9 36

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9 35.gif

9_35

9 37.gif

9_37

9 36.gif Visit 9 36's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 9 36's page at Knotilus!

Visit 9 36's page at the original Knot Atlas!

9 36 Quick Notes


9 36 Further Notes and Views

Knot presentations

Planar diagram presentation X1425 X7,10,8,11 X3948 X9,3,10,2 X11,17,12,16 X5,15,6,14 X15,7,16,6 X13,1,14,18 X17,13,18,12
Gauss code -1, 4, -3, 1, -6, 7, -2, 3, -4, 2, -5, 9, -8, 6, -7, 5, -9, 8
Dowker-Thistlethwaite code 4 8 14 10 2 16 18 6 12
Conway Notation [22,3,2]

Three dimensional invariants

Symmetry type Reversible
Unknotting number 2
3-genus 3
Bridge index 3
Super bridge index [math]\displaystyle{ \{4,7\} }[/math]
Nakanishi index 1
Maximal Thurston-Bennequin number [1][-12]
Hyperbolic Volume 9.88458
A-Polynomial See Data:9 36/A-polynomial

[edit Notes for 9 36'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{ 3 }[/math]
Rasmussen s-Invariant 4

[edit Notes for 9 36's four dimensional invariants]

Polynomial invariants

Alexander polynomial [math]\displaystyle{ -t^3+5 t^2-8 t+9-8 t^{-1} +5 t^{-2} - t^{-3} }[/math]
Conway polynomial [math]\displaystyle{ -z^6-z^4+3 z^2+1 }[/math]
2nd Alexander ideal (db, data sources) [math]\displaystyle{ \{1\} }[/math]
Determinant and Signature { 37, 4 }
Jones polynomial [math]\displaystyle{ -q^9+2 q^8-4 q^7+6 q^6-6 q^5+6 q^4-5 q^3+4 q^2-2 q+1 }[/math]
HOMFLY-PT polynomial (db, data sources) [math]\displaystyle{ -z^6 a^{-4} +z^4 a^{-2} -4 z^4 a^{-4} +2 z^4 a^{-6} +3 z^2 a^{-2} -5 z^2 a^{-4} +6 z^2 a^{-6} -z^2 a^{-8} +2 a^{-2} -3 a^{-4} +4 a^{-6} -2 a^{-8} }[/math]
Kauffman polynomial (db, data sources) [math]\displaystyle{ z^8 a^{-4} +z^8 a^{-6} +2 z^7 a^{-3} +5 z^7 a^{-5} +3 z^7 a^{-7} +z^6 a^{-2} +z^6 a^{-4} +4 z^6 a^{-6} +4 z^6 a^{-8} -7 z^5 a^{-3} -14 z^5 a^{-5} -4 z^5 a^{-7} +3 z^5 a^{-9} -4 z^4 a^{-2} -12 z^4 a^{-4} -17 z^4 a^{-6} -7 z^4 a^{-8} +2 z^4 a^{-10} +6 z^3 a^{-3} +9 z^3 a^{-5} -2 z^3 a^{-9} +z^3 a^{-11} +5 z^2 a^{-2} +12 z^2 a^{-4} +15 z^2 a^{-6} +7 z^2 a^{-8} -z^2 a^{-10} -z a^{-3} -2 z a^{-5} +z a^{-7} +z a^{-9} -z a^{-11} -2 a^{-2} -3 a^{-4} -4 a^{-6} -2 a^{-8} }[/math]
The A2 invariant [math]\displaystyle{ 1+ q^{-4} + q^{-6} - q^{-8} + q^{-10} -2 q^{-12} + q^{-14} + q^{-16} + q^{-18} +2 q^{-20} - q^{-22} - q^{-26} - q^{-28} }[/math]
The G2 invariant [math]\displaystyle{ q^{-2} - q^{-4} +4 q^{-6} -5 q^{-8} +5 q^{-10} -2 q^{-12} -4 q^{-14} +14 q^{-16} -17 q^{-18} +19 q^{-20} -11 q^{-22} -2 q^{-24} +18 q^{-26} -27 q^{-28} +28 q^{-30} -17 q^{-32} + q^{-34} +13 q^{-36} -23 q^{-38} +20 q^{-40} -9 q^{-42} -5 q^{-44} +15 q^{-46} -18 q^{-48} +9 q^{-50} +3 q^{-52} -18 q^{-54} +24 q^{-56} -24 q^{-58} +16 q^{-60} + q^{-62} -18 q^{-64} +32 q^{-66} -33 q^{-68} +28 q^{-70} -9 q^{-72} -10 q^{-74} +25 q^{-76} -28 q^{-78} +24 q^{-80} -7 q^{-82} -7 q^{-84} +19 q^{-86} -16 q^{-88} +6 q^{-90} +6 q^{-92} -16 q^{-94} +18 q^{-96} -11 q^{-98} -2 q^{-100} +11 q^{-102} -18 q^{-104} +20 q^{-106} -15 q^{-108} +4 q^{-110} +3 q^{-112} -12 q^{-114} +12 q^{-116} -13 q^{-118} +10 q^{-120} -5 q^{-122} + q^{-124} +3 q^{-126} -7 q^{-128} +6 q^{-130} -5 q^{-132} +4 q^{-134} -2 q^{-136} + q^{-140} -2 q^{-142} +2 q^{-144} - q^{-146} + q^{-148} }[/math]
Further Quantum Invariants
Further quantum knot invariants for 9_36.

