^{-92} -21 q^{-94} +24 q^{-96} -19 q^{-98} +13 q^{-100} -4 q^{-102} -2 q^{-104} +4 q^{-106} -6 q^{-108} +4 q^{-110} -2 q^{-112} + q^{-114} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 9_26.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ -q^5+2 q^3-q+3 q^{-1} - q^{-3} + q^{-7} -2 q^{-9} +2 q^{-11} -2 q^{-13} + q^{-15} }[/math]
2	[math]\displaystyle{ q^{16}-2 q^{14}-2 q^{12}+6 q^{10}-2 q^8-7 q^6+10 q^4+2 q^2-12+10 q^{-2} +6 q^{-4} -12 q^{-6} +4 q^{-8} +7 q^{-10} -5 q^{-12} -5 q^{-14} +3 q^{-16} +6 q^{-18} -10 q^{-20} - q^{-22} +14 q^{-24} -9 q^{-26} -5 q^{-28} +12 q^{-30} -5 q^{-32} -5 q^{-34} +6 q^{-36} - q^{-38} -2 q^{-40} + q^{-42} }[/math]
3	[math]\displaystyle{ -q^{33}+2 q^{31}+2 q^{29}-3 q^{27}-6 q^{25}+2 q^{23}+13 q^{21}-q^{19}-19 q^{17}-6 q^{15}+26 q^{13}+16 q^{11}-27 q^9-31 q^7+27 q^5+42 q^3-16 q-51 q^{-1} +9 q^{-3} +56 q^{-5} +4 q^{-7} -54 q^{-9} -15 q^{-11} +50 q^{-13} +19 q^{-15} -38 q^{-17} -28 q^{-19} +25 q^{-21} +30 q^{-23} -8 q^{-25} -32 q^{-27} -9 q^{-29} +30 q^{-31} +29 q^{-33} -27 q^{-35} -43 q^{-37} +20 q^{-39} +54 q^{-41} -10 q^{-43} -57 q^{-45} +53 q^{-49} +9 q^{-51} -45 q^{-53} -14 q^{-55} +32 q^{-57} +14 q^{-59} -21 q^{-61} -12 q^{-63} +13 q^{-65} +10 q^{-67} -8 q^{-69} -5 q^{-71} +3 q^{-73} +3 q^{-75} - q^{-77} -2 q^{-79} + q^{-81} }[/math]
4	[math]\displaystyle{ q^{56}-2 q^{54}-2 q^{52}+3 q^{50}+3 q^{48}+6 q^{46}-9 q^{44}-13 q^{42}+2 q^{40}+10 q^{38}+31 q^{36}-9 q^{34}-41 q^{32}-23 q^{30}+8 q^{28}+81 q^{26}+29 q^{24}-55 q^{22}-85 q^{20}-53 q^{18}+115 q^{16}+119 q^{14}+6 q^{12}-130 q^{10}-173 q^8+63 q^6+186 q^4+138 q^2-83-269 q^{-2} -60 q^{-4} +161 q^{-6} +241 q^{-8} +25 q^{-10} -270 q^{-12} -161 q^{-14} +76 q^{-16} +255 q^{-18} +108 q^{-20} -196 q^{-22} -186 q^{-24} -4 q^{-26} +198 q^{-28} +138 q^{-30} -91 q^{-32} -167 q^{-34} -70 q^{-36} +113 q^{-38} +145 q^{-40} +30 q^{-42} -122 q^{-44} -137 q^{-46} -5 q^{-48} +139 q^{-50} +170 q^{-52} -49 q^{-54} -195 q^{-56} -142 q^{-58} +91 q^{-60} +276 q^{-62} +59 q^{-64} -179 q^{-66} -244 q^{-68} -11 q^{-70} +280 q^{-72} +147 q^{-74} -80 q^{-76} -238 q^{-78} -103 q^{-80} +178 q^{-82} +146 q^{-84} +21 q^{-86} -142 q^{-88} -113 q^{-90} +68 q^{-92} +75 q^{-94} +50 q^{-96} -50 q^{-98} -66 q^{-100} +18 q^{-102} +20 q^{-104} +30 q^{-106} -14 q^{-108} -26 q^{-110} +8 q^{-112} +2 q^{-114} +11 q^{-116} -3 q^{-118} -8 q^{-120} +3 q^{-122} +3 q^{-126} - q^{-128} -2 q^{-130} + q^{-132} }[/math]
