9 13: Difference between revisions

Revision as of 20:13, 28 August 2005

9_12

9_14

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9 13 Quick Notes

9 13 Further Notes and Views

Knot presentations

Planar diagram presentation	X₆₂₇₁ X_14,6,15,5 X_16,8,17,7 X_18,10,1,9 X_8,18,9,17 X_10,16,11,15 X_2,14,3,13 X_12,4,13,3 X_4,12,5,11
Gauss code	1, -7, 8, -9, 2, -1, 3, -5, 4, -6, 9, -8, 7, -2, 6, -3, 5, -4
Dowker-Thistlethwaite code	6 12 14 16 18 4 2 10 8
Conway Notation	[3213]

Three dimensional invariants

Symmetry type	Reversible
Unknotting number	$\{2,3\}$
3-genus	2
Bridge index	2
Super bridge index	$\{4,6\}$
Nakanishi index	1
Maximal Thurston-Bennequin number	[3][-14]
Hyperbolic Volume	9.13509
A-Polynomial	See Data:9 13/A-polynomial

[edit Notes for 9 13's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus	$2$
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 13'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^2-9 t+11-9 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+7 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	{ 37, 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}+2 q^{10}-4 q^9+5 q^8-6 q^7+7 q^6-5 q^5+4 q^4-2 q^3+q^2}
HOMFLY-PT 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^4 a^{-4} +2 z^4 a^{-6} +z^4 a^{-8} +2 z^2 a^{-4} +5 z^2 a^{-6} +z^2 a^{-8} -z^2 a^{-10} +3 a^{-6} - a^{-8} - a^{-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} +4 z^7 a^{-9} +2 z^7 a^{-11} +3 z^6 a^{-6} +z^6 a^{-8} +2 z^6 a^{-12} +2 z^5 a^{-5} -2 z^5 a^{-7} -9 z^5 a^{-9} -4 z^5 a^{-11} +z^5 a^{-13} +z^4 a^{-4} -7 z^4 a^{-6} -4 z^4 a^{-8} -z^4 a^{-10} -5 z^4 a^{-12} -3 z^3 a^{-5} +z^3 a^{-7} +9 z^3 a^{-9} +2 z^3 a^{-11} -3 z^3 a^{-13} -2 z^2 a^{-4} +8 z^2 a^{-6} +6 z^2 a^{-8} -2 z^2 a^{-10} +2 z^2 a^{-12} +z a^{-7} -3 z a^{-9} -2 z a^{-11} +2 z a^{-13} -3 a^{-6} - a^{-8} + a^{-10} }
The A2 invariant	$q^{-6}-q^{-8}+q^{-10}+3q^{-16}+q^{-18}+2q^{-20}-q^{-24}-2q^{-28}-q^{-34}$
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} -8 q^{-46} +11 q^{-48} -9 q^{-50} +4 q^{-52} +4 q^{-54} -12 q^{-56} +19 q^{-58} -20 q^{-60} +17 q^{-62} -8 q^{-64} -4 q^{-66} +18 q^{-68} -22 q^{-70} +23 q^{-72} -13 q^{-74} + q^{-76} +10 q^{-78} -15 q^{-80} +13 q^{-82} - q^{-84} -8 q^{-86} +22 q^{-88} -18 q^{-90} +6 q^{-92} +13 q^{-94} -27 q^{-96} +36 q^{-98} -32 q^{-100} +16 q^{-102} +4 q^{-104} -21 q^{-106} +34 q^{-108} -36 q^{-110} +24 q^{-112} -9 q^{-114} -11 q^{-116} +18 q^{-118} -22 q^{-120} +14 q^{-122} -2 q^{-124} -10 q^{-126} +15 q^{-128} -14 q^{-130} +2 q^{-132} +12 q^{-134} -24 q^{-136} +24 q^{-138} -17 q^{-140} + q^{-142} +13 q^{-144} -23 q^{-146} +26 q^{-148} -20 q^{-150} +9 q^{-152} +2 q^{-154} -12 q^{-156} +14 q^{-158} -12 q^{-160} +9 q^{-162} -3 q^{-164} - q^{-166} +3 q^{-168} -4 q^{-170} +3 q^{-172} - q^{-174} + q^{-176} }

Further Quantum Invariants

Further quantum knot invariants for 9_13.