A1 Invariants.

Weight Invariant
1 [math]\displaystyle{ q- q^{-1} +2 q^{-3} - q^{-5} + q^{-7} +2 q^{-13} -2 q^{-15} + q^{-17} - q^{-19} }[/math]
2 [math]\displaystyle{ q^6-q^4-2 q^2+4+ q^{-2} -6 q^{-4} +4 q^{-6} +5 q^{-8} -6 q^{-10} + q^{-12} +7 q^{-14} -4 q^{-16} -3 q^{-18} +5 q^{-20} -4 q^{-24} + q^{-26} +4 q^{-28} -3 q^{-30} -5 q^{-32} +7 q^{-34} -7 q^{-38} +5 q^{-40} + q^{-42} -4 q^{-44} +2 q^{-46} - q^{-50} + q^{-52} }[/math]
3 [math]\displaystyle{ q^{15}-q^{13}-2 q^{11}+5 q^7+3 q^5-7 q^3-8 q+6 q^{-1} +14 q^{-3} -18 q^{-7} -6 q^{-9} +17 q^{-11} +16 q^{-13} -11 q^{-15} -21 q^{-17} +5 q^{-19} +23 q^{-21} +4 q^{-23} -21 q^{-25} -10 q^{-27} +20 q^{-29} +15 q^{-31} -16 q^{-33} -17 q^{-35} +12 q^{-37} +19 q^{-39} -9 q^{-41} -21 q^{-43} +3 q^{-45} +18 q^{-47} +3 q^{-49} -16 q^{-51} -12 q^{-53} +10 q^{-55} +20 q^{-57} -2 q^{-59} -24 q^{-61} -5 q^{-63} +22 q^{-65} +14 q^{-67} -20 q^{-69} -14 q^{-71} +12 q^{-73} +12 q^{-75} -6 q^{-77} -8 q^{-79} +3 q^{-81} +4 q^{-83} -2 q^{-85} - q^{-87} + q^{-89} + q^{-97} - q^{-99} }[/math]
4 [math]\displaystyle{ q^{28}-q^{26}-2 q^{24}+q^{20}+7 q^{18}+q^{16}-7 q^{14}-9 q^{12}-8 q^{10}+17 q^8+19 q^6+5 q^4-17 q^2-39-2 q^{-2} +28 q^{-4} +44 q^{-6} +18 q^{-8} -51 q^{-10} -49 q^{-12} -17 q^{-14} +51 q^{-16} +78 q^{-18} +3 q^{-20} -54 q^{-22} -83 q^{-24} -10 q^{-26} +86 q^{-28} +73 q^{-30} +8 q^{-32} -94 q^{-34} -82 q^{-36} +29 q^{-38} +93 q^{-40} +71 q^{-42} -55 q^{-44} -104 q^{-46} -27 q^{-48} +70 q^{-50} +92 q^{-52} -15 q^{-54} -91 q^{-56} -48 q^{-58} +52 q^{-60} +85 q^{-62} -77 q^{-66} -54 q^{-68} +37 q^{-70} +75 q^{-72} +23 q^{-74} -56 q^{-76} -71 q^{-78} - q^{-80} +55 q^{-82} +69 q^{-84} +4 q^{-86} -75 q^{-88} -69 q^{-90} -6 q^{-92} +99 q^{-94} +89 q^{-96} -29 q^{-98} -105 q^{-100} -89 q^{-102} +60 q^{-104} +126 q^{-106} +38 q^{-108} -69 q^{-110} -113 q^{-112} -2 q^{-114} +81 q^{-116} +55 q^{-118} -8 q^{-120} -67 q^{-122} -22 q^{-124} +25 q^{-126} +26 q^{-128} +11 q^{-130} -22 q^{-132} -8 q^{-134} +4 q^{-136} +3 q^{-138} +7 q^{-140} -6 q^{-142} - q^{-144} + q^{-146} - q^{-148} +3 q^{-150} -2 q^{-152} - q^{-158} + q^{-160} }[/math]