5	[math]\displaystyle{ -q^{85}+2 q^{83}+2 q^{81}-3 q^{79}-3 q^{77}-3 q^{75}+q^{73}+9 q^{71}+13 q^{69}-2 q^{67}-20 q^{65}-22 q^{63}-7 q^{61}+25 q^{59}+49 q^{57}+33 q^{55}-33 q^{53}-87 q^{51}-70 q^{49}+12 q^{47}+118 q^{45}+151 q^{43}+44 q^{41}-140 q^{39}-239 q^{37}-149 q^{35}+90 q^{33}+325 q^{31}+318 q^{29}+23 q^{27}-350 q^{25}-498 q^{23}-242 q^{21}+283 q^{19}+651 q^{17}+513 q^{15}-83 q^{13}-709 q^{11}-812 q^9-203 q^7+644 q^5+1030 q^3+571 q-444 q^{-1} -1161 q^{-3} -901 q^{-5} +162 q^{-7} +1138 q^{-9} +1163 q^{-11} +163 q^{-13} -1012 q^{-15} -1308 q^{-17} -440 q^{-19} +807 q^{-21} +1315 q^{-23} +646 q^{-25} -568 q^{-27} -1239 q^{-29} -763 q^{-31} +370 q^{-33} +1078 q^{-35} +786 q^{-37} -173 q^{-39} -913 q^{-41} -771 q^{-43} +43 q^{-45} +737 q^{-47} +723 q^{-49} +97 q^{-51} -572 q^{-53} -695 q^{-55} -228 q^{-57} +398 q^{-59} +683 q^{-61} +398 q^{-63} -215 q^{-65} -672 q^{-67} -599 q^{-69} -17 q^{-71} +649 q^{-73} +829 q^{-75} +281 q^{-77} -581 q^{-79} -1025 q^{-81} -601 q^{-83} +431 q^{-85} +1174 q^{-87} +912 q^{-89} -205 q^{-91} -1204 q^{-93} -1173 q^{-95} -90 q^{-97} +1103 q^{-99} +1337 q^{-101} +391 q^{-103} -880 q^{-105} -1346 q^{-107} -639 q^{-109} +569 q^{-111} +1213 q^{-113} +788 q^{-115} -255 q^{-117} -967 q^{-119} -799 q^{-121} -8 q^{-123} +667 q^{-125} +701 q^{-127} +174 q^{-129} -392 q^{-131} -536 q^{-133} -233 q^{-135} +181 q^{-137} +354 q^{-139} +215 q^{-141} -45 q^{-143} -209 q^{-145} -164 q^{-147} -4 q^{-149} +104 q^{-151} +98 q^{-153} +24 q^{-155} -43 q^{-157} -59 q^{-159} -16 q^{-161} +22 q^{-163} +23 q^{-165} +8 q^{-167} -7 q^{-169} -10 q^{-171} -6 q^{-173} +5 q^{-175} +7 q^{-177} -2 q^{-179} -3 q^{-181} +3 q^{-189} - q^{-191} -2 q^{-193} + q^{-195} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ -q^6+q^4+1+3 q^{-2} - q^{-4} +2 q^{-6} - q^{-8} -2 q^{-14} + q^{-16} - q^{-18} + q^{-22} }[/math]
1,1	[math]\displaystyle{ q^{20}-4 q^{18}+10 q^{16}-22 q^{14}+42 q^{12}-70 q^{10}+100 q^8-140 q^6+177 q^4-196 q^2+208-188 q^{-2} +157 q^{-4} -86 q^{-6} +2 q^{-8} +94 q^{-10} -193 q^{-12} +280 q^{-14} -354 q^{-16} +392 q^{-18} -402 q^{-20} +372 q^{-22} -312 q^{-24} +228 q^{-26} -133 q^{-28} +34 q^{-30} +58 q^{-32} -124 q^{-34} +170 q^{-36} -190 q^{-38} +194 q^{-40} -176 q^{-42} +146 q^{-44} -118 q^{-46} +86 q^{-48} -58 q^{-50} +36 q^{-52} -20 q^{-54} +10 q^{-56} -4 q^{-58} + q^{-60} }[/math]
2,0	[math]\displaystyle{ q^{18}-q^{16}-2 q^{14}+q^{12}+2 q^{10}-2 q^8-4 q^6+4 q^4+6 q^2-2-2 q^{-2} +8 q^{-4} +2 q^{-6} -4 q^{-8} + q^{-10} +5 q^{-12} -2 q^{-14} -4 q^{-16} +2 q^{-18} -2 q^{-20} -7 q^{-22} + q^{-24} +4 q^{-26} -5 q^{-28} - q^{-30} +7 q^{-32} +3 q^{-34} -4 q^{-36} +5 q^{-40} -4 q^{-44} + q^{-48} - q^{-50} - q^{-52} + q^{-56} }[/math]

