9 11

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9 10.gif

9_10

9 12.gif

9_12

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9 11 Quick Notes


9 11 Further Notes and Views

Knot presentations

Planar diagram presentation X1425 X9,12,10,13 X3,11,4,10 X11,3,12,2 X13,1,14,18 X5,15,6,14 X7,17,8,16 X15,7,16,6 X17,9,18,8
Gauss code -1, 4, -3, 1, -6, 8, -7, 9, -2, 3, -4, 2, -5, 6, -8, 7, -9, 5
Dowker-Thistlethwaite code 4 10 14 16 12 2 18 6 8
Conway Notation [4122]

Three dimensional invariants

Symmetry type Reversible
Unknotting number 2
3-genus 3
Bridge index 2
Super bridge index [math]\displaystyle{ \{4,6\} }[/math]
Nakanishi index 1
Maximal Thurston-Bennequin number [1][-12]
Hyperbolic Volume 8.28859
A-Polynomial See Data:9 11/A-polynomial

[edit Notes for 9 11'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 11's four dimensional invariants]

Polynomial invariants

Alexander polynomial [math]\displaystyle{ -t^3+5 t^2-7 t+7-7 t^{-1} +5 t^{-2} - t^{-3} }[/math]
Conway polynomial [math]\displaystyle{ -z^6-z^4+4 z^2+1 }[/math]
2nd Alexander ideal (db, data sources) [math]\displaystyle{ \{1\} }[/math]
Determinant and Signature { 33, 4 }
Jones polynomial [math]\displaystyle{ -q^9+2 q^8-4 q^7+5 q^6-5 q^5+6 q^4-4 q^3+3 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} -4 z^2 a^{-4} +6 z^2 a^{-6} -z^2 a^{-8} + a^{-2} - a^{-4} +3 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} +4 z^7 a^{-5} +2 z^7 a^{-7} +z^6 a^{-2} -z^6 a^{-4} +z^6 a^{-6} +3 z^6 a^{-8} -8 z^5 a^{-3} -12 z^5 a^{-5} -z^5 a^{-7} +3 z^5 a^{-9} -4 z^4 a^{-2} -5 z^4 a^{-4} -7 z^4 a^{-6} -4 z^4 a^{-8} +2 z^4 a^{-10} +8 z^3 a^{-3} +9 z^3 a^{-5} -3 z^3 a^{-7} -3 z^3 a^{-9} +z^3 a^{-11} +4 z^2 a^{-2} +5 z^2 a^{-4} +6 z^2 a^{-6} +4 z^2 a^{-8} -z^2 a^{-10} -z a^{-3} -2 z a^{-5} +2 z a^{-7} +2 z a^{-9} -z a^{-11} - a^{-2} - a^{-4} -3 a^{-6} -2 a^{-8} }[/math]
The A2 invariant [math]\displaystyle{ 1- q^{-8} +2 q^{-10} +2 q^{-14} + q^{-16} + q^{-20} - q^{-22} - q^{-26} - q^{-28} }[/math]
The G2 invariant [math]\displaystyle{ q^{-2} - q^{-4} +3 q^{-6} -4 q^{-8} +3 q^{-10} - q^{-12} -2 q^{-14} +10 q^{-16} -12 q^{-18} +13 q^{-20} -7 q^{-22} -2 q^{-24} +10 q^{-26} -17 q^{-28} +17 q^{-30} -12 q^{-32} +2 q^{-34} +8 q^{-36} -13 q^{-38} +12 q^{-40} -7 q^{-42} -2 q^{-44} +7 q^{-46} -8 q^{-48} +4 q^{-50} - q^{-52} -5 q^{-54} +16 q^{-56} -12 q^{-58} +10 q^{-60} - q^{-62} -8 q^{-64} +19 q^{-66} -20 q^{-68} +18 q^{-70} -9 q^{-72} +16 q^{-76} -19 q^{-78} +17 q^{-80} -8 q^{-82} -2 q^{-84} +8 q^{-86} -10 q^{-88} +4 q^{-90} -4 q^{-94} +7 q^{-96} -6 q^{-98} +3 q^{-102} -10 q^{-104} +10 q^{-106} -9 q^{-108} +4 q^{-110} - q^{-112} -4 q^{-114} +8 q^{-116} -11 q^{-118} +11 q^{-120} -7 q^{-122} +2 q^{-124} + q^{-126} -6 q^{-128} +6 q^{-130} -6 q^{-132} +5 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_11.

