9 10

9_9

9_11

Visit 9 10's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 10's page at Knotilus! Visit 9 10's page at the original Knot Atlas! 9 10 Quick Notes

9 10 Further Notes and Views

Knot presentations

Planar diagram presentation	X8291 X12,4,13,3 X18,10,1,9 X10,18,11,17 X16,8,17,7 X2,12,3,11 X4,16,5,15 X14,6,15,5 X6,14,7,13
Gauss code	1, -6, 2, -7, 8, -9, 5, -1, 3, -4, 6, -2, 9, -8, 7, -5, 4, -3
Dowker-Thistlethwaite code	8 12 14 16 18 2 6 4 10
Conway Notation	[333]

Three dimensional invariants

Symmetry type Reversible Unknotting number $\{2,3\}$ 3-genus 2 Bridge index 2 Super bridge index Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{4,6\}} Nakanishi index 1 Maximal Thurston-Bennequin number [3][-14] Hyperbolic Volume 8.77346 A-Polynomial See Data:9 10/A-polynomial

[edit Notes for 9 10's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2} Topological 4 genus Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2} Concordance genus Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2} Rasmussen s-Invariant -4

[edit Notes for 9 10's four dimensional invariants]

Polynomial invariants

Alexander polynomial	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 4 t^2-8 t+9-8 t^{-1} +4 t^{-2} }
Conway polynomial	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 4 z^4+8 z^2+1}
2nd Alexander ideal (db, data sources)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{1\}}
Determinant and Signature	{ 33, 4 }
Jones polynomial	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^{11}+q^{10}-3 q^9+5 q^8-5 q^7+6 q^6-5 q^5+4 q^4-2 q^3+q^2}
HOMFLY-PT polynomial (db, data sources)	$z^{4}a^{-4}+2z^{4}a^{-6}+z^{4}a^{-8}+2z^{2}a^{-4}+5z^{2}a^{-6}+2z^{2}a^{-8}-z^{2}a^{-10}+2a^{-6}+a^{-8}-2a^{-10}$
Kauffman polynomial (db, data sources)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z^8 a^{-8} +z^8 a^{-10} +2 z^7 a^{-7} +3 z^7 a^{-9} +z^7 a^{-11} +3 z^6 a^{-6} -z^6 a^{-8} -3 z^6 a^{-10} +z^6 a^{-12} +2 z^5 a^{-5} -3 z^5 a^{-7} -7 z^5 a^{-9} -z^5 a^{-11} +z^5 a^{-13} +z^4 a^{-4} -7 z^4 a^{-6} +3 z^4 a^{-8} +9 z^4 a^{-10} -2 z^4 a^{-12} -3 z^3 a^{-5} +3 z^3 a^{-7} +9 z^3 a^{-9} -z^3 a^{-11} -4 z^3 a^{-13} -2 z^2 a^{-4} +7 z^2 a^{-6} -2 z^2 a^{-8} -11 z^2 a^{-10} -4 z a^{-9} +4 z a^{-13} -2 a^{-6} + a^{-8} +2 a^{-10} }
The A2 invariant	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{-6} - q^{-8} + q^{-10} +2 q^{-16} +2 q^{-20} + q^{-22} + q^{-24} + q^{-26} -2 q^{-28} - q^{-30} - q^{-32} - q^{-34} }
The G2 invariant	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{-30} - q^{-32} +2 q^{-34} -3 q^{-36} +2 q^{-38} - q^{-40} -2 q^{-42} +7 q^{-44} -9 q^{-46} +11 q^{-48} -8 q^{-50} +3 q^{-52} +5 q^{-54} -13 q^{-56} +21 q^{-58} -19 q^{-60} +12 q^{-62} -2 q^{-64} -10 q^{-66} +18 q^{-68} -17 q^{-70} +14 q^{-72} -2 q^{-74} -7 q^{-76} +13 q^{-78} -9 q^{-80} - q^{-82} +12 q^{-84} -19 q^{-86} +18 q^{-88} -9 q^{-90} -4 q^{-92} +21 q^{-94} -28 q^{-96} +31 q^{-98} -20 q^{-100} +6 q^{-102} +11 q^{-104} -21 q^{-106} +26 q^{-108} -18 q^{-110} +12 q^{-112} +2 q^{-114} -10 q^{-116} +15 q^{-118} -10 q^{-120} -2 q^{-122} +9 q^{-124} -16 q^{-126} +10 q^{-128} -3 q^{-130} -12 q^{-132} +18 q^{-134} -20 q^{-136} +15 q^{-138} -10 q^{-140} -9 q^{-142} +13 q^{-144} -16 q^{-146} +13 q^{-148} -9 q^{-150} +2 q^{-152} +3 q^{-154} -4 q^{-156} +6 q^{-158} -6 q^{-160} +4 q^{-162} - q^{-164} + q^{-168} -2 q^{-170} +2 q^{-172} + q^{-176} }

Further Quantum Invariants

Further quantum knot invariants for 9_10.