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^{14}-2 q^{12}+2 q^8-6 q^6+4 q^4+6 q^2-7+6 q^{-2} +9 q^{-4} -10 q^{-6} +3 q^{-8} +8 q^{-10} -6 q^{-12} - q^{-14} +3 q^{-16} -5 q^{-20} -4 q^{-22} +7 q^{-24} -4 q^{-26} -7 q^{-28} +12 q^{-30} - q^{-32} -8 q^{-34} +9 q^{-36} -6 q^{-40} +4 q^{-42} -2 q^{-46} + q^{-48} }[/math]
1,0,0	[math]\displaystyle{ -q^7+q^5-q^3+2 q+3 q^{-3} +2 q^{-7} + q^{-9} -2 q^{-15} -3 q^{-19} + q^{-21} - q^{-23} + q^{-25} + q^{-29} }[/math]

A4 Invariants.

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

B2 Invariants.

Weight	Invariant
0,1	[math]\displaystyle{ -q^{14}+2 q^{12}-4 q^{10}+6 q^8-8 q^6+10 q^4-10 q^2+11-8 q^{-2} +7 q^{-4} -3 q^{-8} +10 q^{-10} -14 q^{-12} +19 q^{-14} -21 q^{-16} +20 q^{-18} -19 q^{-20} +14 q^{-22} -11 q^{-24} +4 q^{-26} + q^{-28} -6 q^{-30} +9 q^{-32} -10 q^{-34} +11 q^{-36} -10 q^{-38} +8 q^{-40} -6 q^{-42} +4 q^{-44} -2 q^{-46} + q^{-48} }[/math]
1,0	[math]\displaystyle{ q^{24}-2 q^{20}-2 q^{18}+2 q^{16}+4 q^{14}-2 q^{12}-7 q^{10}-2 q^8+9 q^6+8 q^4-4 q^2-10+ q^{-2} +12 q^{-4} +8 q^{-6} -8 q^{-8} -9 q^{-10} +3 q^{-12} +10 q^{-14} -8 q^{-18} -2 q^{-20} +7 q^{-22} +3 q^{-24} -7 q^{-26} -5 q^{-28} +5 q^{-30} +6 q^{-32} -5 q^{-34} -8 q^{-36} +2 q^{-38} +9 q^{-40} -10 q^{-44} -4 q^{-46} +10 q^{-48} +9 q^{-50} -5 q^{-52} -11 q^{-54} - q^{-56} +10 q^{-58} +6 q^{-60} -5 q^{-62} -7 q^{-64} +5 q^{-68} +2 q^{-70} -2 q^{-72} -2 q^{-74} + q^{-78} }[/math]

D4 Invariants.