A1 Invariants.

Weight Invariant
1 [math]\displaystyle{ q- q^{-1} + q^{-3} - q^{-5} +2 q^{-7} + q^{-9} + q^{-13} -2 q^{-15} + q^{-17} - q^{-19} }[/math]
2 [math]\displaystyle{ q^6-q^4-2 q^2+3+ q^{-2} -4 q^{-4} +3 q^{-6} +4 q^{-8} -4 q^{-10} + q^{-12} +4 q^{-14} -4 q^{-16} - q^{-18} +4 q^{-20} -2 q^{-24} +2 q^{-26} +3 q^{-28} -2 q^{-30} -4 q^{-32} +4 q^{-34} - q^{-36} -5 q^{-38} +4 q^{-40} - q^{-42} -3 q^{-44} +3 q^{-46} - q^{-50} + q^{-52} }[/math]
3 [math]\displaystyle{ q^{15}-q^{13}-2 q^{11}+4 q^7+3 q^5-5 q^3-6 q+4 q^{-1} +9 q^{-3} -11 q^{-7} -4 q^{-9} +11 q^{-11} +8 q^{-13} -7 q^{-15} -10 q^{-17} +4 q^{-19} +13 q^{-21} + q^{-23} -9 q^{-25} -3 q^{-27} +10 q^{-29} +4 q^{-31} -8 q^{-33} -8 q^{-35} +8 q^{-37} +8 q^{-39} -7 q^{-41} -9 q^{-43} +4 q^{-45} +10 q^{-47} -8 q^{-51} -6 q^{-53} +6 q^{-55} +8 q^{-57} - q^{-59} -14 q^{-61} -2 q^{-63} +10 q^{-65} +5 q^{-67} -9 q^{-69} -4 q^{-71} +6 q^{-73} +2 q^{-75} -3 q^{-77} +3 q^{-81} - q^{-83} -2 q^{-85} +2 q^{-87} +2 q^{-89} - q^{-91} - q^{-93} + q^{-97} - q^{-99} }[/math]
4 [math]\displaystyle{ q^{28}-q^{26}-2 q^{24}+q^{20}+6 q^{18}+q^{16}-5 q^{14}-7 q^{12}-7 q^{10}+12 q^8+13 q^6+3 q^4-10 q^2-24+16 q^{-4} +23 q^{-6} +9 q^{-8} -29 q^{-10} -21 q^{-12} -4 q^{-14} +25 q^{-16} +33 q^{-18} -5 q^{-20} -21 q^{-22} -31 q^{-24} +35 q^{-28} +21 q^{-30} +3 q^{-32} -36 q^{-34} -28 q^{-36} +14 q^{-38} +30 q^{-40} +30 q^{-42} -20 q^{-44} -37 q^{-46} -6 q^{-48} +23 q^{-50} +36 q^{-52} -9 q^{-54} -36 q^{-56} -13 q^{-58} +20 q^{-60} +30 q^{-62} -7 q^{-64} -32 q^{-66} -16 q^{-68} +19 q^{-70} +29 q^{-72} +3 q^{-74} -25 q^{-76} -27 q^{-78} +6 q^{-80} +22 q^{-82} +24 q^{-84} + q^{-86} -29 q^{-88} -25 q^{-90} -6 q^{-92} +38 q^{-94} +35 q^{-96} -11 q^{-98} -37 q^{-100} -38 q^{-102} +22 q^{-104} +47 q^{-106} +14 q^{-108} -18 q^{-110} -45 q^{-112} - q^{-114} +28 q^{-116} +17 q^{-118} +5 q^{-120} -26 q^{-122} -7 q^{-124} +7 q^{-126} +5 q^{-128} +11 q^{-130} -8 q^{-132} -2 q^{-134} -2 q^{-136} -3 q^{-138} +7 q^{-140} -2 q^{-142} + q^{-144} - q^{-146} -3 q^{-148} +2 q^{-150} - q^{-152} + q^{-154} - q^{-158} + q^{-160} }[/math]