A1 Invariants.

Weight	Invariant
1	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{-3} - q^{-5} +2 q^{-7} - q^{-9} + q^{-11} + q^{-13} +2 q^{-17} -2 q^{-19} - q^{-23} }
2	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{-6} - q^{-8} +4 q^{-12} -2 q^{-14} -3 q^{-16} +7 q^{-18} - q^{-20} -6 q^{-22} +7 q^{-24} + q^{-26} -5 q^{-28} +2 q^{-30} +2 q^{-32} -3 q^{-36} +4 q^{-38} +3 q^{-40} -7 q^{-42} + q^{-44} +5 q^{-46} -7 q^{-48} -2 q^{-50} +4 q^{-52} -3 q^{-54} - q^{-56} +2 q^{-58} + q^{-64} }
3	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{-9} - q^{-11} +2 q^{-15} +2 q^{-17} -2 q^{-19} -3 q^{-21} +4 q^{-23} +7 q^{-25} -4 q^{-27} -10 q^{-29} +2 q^{-31} +17 q^{-33} +2 q^{-35} -20 q^{-37} -6 q^{-39} +21 q^{-41} +12 q^{-43} -20 q^{-45} -14 q^{-47} +17 q^{-49} +15 q^{-51} -9 q^{-53} -14 q^{-55} +3 q^{-57} +9 q^{-59} +3 q^{-61} -9 q^{-63} -9 q^{-65} +6 q^{-67} +15 q^{-69} -2 q^{-71} -20 q^{-73} + q^{-75} +20 q^{-77} +3 q^{-79} -23 q^{-81} -9 q^{-83} +16 q^{-85} +13 q^{-87} -16 q^{-89} -13 q^{-91} +8 q^{-93} +14 q^{-95} -2 q^{-97} -10 q^{-99} + q^{-101} +7 q^{-103} +2 q^{-105} -3 q^{-107} + q^{-111} + q^{-113} - q^{-115} - q^{-123} }
4	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{-12} - q^{-14} +2 q^{-18} +2 q^{-22} -3 q^{-24} - q^{-26} +5 q^{-28} +5 q^{-32} -8 q^{-34} -6 q^{-36} +9 q^{-38} +6 q^{-40} +14 q^{-42} -17 q^{-44} -24 q^{-46} + q^{-48} +19 q^{-50} +47 q^{-52} -10 q^{-54} -51 q^{-56} -33 q^{-58} +14 q^{-60} +83 q^{-62} +23 q^{-64} -54 q^{-66} -70 q^{-68} -17 q^{-70} +90 q^{-72} +55 q^{-74} -28 q^{-76} -73 q^{-78} -42 q^{-80} +56 q^{-82} +55 q^{-84} +6 q^{-86} -43 q^{-88} -44 q^{-90} +11 q^{-92} +35 q^{-94} +26 q^{-96} -10 q^{-98} -34 q^{-100} -28 q^{-102} +15 q^{-104} +44 q^{-106} +17 q^{-108} -28 q^{-110} -58 q^{-112} +60 q^{-116} +42 q^{-118} -19 q^{-120} -84 q^{-122} -19 q^{-124} +59 q^{-126} +62 q^{-128} +5 q^{-130} -86 q^{-132} -45 q^{-134} +27 q^{-136} +66 q^{-138} +43 q^{-140} -53 q^{-142} -52 q^{-144} -10 q^{-146} +38 q^{-148} +55 q^{-150} -8 q^{-152} -29 q^{-154} -28 q^{-156} +2 q^{-158} +32 q^{-160} +9 q^{-162} -4 q^{-164} -16 q^{-166} -9 q^{-168} +8 q^{-170} +3 q^{-172} +4 q^{-174} -3 q^{-176} -4 q^{-178} + q^{-180} -2 q^{-182} +2 q^{-184} - q^{-188} + q^{-190} - q^{-192} + q^{-200} }
5	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{-15} - q^{-17} +2 q^{-21} + q^{-27} - q^{-29} - q^{-31} +3 q^{-33} +3 q^{-35} -2 q^{-37} - q^{-41} +4 q^{-45} +6 q^{-47} - q^{-49} -9 q^{-51} -11 q^{-53} -4 q^{-55} +14 q^{-57} +26 q^{-59} +22 q^{-61} -9 q^{-63} -49 q^{-65} -52 q^{-67} -8 q^{-69} +62 q^{-71} +99 q^{-73} +56 q^{-75} -60 q^{-77} -153 q^{-79} -118 q^{-81} +31 q^{-83} +186 q^{-85} +199 q^{-87} +30 q^{-89} -197 q^{-91} -273 q^{-93} -104 q^{-95} +171 q^{-97} +314 q^{-99} +183 q^{-101} -115 q^{-103} -323 q^{-105} -243 q^{-107} +53 q^{-109} +287 q^{-111} +266 q^{-113} +15 q^{-115} -227 q^{-117} -259 q^{-119} -71 q^{-121} +158 q^{-123} +220 q^{-125} +96 q^{-127} -80 q^{-129} -173 q^{-131} -115 q^{-133} +26 q^{-135} +121 q^{-137} +115 q^{-139} +26 q^{-141} -78 q^{-143} -122 q^{-145} -65 q^{-147} +45 q^{-149} +124 q^{-151} +100 q^{-153} -22 q^{-155} -140 q^{-157} -139 q^{-159} +5 q^{-161} +164 q^{-163} +177 q^{-165} +17 q^{-167} -179 q^{-169} -224 q^{-171} -48 q^{-173} +193 q^{-175} +264 q^{-177} +89 q^{-179} -176 q^{-181} -302 q^{-183} -140 q^{-185} +144 q^{-187} +300 q^{-189} +204 q^{-191} -76 q^{-193} -287 q^{-195} -239 q^{-197} +2 q^{-199} +221 q^{-201} +257 q^{-203} +86 q^{-205} -151 q^{-207} -235 q^{-209} -128 q^{-211} +58 q^{-213} +184 q^{-215} +157 q^{-217} +16 q^{-219} -118 q^{-221} -140 q^{-223} -62 q^{-225} +48 q^{-227} +102 q^{-229} +76 q^{-231} -3 q^{-233} -64 q^{-235} -64 q^{-237} -23 q^{-239} +22 q^{-241} +44 q^{-243} +25 q^{-245} -3 q^{-247} -20 q^{-249} -20 q^{-251} -7 q^{-253} +9 q^{-255} +11 q^{-257} +6 q^{-259} +2 q^{-261} -5 q^{-263} -6 q^{-265} + q^{-269} + q^{-271} +4 q^{-273} -2 q^{-277} - q^{-283} + q^{-285} + q^{-287} - q^{-295} }