Weight

Invariant

1,0,0,0

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

G2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ q^{32}-2 q^{30}+4 q^{28}-7 q^{26}+5 q^{24}-4 q^{22}-4 q^{20}+16 q^{18}-23 q^{16}+28 q^{14}-23 q^{12}+8 q^{10}+15 q^8-39 q^6+53 q^4-49 q^2+30+2 q^{-2} -31 q^{-4} +51 q^{-6} -49 q^{-8} +35 q^{-10} -5 q^{-12} -23 q^{-14} +35 q^{-16} -28 q^{-18} +6 q^{-20} +25 q^{-22} -40 q^{-24} +44 q^{-26} -23 q^{-28} -10 q^{-30} +46 q^{-32} -73 q^{-34} +76 q^{-36} -53 q^{-38} +12 q^{-40} +33 q^{-42} -67 q^{-44} +78 q^{-46} -62 q^{-48} +28 q^{-50} +5 q^{-52} -38 q^{-54} +44 q^{-56} -31 q^{-58} +4 q^{-60} +21 q^{-62} -33 q^{-64} +26 q^{-66} -4 q^{-68} -25 q^{-70} +44 q^{-72} -50 q^{-74} +40 q^{-76} -15 q^{-78} -15 q^{-80} +38 q^{-82} -46 q^{-84} +44 q^{-86} -27 q^{-88} +9 q^{-90} +8 q^{-92} -21 q^{-94} +24 q^{-96} -19 q^{-98} +13 q^{-100} -4 q^{-102} -2 q^{-104} +4 q^{-106} -6 q^{-108} +4 q^{-110} -2 q^{-112} + q^{-114} }[/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["9 26"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ t^3-5 t^2+11 t-13+11 t^{-1} -5 t^{-2} + t^{-3} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ z^6+z^4+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]=

{ 47, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ q^7-3 q^6+5 q^5-7 q^4+8 q^3-8 q^2+7 q-4+3 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} +4 z^4 a^{-2} -2 z^4 a^{-4} -z^4+6 z^2 a^{-2} -5 z^2 a^{-4} +z^2 a^{-6} -2 z^2+3 a^{-2} -3 a^{-4} + a^{-6} }[/math]

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ z^8 a^{-2} +z^8 a^{-4} +3 z^7 a^{-1} +6 z^7 a^{-3} +3 z^7 a^{-5} +5 z^6 a^{-2} +6 z^6 a^{-4} +4 z^6 a^{-6} +3 z^6+a z^5-6 z^5 a^{-1} -11 z^5 a^{-3} -z^5 a^{-5} +3 z^5 a^{-7} -16 z^4 a^{-2} -14 z^4 a^{-4} -5 z^4 a^{-6} +z^4 a^{-8} -8 z^4-2 a z^3+3 z^3 a^{-1} +7 z^3 a^{-3} -2 z^3 a^{-5} -4 z^3 a^{-7} +13 z^2 a^{-2} +11 z^2 a^{-4} +2 z^2 a^{-6} -z^2 a^{-8} +5 z^2-z a^{-1} -z a^{-3} +z a^{-5} +z a^{-7} -3 a^{-2} -3 a^{-4} - a^{-6} }[/math]

Vassiliev invariants

V2 and V3:

(0, -1)