5 [math]\displaystyle{ q^{45}-q^{43}-2 q^{41}+q^{37}+3 q^{35}+4 q^{33}+q^{31}-7 q^{29}-9 q^{27}-5 q^{25}+3 q^{23}+15 q^{21}+18 q^{19}+5 q^{17}-18 q^{15}-28 q^{13}-20 q^{11}+5 q^9+35 q^7+41 q^5+16 q^3-27 q-53 q^{-1} -43 q^{-3} - q^{-5} +51 q^{-7} +67 q^{-9} +35 q^{-11} -27 q^{-13} -69 q^{-15} -66 q^{-17} -16 q^{-19} +51 q^{-21} +84 q^{-23} +59 q^{-25} -9 q^{-27} -73 q^{-29} -89 q^{-31} -43 q^{-33} +41 q^{-35} +100 q^{-37} +92 q^{-39} +10 q^{-41} -83 q^{-43} -118 q^{-45} -67 q^{-47} +48 q^{-49} +132 q^{-51} +111 q^{-53} -7 q^{-55} -120 q^{-57} -139 q^{-59} -44 q^{-61} +98 q^{-63} +156 q^{-65} +73 q^{-67} -67 q^{-69} -145 q^{-71} -97 q^{-73} +40 q^{-75} +135 q^{-77} +98 q^{-79} -23 q^{-81} -112 q^{-83} -94 q^{-85} +12 q^{-87} +99 q^{-89} +81 q^{-91} -16 q^{-93} -90 q^{-95} -73 q^{-97} +16 q^{-99} +86 q^{-101} +69 q^{-103} -18 q^{-105} -88 q^{-107} -75 q^{-109} +4 q^{-111} +75 q^{-113} +87 q^{-115} +31 q^{-117} -52 q^{-119} -90 q^{-121} -74 q^{-123} +5 q^{-125} +91 q^{-127} +114 q^{-129} +55 q^{-131} -56 q^{-133} -148 q^{-135} -122 q^{-137} +18 q^{-139} +149 q^{-141} +166 q^{-143} +51 q^{-145} -127 q^{-147} -199 q^{-149} -96 q^{-151} +86 q^{-153} +190 q^{-155} +132 q^{-157} -37 q^{-159} -167 q^{-161} -141 q^{-163} - q^{-165} +127 q^{-167} +128 q^{-169} +23 q^{-171} -82 q^{-173} -104 q^{-175} -34 q^{-177} +51 q^{-179} +74 q^{-181} +33 q^{-183} -25 q^{-185} -47 q^{-187} -29 q^{-189} +8 q^{-191} +28 q^{-193} +22 q^{-195} - q^{-197} -15 q^{-199} -14 q^{-201} -4 q^{-203} +6 q^{-205} +9 q^{-207} +4 q^{-209} -4 q^{-211} -3 q^{-213} - q^{-215} - q^{-217} +2 q^{-219} +2 q^{-221} - q^{-223} + q^{-227} - q^{-229} + q^{-233} - q^{-235} }[/math]