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} +2 q^{-11} + q^{-13} - q^{-15} + q^{-17} -2 q^{-19} + q^{-21} - 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} -2 q^{-16} +7 q^{-18} -3 q^{-20} -5 q^{-22} +9 q^{-24} - q^{-26} -6 q^{-28} +5 q^{-30} +2 q^{-32} -2 q^{-34} -3 q^{-36} +4 q^{-38} + q^{-40} -8 q^{-42} +3 q^{-44} +4 q^{-46} -8 q^{-48} + q^{-50} +6 q^{-52} -5 q^{-54} - q^{-56} +4 q^{-58} - q^{-60} - q^{-62} + 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} -2 q^{-21} +4 q^{-23} +5 q^{-25} -5 q^{-27} -6 q^{-29} +7 q^{-31} +12 q^{-33} -8 q^{-35} -17 q^{-37} +7 q^{-39} +24 q^{-41} -3 q^{-43} -27 q^{-45} - q^{-47} +27 q^{-49} +5 q^{-51} -22 q^{-53} -11 q^{-55} +16 q^{-57} +12 q^{-59} -7 q^{-61} -12 q^{-63} -4 q^{-65} +14 q^{-67} +8 q^{-69} -14 q^{-71} -18 q^{-73} +13 q^{-75} +19 q^{-77} -10 q^{-79} -24 q^{-81} +7 q^{-83} +25 q^{-85} - q^{-87} -24 q^{-89} -4 q^{-91} +21 q^{-93} +11 q^{-95} -15 q^{-97} -13 q^{-99} +10 q^{-101} +12 q^{-103} -3 q^{-105} -10 q^{-107} +6 q^{-111} + q^{-113} -3 q^{-115} - q^{-117} + q^{-119} + q^{-121} - 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} +5 q^{-28} -2 q^{-30} +4 q^{-32} -6 q^{-34} +11 q^{-38} -2 q^{-40} +2 q^{-42} -19 q^{-44} -4 q^{-46} +25 q^{-48} +14 q^{-50} +9 q^{-52} -45 q^{-54} -34 q^{-56} +28 q^{-58} +50 q^{-60} +47 q^{-62} -57 q^{-64} -82 q^{-66} -5 q^{-68} +66 q^{-70} +97 q^{-72} -31 q^{-74} -100 q^{-76} -48 q^{-78} +39 q^{-80} +106 q^{-82} +12 q^{-84} -66 q^{-86} -63 q^{-88} -5 q^{-90} +70 q^{-92} +38 q^{-94} -15 q^{-96} -47 q^{-98} -33 q^{-100} +16 q^{-102} +47 q^{-104} +29 q^{-106} -32 q^{-108} -54 q^{-110} -22 q^{-112} +57 q^{-114} +62 q^{-116} -21 q^{-118} -69 q^{-120} -53 q^{-122} +60 q^{-124} +89 q^{-126} -71 q^{-130} -84 q^{-132} +38 q^{-134} +95 q^{-136} +39 q^{-138} -40 q^{-140} -100 q^{-142} -10 q^{-144} +62 q^{-146} +61 q^{-148} +14 q^{-150} -73 q^{-152} -43 q^{-154} +7 q^{-156} +45 q^{-158} +45 q^{-160} -25 q^{-162} -32 q^{-164} -21 q^{-166} +10 q^{-168} +33 q^{-170} +2 q^{-172} -7 q^{-174} -16 q^{-176} -5 q^{-178} +12 q^{-180} +2 q^{-182} +2 q^{-184} -4 q^{-186} -3 q^{-188} +3 q^{-190} + q^{-194} - q^{-196} - q^{-198} + 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} +3 q^{-33} + q^{-35} -3 q^{-37} +2 q^{-39} +3 q^{-41} +3 q^{-43} +2 q^{-45} -4 q^{-47} -11 q^{-49} -5 q^{-51} +8 q^{-53} +20 q^{-55} +19 q^{-57} -3 q^{-59} -31 q^{-61} -41 q^{-63} -17 q^{-65} +37 q^{-67} +76 q^{-69} +55 q^{-71} -22 q^{-73} -105 q^{-75} -120 q^{-77} -24 q^{-79} +126 q^{-81} +195 q^{-83} +100 q^{-85} -107 q^{-87} -264 q^{-89} -210 q^{-91} +54 q^{-93} +305 q^{-95} +313 q^{-97} +39 q^{-99} -299 q^{-101} -394 q^{-103} -148 q^{-105} +245 q^{-107} +429 q^{-109} +237 q^{-111} -158 q^{-113} -408 q^{-115} -294 q^{-117} +54 q^{-119} +337 q^{-121} +315 q^{-123} +28 q^{-125} -245 q^{-127} -282 q^{-129} -94 q^{-131} +139 q^{-133} +235 q^{-135} +129 q^{-137} -59 q^{-139} -174 q^{-141} -144 q^{-143} -9 q^{-145} +115 q^{-147} +159 q^{-149} +67 q^{-151} -91 q^{-153} -167 q^{-155} -104 q^{-157} +60 q^{-159} +199 q^{-161} +152 q^{-163} -59 q^{-165} -232 q^{-167} -192 q^{-169} +37 q^{-171} +268 q^{-173} +258 q^{-175} -16 q^{-177} -292 q^{-179} -309 q^{-181} -38 q^{-183} +295 q^{-185} +368 q^{-187} +106 q^{-189} -254 q^{-191} -395 q^{-193} -189 q^{-195} +181 q^{-197} +388 q^{-199} +258 q^{-201} -75 q^{-203} -334 q^{-205} -308 q^{-207} -38 q^{-209} +241 q^{-211} +304 q^{-213} +128 q^{-215} -122 q^{-217} -258 q^{-219} -188 q^{-221} +15 q^{-223} +178 q^{-225} +193 q^{-227} +66 q^{-229} -84 q^{-231} -159 q^{-233} -110 q^{-235} +14 q^{-237} +102 q^{-239} +103 q^{-241} +35 q^{-243} -44 q^{-245} -79 q^{-247} -49 q^{-249} +8 q^{-251} +44 q^{-253} +42 q^{-255} +10 q^{-257} -17 q^{-259} -26 q^{-261} -15 q^{-263} +5 q^{-265} +13 q^{-267} +8 q^{-269} -2 q^{-273} -5 q^{-275} -2 q^{-277} +3 q^{-279} + q^{-281} - q^{-283} - q^{-289} + q^{-291} + q^{-293} - q^{-295} }