A2 Invariants.

Weight	Invariant
1,0	$q^{-6}-q^{-8}+q^{-10}+2q^{-16}+2q^{-20}+q^{-22}+q^{-24}+q^{-26}-2q^{-28}-q^{-30}-q^{-32}-q^{-34}$
1,1	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{-12} -2 q^{-14} +4 q^{-16} -8 q^{-18} +17 q^{-20} -22 q^{-22} +32 q^{-24} -42 q^{-26} +58 q^{-28} -56 q^{-30} +60 q^{-32} -54 q^{-34} +43 q^{-36} -16 q^{-38} -12 q^{-40} +44 q^{-42} -69 q^{-44} +100 q^{-46} -114 q^{-48} +118 q^{-50} -119 q^{-52} +94 q^{-54} -80 q^{-56} +46 q^{-58} -24 q^{-60} -8 q^{-62} +36 q^{-64} -46 q^{-66} +56 q^{-68} -56 q^{-70} +56 q^{-72} -44 q^{-74} +30 q^{-76} -28 q^{-78} +16 q^{-80} -12 q^{-82} +8 q^{-84} -4 q^{-86} +4 q^{-88} + q^{-92} }
2,0	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{-12} - q^{-14} - q^{-16} +3 q^{-18} +2 q^{-20} -3 q^{-22} - q^{-24} +4 q^{-26} +3 q^{-28} -2 q^{-30} + q^{-32} +4 q^{-34} - q^{-36} -2 q^{-38} +2 q^{-40} + q^{-42} +4 q^{-46} +4 q^{-48} +2 q^{-54} -6 q^{-58} - q^{-60} -3 q^{-64} -6 q^{-66} -3 q^{-68} - q^{-72} - q^{-74} +2 q^{-78} + q^{-80} +2 q^{-82} + q^{-84} + q^{-86} }

A3 Invariants.