5 [math]\displaystyle{ q^{45}-q^{43}-2 q^{41}+q^{37}+3 q^{35}+5 q^{33}+q^{31}-9 q^{29}-11 q^{27}-6 q^{25}+5 q^{23}+21 q^{21}+25 q^{19}+6 q^{17}-27 q^{15}-44 q^{13}-34 q^{11}+8 q^9+59 q^7+76 q^5+35 q^3-45 q-106 q^{-1} -100 q^{-3} -15 q^{-5} +100 q^{-7} +164 q^{-9} +110 q^{-11} -40 q^{-13} -176 q^{-15} -207 q^{-17} -84 q^{-19} +125 q^{-21} +269 q^{-23} +221 q^{-25} +2 q^{-27} -242 q^{-29} -333 q^{-31} -173 q^{-33} +140 q^{-35} +375 q^{-37} +336 q^{-39} +30 q^{-41} -322 q^{-43} -450 q^{-45} -221 q^{-47} +200 q^{-49} +482 q^{-51} +381 q^{-53} -35 q^{-55} -433 q^{-57} -485 q^{-59} -128 q^{-61} +338 q^{-63} +519 q^{-65} +253 q^{-67} -222 q^{-69} -494 q^{-71} -330 q^{-73} +118 q^{-75} +436 q^{-77} +348 q^{-79} -47 q^{-81} -372 q^{-83} -329 q^{-85} +8 q^{-87} +313 q^{-89} +296 q^{-91} -3 q^{-93} -278 q^{-95} -265 q^{-97} +8 q^{-99} +260 q^{-101} +253 q^{-103} +5 q^{-105} -247 q^{-107} -268 q^{-109} -46 q^{-111} +216 q^{-113} +294 q^{-115} +133 q^{-117} -138 q^{-119} -318 q^{-121} -258 q^{-123} +15 q^{-125} +301 q^{-127} +383 q^{-129} +166 q^{-131} -219 q^{-133} -481 q^{-135} -364 q^{-137} +81 q^{-139} +496 q^{-141} +531 q^{-143} +115 q^{-145} -430 q^{-147} -633 q^{-149} -289 q^{-151} +295 q^{-153} +616 q^{-155} +423 q^{-157} -124 q^{-159} -530 q^{-161} -459 q^{-163} -22 q^{-165} +380 q^{-167} +414 q^{-169} +112 q^{-171} -228 q^{-173} -320 q^{-175} -139 q^{-177} +112 q^{-179} +213 q^{-181} +116 q^{-183} -39 q^{-185} -118 q^{-187} -82 q^{-189} +6 q^{-191} +60 q^{-193} +48 q^{-195} + q^{-197} -25 q^{-199} -22 q^{-201} -4 q^{-203} +9 q^{-205} +11 q^{-207} +2 q^{-209} -7 q^{-211} -3 q^{-213} +2 q^{-215} +2 q^{-219} + q^{-221} -3 q^{-223} - q^{-225} +2 q^{-227} + q^{-233} - q^{-235} }[/math]