A2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ 1- q^{-8} +2 q^{-10} +2 q^{-14} + q^{-16} + q^{-20} - q^{-22} - q^{-26} - q^{-28} }[/math]
1,1 [math]\displaystyle{ q^4-2 q^2+6-12 q^{-2} +19 q^{-4} -30 q^{-6} +40 q^{-8} -44 q^{-10} +49 q^{-12} -44 q^{-14} +38 q^{-16} -18 q^{-18} +3 q^{-20} +18 q^{-22} -34 q^{-24} +54 q^{-26} -68 q^{-28} +74 q^{-30} -76 q^{-32} +74 q^{-34} -63 q^{-36} +50 q^{-38} -32 q^{-40} +14 q^{-42} -2 q^{-44} -10 q^{-46} +14 q^{-48} -22 q^{-50} +25 q^{-52} -26 q^{-54} +24 q^{-56} -28 q^{-58} +26 q^{-60} -22 q^{-62} +18 q^{-64} -14 q^{-66} +11 q^{-68} -6 q^{-70} +4 q^{-72} -2 q^{-74} + q^{-76} }[/math]
2,0 [math]\displaystyle{ q^4-1- q^{-2} + q^{-4} + q^{-6} -2 q^{-8} +4 q^{-12} +3 q^{-14} - q^{-16} + q^{-18} +2 q^{-20} - q^{-22} - q^{-24} + q^{-28} -2 q^{-30} +3 q^{-32} +3 q^{-34} + q^{-38} +4 q^{-40} -4 q^{-44} - q^{-46} -2 q^{-50} -4 q^{-52} - q^{-54} - q^{-58} + q^{-66} + q^{-68} + q^{-70} }[/math]

A3 Invariants.