Weight	Invariant
0,1,0	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{-12} - q^{-14} +2 q^{-18} -3 q^{-20} + q^{-22} +6 q^{-24} -4 q^{-26} + q^{-28} +7 q^{-30} -2 q^{-32} + q^{-34} +6 q^{-36} + q^{-38} +2 q^{-44} -2 q^{-46} -4 q^{-48} +3 q^{-50} -8 q^{-54} +2 q^{-56} - q^{-58} -7 q^{-60} + q^{-62} - q^{-66} +2 q^{-68} +2 q^{-70} + q^{-74} }
1,0,0	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{-9} - q^{-11} + q^{-13} - q^{-15} + q^{-17} +2 q^{-21} + q^{-23} + q^{-25} +2 q^{-27} + q^{-29} +2 q^{-31} + q^{-35} -2 q^{-37} - q^{-39} -2 q^{-41} - q^{-43} - q^{-45} }

A4 Invariants.

Weight	Invariant
0,1,0,0	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{-18} - q^{-20} + q^{-24} - q^{-26} - q^{-28} +3 q^{-30} +2 q^{-32} -2 q^{-34} + q^{-36} +5 q^{-38} +2 q^{-40} -2 q^{-42} +4 q^{-44} +8 q^{-46} + q^{-50} +6 q^{-52} +3 q^{-54} - q^{-56} +4 q^{-58} +5 q^{-60} - q^{-64} +2 q^{-66} -4 q^{-68} -11 q^{-70} -7 q^{-72} -5 q^{-74} -10 q^{-76} -8 q^{-78} +2 q^{-82} + q^{-84} +2 q^{-86} +5 q^{-88} +3 q^{-90} + q^{-92} + q^{-94} + q^{-96} }
1,0,0,0	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{-12} - q^{-14} + q^{-16} - q^{-18} + q^{-22} +2 q^{-26} + q^{-28} +2 q^{-30} + q^{-32} +2 q^{-34} + q^{-36} +2 q^{-38} + q^{-40} + q^{-44} -2 q^{-46} - q^{-48} -2 q^{-50} -2 q^{-52} - q^{-54} - q^{-56} }

B2 Invariants.

Weight	Invariant
0,1	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{-12} - q^{-14} +2 q^{-16} -4 q^{-18} +5 q^{-20} -5 q^{-22} +6 q^{-24} -4 q^{-26} +5 q^{-28} - q^{-30} +5 q^{-34} -6 q^{-36} +9 q^{-38} -10 q^{-40} +10 q^{-42} -10 q^{-44} +8 q^{-46} -6 q^{-48} +3 q^{-50} -2 q^{-54} +4 q^{-56} -5 q^{-58} +5 q^{-60} -5 q^{-62} +4 q^{-64} -3 q^{-66} +2 q^{-68} -2 q^{-70} - q^{-74} }
1,0	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{-18} - q^{-22} - q^{-24} + q^{-26} +3 q^{-28} -4 q^{-32} -2 q^{-34} +4 q^{-36} +6 q^{-38} - q^{-40} -5 q^{-42} -2 q^{-44} +6 q^{-46} +5 q^{-48} -2 q^{-50} -4 q^{-52} +3 q^{-54} +5 q^{-56} + q^{-58} -3 q^{-60} +4 q^{-64} +2 q^{-66} -3 q^{-68} -3 q^{-70} +2 q^{-72} +3 q^{-74} -2 q^{-76} -4 q^{-78} + q^{-80} +5 q^{-82} -6 q^{-86} -5 q^{-88} +3 q^{-90} +4 q^{-92} -3 q^{-94} -7 q^{-96} -3 q^{-98} +3 q^{-100} +2 q^{-102} - q^{-104} -2 q^{-106} +2 q^{-110} +2 q^{-112} + q^{-120} }

D4 Invariants.