A2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ 1+ q^{-4} + q^{-6} - q^{-8} + q^{-10} -2 q^{-12} + q^{-14} + q^{-16} + q^{-18} +2 q^{-20} - q^{-22} - q^{-26} - q^{-28} }[/math]
1,1 [math]\displaystyle{ q^4-2 q^2+8-16 q^{-2} +29 q^{-4} -44 q^{-6} +60 q^{-8} -74 q^{-10} +80 q^{-12} -72 q^{-14} +62 q^{-16} -32 q^{-18} +3 q^{-20} +38 q^{-22} -70 q^{-24} +100 q^{-26} -123 q^{-28} +132 q^{-30} -134 q^{-32} +118 q^{-34} -96 q^{-36} +64 q^{-38} -30 q^{-40} +28 q^{-44} -44 q^{-46} +52 q^{-48} -52 q^{-50} +46 q^{-52} -44 q^{-54} +34 q^{-56} -28 q^{-58} +24 q^{-60} -20 q^{-62} +16 q^{-64} -12 q^{-66} +9 q^{-68} -6 q^{-70} +4 q^{-72} -2 q^{-74} + q^{-76} }[/math]
2,0 [math]\displaystyle{ q^4-1+2 q^{-4} + q^{-6} -2 q^{-8} - q^{-10} +4 q^{-12} + q^{-14} -2 q^{-16} +2 q^{-18} +3 q^{-20} - q^{-24} +2 q^{-26} +2 q^{-28} - q^{-30} +2 q^{-32} + q^{-34} -4 q^{-36} -3 q^{-38} +2 q^{-40} -2 q^{-42} -3 q^{-44} +2 q^{-46} +4 q^{-48} + q^{-50} -2 q^{-52} + q^{-54} -3 q^{-58} -2 q^{-60} - q^{-62} + q^{-66} + q^{-68} + q^{-70} }[/math]

A3 Invariants.

Weight Invariant
0,1,0 [math]\displaystyle{ 1- q^{-2} +2 q^{-4} +2 q^{-6} -2 q^{-8} +4 q^{-10} + q^{-12} -4 q^{-14} +4 q^{-16} - q^{-18} -5 q^{-20} +4 q^{-22} +2 q^{-24} -3 q^{-26} +3 q^{-28} +3 q^{-30} + q^{-32} - q^{-34} + q^{-36} +3 q^{-38} -5 q^{-40} -2 q^{-42} +4 q^{-44} -6 q^{-46} -2 q^{-48} +5 q^{-50} -3 q^{-52} -2 q^{-54} +3 q^{-56} - q^{-60} + q^{-62} }[/math]
1,0,0 [math]\displaystyle{ q^{-1} +2 q^{-5} +2 q^{-9} - q^{-11} -2 q^{-15} - q^{-17} + q^{-21} +3 q^{-23} + q^{-25} +3 q^{-27} - q^{-29} -2 q^{-33} - q^{-35} - q^{-37} }[/math]

A4 Invariants.

Weight Invariant
0,1,0,0 [math]\displaystyle{ q^{-2} + q^{-6} +3 q^{-8} + q^{-10} + q^{-12} +3 q^{-14} -3 q^{-18} + q^{-22} -3 q^{-24} - q^{-26} +6 q^{-28} +5 q^{-30} -4 q^{-32} +2 q^{-34} +5 q^{-36} -5 q^{-38} -5 q^{-40} +4 q^{-42} + q^{-44} - q^{-46} +5 q^{-48} +6 q^{-50} - q^{-52} -2 q^{-54} +2 q^{-56} -3 q^{-58} -8 q^{-60} -2 q^{-62} + q^{-64} -4 q^{-66} -2 q^{-68} +2 q^{-70} +2 q^{-72} + q^{-78} + q^{-80} }[/math]
1,0,0,0 [math]\displaystyle{ q^{-2} +2 q^{-6} + q^{-8} + q^{-10} +2 q^{-12} - q^{-14} -3 q^{-18} - q^{-20} -2 q^{-22} + q^{-26} +3 q^{-28} +3 q^{-30} +2 q^{-32} +3 q^{-34} - q^{-36} -2 q^{-40} -2 q^{-42} - q^{-44} - q^{-46} }[/math]