Weight Invariant
0,1,0 [math]\displaystyle{ 1- q^{-2} + q^{-4} + q^{-6} -2 q^{-8} +3 q^{-10} + q^{-12} -2 q^{-14} +3 q^{-16} -4 q^{-20} +3 q^{-22} +3 q^{-24} -2 q^{-26} +5 q^{-28} +4 q^{-30} +2 q^{-32} - q^{-34} -6 q^{-40} -3 q^{-42} +2 q^{-44} -4 q^{-46} -2 q^{-48} +5 q^{-50} -2 q^{-52} -2 q^{-54} +3 q^{-56} - q^{-60} + q^{-62} }[/math]
1,0,0 [math]\displaystyle{ q^{-1} + q^{-5} - q^{-7} + q^{-9} - q^{-11} + q^{-13} + q^{-17} + q^{-19} + q^{-21} +2 q^{-23} +2 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^{-8} + q^{-14} + q^{-16} +2 q^{-20} +2 q^{-22} - q^{-24} - q^{-26} +2 q^{-28} +2 q^{-30} -4 q^{-32} +2 q^{-34} +6 q^{-36} +2 q^{-38} + q^{-40} +7 q^{-42} +4 q^{-44} + q^{-48} -5 q^{-52} -6 q^{-54} -2 q^{-56} -4 q^{-58} -6 q^{-60} - q^{-62} +2 q^{-64} -2 q^{-66} - q^{-68} +2 q^{-70} +2 q^{-72} + q^{-78} + q^{-80} }[/math]
1,0,0,0 [math]\displaystyle{ q^{-2} + q^{-6} + q^{-12} - q^{-14} + q^{-16} - q^{-18} + q^{-20} + q^{-24} + q^{-26} +2 q^{-28} +2 q^{-30} + q^{-32} +2 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} +3 q^{-4} -3 q^{-6} +4 q^{-8} -5 q^{-10} +5 q^{-12} -4 q^{-14} +3 q^{-16} -2 q^{-18} +3 q^{-22} -5 q^{-24} +8 q^{-26} -7 q^{-28} +10 q^{-30} -8 q^{-32} +9 q^{-34} -6 q^{-36} +4 q^{-38} -2 q^{-40} - q^{-42} +2 q^{-44} -4 q^{-46} +4 q^{-48} -5 q^{-50} +4 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} +2 q^{-6} +2 q^{-8} -2 q^{-10} -3 q^{-12} + q^{-14} +5 q^{-16} +2 q^{-18} -4 q^{-20} -3 q^{-22} +3 q^{-24} +5 q^{-26} -5 q^{-30} -3 q^{-32} +3 q^{-34} +4 q^{-36} -3 q^{-40} +2 q^{-42} +4 q^{-44} +2 q^{-46} -2 q^{-48} + q^{-50} +4 q^{-52} +2 q^{-54} -4 q^{-56} -3 q^{-58} +2 q^{-60} +2 q^{-62} -3 q^{-64} -5 q^{-66} - q^{-68} +3 q^{-70} + q^{-72} -4 q^{-74} -4 q^{-76} + q^{-78} +5 q^{-80} + q^{-82} -3 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} +2 q^{-6} -2 q^{-8} +4 q^{-10} -3 q^{-12} +4 q^{-14} -3 q^{-16} +5 q^{-18} -3 q^{-20} +2 q^{-22} -2 q^{-24} + q^{-26} -2 q^{-30} +3 q^{-32} -3 q^{-34} +7 q^{-36} -4 q^{-38} +10 q^{-40} -3 q^{-42} +10 q^{-44} -5 q^{-46} +7 q^{-48} -5 q^{-50} +2 q^{-52} -5 q^{-54} -3 q^{-56} -3 q^{-58} -3 q^{-60} + q^{-62} -4 q^{-64} +2 q^{-66} -3 q^{-68} +5 q^{-70} -3 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} +3 q^{-6} -4 q^{-8} +3 q^{-10} - q^{-12} -2 q^{-14} +10 q^{-16} -12 q^{-18} +13 q^{-20} -7 q^{-22} -2 q^{-24} +10 q^{-26} -17 q^{-28} +17 q^{-30} -12 q^{-32} +2 q^{-34} +8 q^{-36} -13 q^{-38} +12 q^{-40} -7 q^{-42} -2 q^{-44} +7 q^{-46} -8 q^{-48} +4 q^{-50} - q^{-52} -5 q^{-54} +16 q^{-56} -12 q^{-58} +10 q^{-60} - q^{-62} -8 q^{-64} +19 q^{-66} -20 q^{-68} +18 q^{-70} -9 q^{-72} +16 q^{-76} -19 q^{-78} +17 q^{-80} -8 q^{-82} -2 q^{-84} +8 q^{-86} -10 q^{-88} +4 q^{-90} -4 q^{-94} +7 q^{-96} -6 q^{-98} +3 q^{-102} -10 q^{-104} +10 q^{-106} -9 q^{-108} +4 q^{-110} - q^{-112} -4 q^{-114} +8 q^{-116} -11 q^{-118} +11 q^{-120} -7 q^{-122} +2 q^{-124} + q^{-126} -6 q^{-128} +6 q^{-130} -6 q^{-132} +5 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 11"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ -t^3+5 t^2-7 t+7-7 t^{-1} +5 t^{-2} - t^{-3} }[/math]
In[5]:= Conway[K][z]
Out[5]= [math]\displaystyle{ -z^6-z^4+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{ \{1\} }[/math]
In[7]:= {KnotDet[K], KnotSignature[K]}
Out[7]= { 33, 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+5 q^6-5 q^5+6 q^4-4 q^3+3 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} -4 z^2 a^{-4} +6 z^2 a^{-6} -z^2 a^{-8} + a^{-2} - a^{-4} +3 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} +4 z^7 a^{-5} +2 z^7 a^{-7} +z^6 a^{-2} -z^6 a^{-4} +z^6 a^{-6} +3 z^6 a^{-8} -8 z^5 a^{-3} -12 z^5 a^{-5} -z^5 a^{-7} +3 z^5 a^{-9} -4 z^4 a^{-2} -5 z^4 a^{-4} -7 z^4 a^{-6} -4 z^4 a^{-8} +2 z^4 a^{-10} +8 z^3 a^{-3} +9 z^3 a^{-5} -3 z^3 a^{-7} -3 z^3 a^{-9} +z^3 a^{-11} +4 z^2 a^{-2} +5 z^2 a^{-4} +6 z^2 a^{-6} +4 z^2 a^{-8} -z^2 a^{-10} -z a^{-3} -2 z a^{-5} +2 z a^{-7} +2 z a^{-9} -z a^{-11} - a^{-2} - a^{-4} -3 a^{-6} -2 a^{-8} }[/math]