Weight	Invariant
1,0,0,0	$q^{-18}-q^{-20}+q^{-22}-2q^{-24}+3q^{-26}-4q^{-28}+4q^{-30}-3q^{-32}+6q^{-34}-3q^{-36}+4q^{-38}-q^{-40}+5q^{-42}+2q^{-44}-q^{-46}+5q^{-48}-2q^{-50}+9q^{-52}-5q^{-54}+8q^{-56}-7q^{-58}+9q^{-60}-7q^{-62}+5q^{-64}-8q^{-66}+2q^{-68}-3q^{-70}-q^{-72}-2q^{-74}-4q^{-76}+2q^{-78}-5q^{-80}+2q^{-82}-6q^{-84}+3q^{-86}-4q^{-88}+2q^{-90}-2q^{-92}+3q^{-94}+2q^{-98}+q^{-102}$

G2 Invariants.

Weight

Invariant

1,0

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{-30} - q^{-32} +2 q^{-34} -3 q^{-36} +2 q^{-38} - q^{-40} -2 q^{-42} +7 q^{-44} -9 q^{-46} +11 q^{-48} -8 q^{-50} +3 q^{-52} +5 q^{-54} -13 q^{-56} +21 q^{-58} -19 q^{-60} +12 q^{-62} -2 q^{-64} -10 q^{-66} +18 q^{-68} -17 q^{-70} +14 q^{-72} -2 q^{-74} -7 q^{-76} +13 q^{-78} -9 q^{-80} - q^{-82} +12 q^{-84} -19 q^{-86} +18 q^{-88} -9 q^{-90} -4 q^{-92} +21 q^{-94} -28 q^{-96} +31 q^{-98} -20 q^{-100} +6 q^{-102} +11 q^{-104} -21 q^{-106} +26 q^{-108} -18 q^{-110} +12 q^{-112} +2 q^{-114} -10 q^{-116} +15 q^{-118} -10 q^{-120} -2 q^{-122} +9 q^{-124} -16 q^{-126} +10 q^{-128} -3 q^{-130} -12 q^{-132} +18 q^{-134} -20 q^{-136} +15 q^{-138} -10 q^{-140} -9 q^{-142} +13 q^{-144} -16 q^{-146} +13 q^{-148} -9 q^{-150} +2 q^{-152} +3 q^{-154} -4 q^{-156} +6 q^{-158} -6 q^{-160} +4 q^{-162} - q^{-164} + q^{-168} -2 q^{-170} +2 q^{-172} + q^{-176} }

.

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 10"];

In[4]:=

Alexander[K][t]

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

Out[4]=

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 4 t^2-8 t+9-8 t^{-1} +4 t^{-2} }

In[5]:=

Conway[K][z]

Out[5]=

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 4 z^4+8 z^2+1}

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]=

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{1\}}

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 33, 4 }

In[8]:=

Jones[K][q]

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

Out[8]=

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^{11}+q^{10}-3 q^9+5 q^8-5 q^7+6 q^6-5 q^5+4 q^4-2 q^3+q^2}

In[9]:=

HOMFLYPT[K][a, z]

KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.

Out[9]=

$z^{4}a^{-4}+2z^{4}a^{-6}+z^{4}a^{-8}+2z^{2}a^{-4}+5z^{2}a^{-6}+2z^{2}a^{-8}-z^{2}a^{-10}+2a^{-6}+a^{-8}-2a^{-10}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z^8 a^{-8} +z^8 a^{-10} +2 z^7 a^{-7} +3 z^7 a^{-9} +z^7 a^{-11} +3 z^6 a^{-6} -z^6 a^{-8} -3 z^6 a^{-10} +z^6 a^{-12} +2 z^5 a^{-5} -3 z^5 a^{-7} -7 z^5 a^{-9} -z^5 a^{-11} +z^5 a^{-13} +z^4 a^{-4} -7 z^4 a^{-6} +3 z^4 a^{-8} +9 z^4 a^{-10} -2 z^4 a^{-12} -3 z^3 a^{-5} +3 z^3 a^{-7} +9 z^3 a^{-9} -z^3 a^{-11} -4 z^3 a^{-13} -2 z^2 a^{-4} +7 z^2 a^{-6} -2 z^2 a^{-8} -11 z^2 a^{-10} -4 z a^{-9} +4 z a^{-13} -2 a^{-6} + a^{-8} +2 a^{-10} }

Vassiliev invariants

V2 and V3:

(8, 22)

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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 32} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 176} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 512} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{3856}{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{656}{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 5632} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{30752}{3}} ${\frac {5504}{3}}$ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1520} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{16384}{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 15488} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{123392}{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{20992}{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1241164}{15}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{6064}{15}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1589776}{45}} ${\frac {5108}{9}}$ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{72844}{15}}