B2 Invariants.

Weight Invariant
0,1 [math]\displaystyle{ 1- q^{-2} +4 q^{-4} -4 q^{-6} +6 q^{-8} -6 q^{-10} +7 q^{-12} -6 q^{-14} +4 q^{-16} -3 q^{-18} - q^{-20} +4 q^{-22} -8 q^{-24} +11 q^{-26} -11 q^{-28} +13 q^{-30} -11 q^{-32} +11 q^{-34} -7 q^{-36} +5 q^{-38} - q^{-40} -2 q^{-42} +4 q^{-44} -6 q^{-46} +6 q^{-48} -7 q^{-50} +5 q^{-52} -4 q^{-54} +3 q^{-56} -2 q^{-58} + q^{-60} - q^{-62} }[/math]
1,0 [math]\displaystyle{ q^2- q^{-2} - q^{-4} +3 q^{-6} +3 q^{-8} -2 q^{-10} -4 q^{-12} + q^{-14} +6 q^{-16} +3 q^{-18} -6 q^{-20} -5 q^{-22} +4 q^{-24} +7 q^{-26} -7 q^{-30} -3 q^{-32} +5 q^{-34} +5 q^{-36} -2 q^{-38} -4 q^{-40} +2 q^{-42} +4 q^{-44} -4 q^{-48} +5 q^{-52} +2 q^{-54} -5 q^{-56} -3 q^{-58} +4 q^{-60} +5 q^{-62} -3 q^{-64} -6 q^{-66} +6 q^{-70} + q^{-72} -6 q^{-74} -5 q^{-76} +2 q^{-78} +6 q^{-80} -4 q^{-84} -3 q^{-86} + q^{-88} +3 q^{-90} + q^{-92} - q^{-94} - q^{-96} + q^{-100} }[/math]

D4 Invariants.

Weight Invariant
1,0,0,0 [math]\displaystyle{ q^{-2} - q^{-4} +3 q^{-6} -2 q^{-8} +6 q^{-10} -3 q^{-12} +6 q^{-14} -4 q^{-16} +7 q^{-18} -5 q^{-20} +2 q^{-22} -4 q^{-24} - q^{-28} -4 q^{-30} +4 q^{-32} -6 q^{-34} +9 q^{-36} -7 q^{-38} +12 q^{-40} -6 q^{-42} +12 q^{-44} -7 q^{-46} +9 q^{-48} -5 q^{-50} +4 q^{-52} -4 q^{-54} -2 q^{-56} - q^{-58} -4 q^{-60} +3 q^{-62} -6 q^{-64} +3 q^{-66} -5 q^{-68} +6 q^{-70} -4 q^{-72} +2 q^{-74} -3 q^{-76} +3 q^{-78} - q^{-80} + q^{-82} - q^{-84} + q^{-86} }[/math]