Vassiliev invariants

V2 and V3: (4, 9)
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{ 16 }[/math] [math]\displaystyle{ 72 }[/math] [math]\displaystyle{ 128 }[/math] [math]\displaystyle{ \frac{1160}{3} }[/math] [math]\displaystyle{ \frac{208}{3} }[/math] [math]\displaystyle{ 1152 }[/math] [math]\displaystyle{ 2288 }[/math] [math]\displaystyle{ 384 }[/math] [math]\displaystyle{ 392 }[/math] [math]\displaystyle{ \frac{2048}{3} }[/math] [math]\displaystyle{ 2592 }[/math] [math]\displaystyle{ \frac{18560}{3} }[/math] [math]\displaystyle{ \frac{3328}{3} }[/math] [math]\displaystyle{ \frac{204662}{15} }[/math] [math]\displaystyle{ -\frac{5408}{15} }[/math] [math]\displaystyle{ \frac{284288}{45} }[/math] [math]\displaystyle{ \frac{1450}{9} }[/math] [math]\displaystyle{ \frac{13382}{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]4 is the signature of 9 11. 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      21  1
11     33   0
9    32    1
7   13     2
5  23      -1
3 12       1
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, 11]]
Out[2]=  
9
In[3]:=
PD[Knot[9, 11]]
Out[3]=  
PD[X[1, 4, 2, 5], X[9, 12, 10, 13], X[3, 11, 4, 10], X[11, 3, 12, 2], 

 X[13, 1, 14, 18], X[5, 15, 6, 14], X[7, 17, 8, 16], X[15, 7, 16, 6], 



  X[17, 9, 18, 8]]
In[4]:=
GaussCode[Knot[9, 11]]
Out[4]=  
GaussCode[-1, 4, -3, 1, -6, 8, -7, 9, -2, 3, -4, 2, -5, 6, -8, 7, -9, 5]
In[5]:=
BR[Knot[9, 11]]
Out[5]=  
BR[4, {1, 1, 1, 1, -2, 1, 3, -2, 3}]
In[6]:=
alex = Alexander[Knot[9, 11]][t]
Out[6]=  
     -3   5    7            2    3

7 - t   + -- - - - 7 t + 5 t  - t



          2   t


          t
In[7]:=
Conway[Knot[9, 11]][z]
Out[7]=  
       2    4    6
1 + 4 z  - z  - z
In[8]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=  
{Knot[9, 11], Knot[11, NonAlternating, 95]}
In[9]:=
{KnotDet[Knot[9, 11]], KnotSignature[Knot[9, 11]]}
Out[9]=  
{33, 4}
In[10]:=
J=Jones[Knot[9, 11]][q]
Out[10]=  
             2      3      4      5      6      7      8    9
1 - 2 q + 3 q  - 4 q  + 6 q  - 5 q  + 5 q  - 4 q  + 2 q  - q
In[11]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=  
{Knot[9, 11]}
In[12]:=
A2Invariant[Knot[9, 11]][q]
Out[12]=  
     8      10      14    16    20    22    26    28
1 - q  + 2 q   + 2 q   + q   + q   - q   - q   - q
In[13]:=
Kauffman[Knot[9, 11]][a, z]
Out[13]=  
                                                    2       2      2

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



8    6                11    9     7     5     3    10     8      6



a    a                a     a     a     a     a    a      a      a



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

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

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


   a      a     a    a
In[14]:=
{Vassiliev[2][Knot[9, 11]], Vassiliev[3][Knot[9, 11]]}
Out[14]=  
{0, 9}
In[15]:=
Kh[Knot[9, 11]][q, t]
Out[15]=  
                          3

  3      5    1     q   q       5      7        7  2      9  2



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

  15  6    17  6    19  7


  q   t  + q   t  + q   t
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