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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t^rq^j} are shown, along with their alternating sums Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \chi} (fixed Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j} , alternation over Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r} ). The squares with yellow highlighting are those on the "critical diagonals", where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j-2r=s+1} or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j-2r=s-1} , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s=} 4 is the signature of 9 10. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ 0 1 2 3 4 5 6 7 8 9 χ 23                   1 -1 21                     0 19               3 1   -2 17             2       2 15           3 3       0 13         3 2         1 11       2 3           1 9     2 3             -1 7     2               2 5 1 2                 -1 3 1                   1

Integral Khovanov Homology (db, data source)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i=3} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i=5} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=0} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}} $r=1$ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{2}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{2}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=3} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{2}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=4} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=5} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=6} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{2}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=7} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}_2^{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=8} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=9} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}_2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}}

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \textrm{Include}(\textrm{ColouredJonesM.mhtml})}

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 17, 2005, 14:44:34)...
In[2]:=	Crossings[Knot[9, 10]]
Out[2]=	9
In[3]:=	PD[Knot[9, 10]]
Out[3]=	PD[X[8, 2, 9, 1], X[12, 4, 13, 3], X[18, 10, 1, 9], X[10, 18, 11, 17], X[16, 8, 17, 7], X[2, 12, 3, 11], X[4, 16, 5, 15], X[14, 6, 15, 5], X[6, 14, 7, 13]]
In[4]:=	GaussCode[Knot[9, 10]]
Out[4]=	GaussCode[1, -6, 2, -7, 8, -9, 5, -1, 3, -4, 6, -2, 9, -8, 7, -5, 4, -3]
In[5]:=	BR[Knot[9, 10]]
Out[5]=	BR[4, {1, 1, 2, -1, 2, 2, 2, 2, 3, -2, 3}]
In[6]:=	alex = Alexander[Knot[9, 10]][t]
Out[6]=	4 8 2 9 + -- - - - 8 t + 4 t 2 t t
In[7]:=	Conway[Knot[9, 10]][z]
Out[7]=	2 4 1 + 8 z + 4 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[9, 10]}
In[9]:=	{KnotDet[Knot[9, 10]], KnotSignature[Knot[9, 10]]}
Out[9]=	{33, 4}
In[10]:=	J=Jones[Knot[9, 10]][q]
Out[10]=	2 3 4 5 6 7 8 9 10 11 q - 2 q + 4 q - 5 q + 6 q - 5 q + 5 q - 3 q + q - q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[9, 10]}
In[12]:=	A2Invariant[Knot[9, 10]][q]
Out[12]=	6 8 10 16 20 22 24 26 28 30 32 q - q + q + 2 q + 2 q + q + q + q - 2 q - q - q - 34 q
In[13]:=	Kauffman[Knot[9, 10]][a, z]
Out[13]=	2 2 2 2 3 3 2 -8 2 4 z 4 z 11 z 2 z 7 z 2 z 4 z z --- + a - -- + --- - --- - ----- - ---- + ---- - ---- - ---- - --- + 10 6 13 9 10 8 6 4 13 11 a a a a a a a a a a 3 3 3 4 4 4 4 4 5 5 9 z 3 z 3 z 2 z 9 z 3 z 7 z z z z ---- + ---- - ---- - ---- + ---- + ---- - ---- + -- + --- - --- - 9 7 5 12 10 8 6 4 13 11 a a a a a a a a a a 5 5 5 6 6 6 6 7 7 7 7 z 3 z 2 z z 3 z z 3 z z 3 z 2 z ---- - ---- + ---- + --- - ---- - -- + ---- + --- + ---- + ---- + 9 7 5 12 10 8 6 11 9 7 a a a a a a a a a a 8 8 z z --- + -- 10 8 a a
In[14]:=	{Vassiliev[2][Knot[9, 10]], Vassiliev[3][Knot[9, 10]]}
Out[14]=	{0, 22}
In[15]:=	Kh[Knot[9, 10]][q, t]
Out[15]=	3 5 5 7 2 9 2 9 3 11 3 11 4 q + q + 2 q t + 2 q t + 2 q t + 3 q t + 2 q t + 3 q t + 13 4 13 5 15 5 15 6 17 6 19 7 3 q t + 2 q t + 3 q t + 3 q t + 2 q t + 3 q t + 19 8 23 9 q t + q t