G2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ q^{-2} - q^{-4} +4 q^{-6} -5 q^{-8} +5 q^{-10} -2 q^{-12} -4 q^{-14} +14 q^{-16} -17 q^{-18} +19 q^{-20} -11 q^{-22} -2 q^{-24} +18 q^{-26} -27 q^{-28} +28 q^{-30} -17 q^{-32} + q^{-34} +13 q^{-36} -23 q^{-38} +20 q^{-40} -9 q^{-42} -5 q^{-44} +15 q^{-46} -18 q^{-48} +9 q^{-50} +3 q^{-52} -18 q^{-54} +24 q^{-56} -24 q^{-58} +16 q^{-60} + q^{-62} -18 q^{-64} +32 q^{-66} -33 q^{-68} +28 q^{-70} -9 q^{-72} -10 q^{-74} +25 q^{-76} -28 q^{-78} +24 q^{-80} -7 q^{-82} -7 q^{-84} +19 q^{-86} -16 q^{-88} +6 q^{-90} +6 q^{-92} -16 q^{-94} +18 q^{-96} -11 q^{-98} -2 q^{-100} +11 q^{-102} -18 q^{-104} +20 q^{-106} -15 q^{-108} +4 q^{-110} +3 q^{-112} -12 q^{-114} +12 q^{-116} -13 q^{-118} +10 q^{-120} -5 q^{-122} + q^{-124} +3 q^{-126} -7 q^{-128} +6 q^{-130} -5 q^{-132} +4 q^{-134} -2 q^{-136} + q^{-140} -2 q^{-142} +2 q^{-144} - q^{-146} + q^{-148} }[/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 36"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ -t^3+5 t^2-8 t+9-8 t^{-1} +5 t^{-2} - t^{-3} }[/math]
In[5]:= Conway[K][z]
Out[5]= [math]\displaystyle{ -z^6-z^4+3 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]= { 37, 4 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ -q^9+2 q^8-4 q^7+6 q^6-6 q^5+6 q^4-5 q^3+4 q^2-2 q+1 }[/math]
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= [math]\displaystyle{ -z^6 a^{-4} +z^4 a^{-2} -4 z^4 a^{-4} +2 z^4 a^{-6} +3 z^2 a^{-2} -5 z^2 a^{-4} +6 z^2 a^{-6} -z^2 a^{-8} +2 a^{-2} -3 a^{-4} +4 a^{-6} -2 a^{-8} }[/math]
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= [math]\displaystyle{ z^8 a^{-4} +z^8 a^{-6} +2 z^7 a^{-3} +5 z^7 a^{-5} +3 z^7 a^{-7} +z^6 a^{-2} +z^6 a^{-4} +4 z^6 a^{-6} +4 z^6 a^{-8} -7 z^5 a^{-3} -14 z^5 a^{-5} -4 z^5 a^{-7} +3 z^5 a^{-9} -4 z^4 a^{-2} -12 z^4 a^{-4} -17 z^4 a^{-6} -7 z^4 a^{-8} +2 z^4 a^{-10} +6 z^3 a^{-3} +9 z^3 a^{-5} -2 z^3 a^{-9} +z^3 a^{-11} +5 z^2 a^{-2} +12 z^2 a^{-4} +15 z^2 a^{-6} +7 z^2 a^{-8} -z^2 a^{-10} -z a^{-3} -2 z a^{-5} +z a^{-7} +z a^{-9} -z a^{-11} -2 a^{-2} -3 a^{-4} -4 a^{-6} -2 a^{-8} }[/math]

Vassiliev invariants

V2 and V3: (3, 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{ 12 }[/math] [math]\displaystyle{ 56 }[/math] [math]\displaystyle{ 72 }[/math] [math]\displaystyle{ 270 }[/math] [math]\displaystyle{ 42 }[/math] [math]\displaystyle{ 672 }[/math] [math]\displaystyle{ \frac{4304}{3} }[/math] [math]\displaystyle{ \frac{704}{3} }[/math] [math]\displaystyle{ 216 }[/math] [math]\displaystyle{ 288 }[/math] [math]\displaystyle{ 1568 }[/math] [math]\displaystyle{ 3240 }[/math] [math]\displaystyle{ 504 }[/math] [math]\displaystyle{ \frac{78751}{10} }[/math] [math]\displaystyle{ \frac{138}{5} }[/math] [math]\displaystyle{ \frac{48742}{15} }[/math] [math]\displaystyle{ \frac{577}{6} }[/math] [math]\displaystyle{ \frac{4191}{10} }[/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]4 is the signature of 9 36. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\ r
  \  
j \
 -2-101234567χ
19         1-1
17        1 1
15       31 -2
13      31  2
11     33   0
9    33    0
7   23     1
5  23      -1
3 13       2
1 1        -1
-11         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[9, 36]]
Out[2]=  
9
In[3]:=
PD[Knot[9, 36]]
Out[3]=  
PD[X[1, 4, 2, 5], X[7, 10, 8, 11], X[3, 9, 4, 8], X[9, 3, 10, 2], 