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{ 0 }[/math] [math]\displaystyle{ -8 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ -32 }[/math] [math]\displaystyle{ -8 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ -\frac{176}{3} }[/math] [math]\displaystyle{ \frac{64}{3} }[/math] [math]\displaystyle{ -40 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ 0 }[/math] [math]\displaystyle{ \frac{248}{3} }[/math] [math]\displaystyle{ -\frac{88}{3} }[/math] [math]\displaystyle{ -\frac{112}{3} }[/math] [math]\displaystyle{ 0 }[/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]2 is the signature of 9 26. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -3 -2 -1 0 1 2 3 4 5 6 χ 15                   1 1 13                 2   -2 11               3 1   2 9             4 2     -2 7           4 3       1 5         4 4         0 3       3 4           -1 1     2 5             3 -1   1 2               -1 -3   2                 2 -5 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=3 }[/math] [math]\displaystyle{ r=-3 }[/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}^{5}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=6 }[/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[9, 26]]
Out[2]=	9
In[3]:=	PD[Knot[9, 26]]
Out[3]=	PD[X[1, 4, 2, 5], X[5, 12, 6, 13], X[3, 11, 4, 10], X[11, 3, 12, 2], X[13, 18, 14, 1], X[7, 15, 8, 14], X[17, 7, 18, 6], X[9, 17, 10, 16], X[15, 9, 16, 8]]
In[4]:=	GaussCode[Knot[9, 26]]
Out[4]=	GaussCode[-1, 4, -3, 1, -2, 7, -6, 9, -8, 3, -4, 2, -5, 6, -9, 8, -7, 5]
In[5]:=	BR[Knot[9, 26]]
Out[5]=	BR[4, {1, 1, 1, -2, 1, -2, 3, -2, 3}]
In[6]:=	alex = Alexander[Knot[9, 26]][t]
Out[6]=	-3 5 11 2 3 -13 + t - -- + -- + 11 t - 5 t + t 2 t t
In[7]:=	Conway[Knot[9, 26]][z]
Out[7]=	4 6 1 + z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[9, 26], Knot[11, NonAlternating, 25]}
In[9]:=	{KnotDet[Knot[9, 26]], KnotSignature[Knot[9, 26]]}
Out[9]=	{47, 2}
In[10]:=	J=Jones[Knot[9, 26]][q]
Out[10]=	-2 3 2 3 4 5 6 7 -4 - q + - + 7 q - 8 q + 8 q - 7 q + 5 q - 3 q + q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[9, 26]}
In[12]:=	A2Invariant[Knot[9, 26]][q]
Out[12]=	-6 -4 2 4 6 8 14 16 18 22 1 - q + q + 3 q - q + 2 q - q - 2 q + q - q + q
In[13]:=	Kauffman[Knot[9, 26]][a, z]
Out[13]=	2 2 2 2 -6 3 3 z z z z 2 z 2 z 11 z 13 z -a - -- - -- + -- + -- - -- - - + 5 z - -- + ---- + ----- + ----- - 4 2 7 5 3 a 8 6 4 2 a a a a a a a a a 3 3 3 3 4 4 4 4 z 2 z 7 z 3 z 3 4 z 5 z 14 z ---- - ---- + ---- + ---- - 2 a z - 8 z + -- - ---- - ----- - 7 5 3 a 8 6 4 a a a a a a 4 5 5 5 5 6 6 6 16 z 3 z z 11 z 6 z 5 6 4 z 6 z 5 z ----- + ---- - -- - ----- - ---- + a z + 3 z + ---- + ---- + ---- + 2 7 5 3 a 6 4 2 a a a a a a a 7 7 7 8 8 3 z 6 z 3 z z z ---- + ---- + ---- + -- + -- 5 3 a 4 2 a a a a
In[14]:=	{Vassiliev[2][Knot[9, 26]], Vassiliev[3][Knot[9, 26]]}
Out[14]=	{0, -1}
In[15]:=	Kh[Knot[9, 26]][q, t]
Out[15]=	3 1 2 1 2 2 q 3 5 5 q + 3 q + ----- + ----- + ---- + --- + --- + 4 q t + 4 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 11 5 4 q t + 4 q t + 3 q t + 4 q t + 2 q t + 3 q t + q t + 13 5 15 6 2 q t + q t