 X[11, 17, 12, 16], X[5, 15, 6, 14], X[15, 7, 16, 6], 



  X[13, 1, 14, 18], X[17, 13, 18, 12]]
In[4]:=
GaussCode[Knot[9, 36]]
Out[4]=  
GaussCode[-1, 4, -3, 1, -6, 7, -2, 3, -4, 2, -5, 9, -8, 6, -7, 5, -9, 8]
In[5]:=
BR[Knot[9, 36]]
Out[5]=  
BR[4, {1, 1, 1, -2, 1, 1, 3, -2, 3}]
In[6]:=
alex = Alexander[Knot[9, 36]][t]
Out[6]=  
     -3   5    8            2    3

9 - t   + -- - - - 8 t + 5 t  - t



          2   t


          t
In[7]:=
Conway[Knot[9, 36]][z]
Out[7]=  
       2    4    6
1 + 3 z  - z  - z
In[8]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=  
{Knot[9, 36]}
In[9]:=
{KnotDet[Knot[9, 36]], KnotSignature[Knot[9, 36]]}
Out[9]=  
{37, 4}
In[10]:=
J=Jones[Knot[9, 36]][q]
Out[10]=  
             2      3      4      5      6      7      8    9
1 - 2 q + 4 q  - 5 q  + 6 q  - 6 q  + 6 q  - 4 q  + 2 q  - q
In[11]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=  
{Knot[9, 36], Knot[11, NonAlternating, 16]}
In[12]:=
A2Invariant[Knot[9, 36]][q]
Out[12]=  
     4    6    8    10      12    14    16    18      20    22    26

1 + q  + q  - q  + q   - 2 q   + q   + q   + q   + 2 q   - q   - q   - 



  28


  q
In[13]:=
Kauffman[Knot[9, 36]][a, z]
Out[13]=  
                                                2       2       2

-2   4    3    2     z    z    z    2 z   z    z     7 z    15 z
-- - -- - -- - -- - --- + -- + -- - --- - -- - --- + ---- + ----- + 



8    6    4    2    11    9    7    5     3    10     8      6



a    a    a    a    a     a    a    a     a    a      a      a



     2      2    3       3      3      3      4      4       4
 12 z    5 z    z     2 z    9 z    6 z    2 z    7 z    17 z
 ----- + ---- + --- - ---- + ---- + ---- + ---- - ---- - ----- - 
   4       2     11     9      5      3     10      8      6
  a       a     a      a      a      a     a       a      a

     4      4      5      5       5      5      6      6    6    6
 12 z    4 z    3 z    4 z    14 z    7 z    4 z    4 z    z    z
 ----- - ---- + ---- - ---- - ----- - ---- + ---- + ---- + -- + -- + 
   4       2      9      7      5       3      8      6     4    2
  a       a      a      a      a       a      a      a     a    a

    7      7      7    8    8
 3 z    5 z    2 z    z    z
 ---- + ---- + ---- + -- + --
   7      5      3     6    4


   a      a      a     a    a
In[14]:=
{Vassiliev[2][Knot[9, 36]], Vassiliev[3][Knot[9, 36]]}
Out[14]=  
{0, 7}
In[15]:=
Kh[Knot[9, 36]][q, t]
Out[15]=  
                          3

  3      5    1     q   q       5        7        7  2      9  2



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



                2   t   t
             q t

    9  3      11  3      11  4      13  4    13  5      15  5
 3 q  t  + 3 q   t  + 3 q   t  + 3 q   t  + q   t  + 3 q   t  + 

  15  6    17  6    19  7


  q   t  + q   t  + q   t
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