A2 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^{-6} - q^{-8} + q^{-10} +3 q^{-16} + q^{-18} +2 q^{-20} - q^{-24} -2 q^{-28} - 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} -20 q^{-22} +32 q^{-24} -42 q^{-26} +56 q^{-28} -58 q^{-30} +70 q^{-32} -70 q^{-34} +69 q^{-36} -46 q^{-38} +30 q^{-40} -40 q^{-44} +74 q^{-46} -114 q^{-48} +130 q^{-50} -153 q^{-52} +148 q^{-54} -140 q^{-56} +120 q^{-58} -87 q^{-60} +52 q^{-62} -12 q^{-64} -18 q^{-66} +47 q^{-68} -72 q^{-70} +80 q^{-72} -78 q^{-74} +70 q^{-76} -62 q^{-78} +46 q^{-80} -30 q^{-82} +21 q^{-84} -12 q^{-86} +6 q^{-88} -2 q^{-90} + q^{-92} }
2,0	$q^{-12}-q^{-14}-q^{-16}+3q^{-18}+2q^{-20}-3q^{-22}+5q^{-26}+2q^{-28}-3q^{-30}+q^{-32}+5q^{-34}-q^{-36}-2q^{-38}+5q^{-40}+3q^{-42}+q^{-44}+2q^{-46}+2q^{-48}-3q^{-50}-3q^{-52}-3q^{-56}-7q^{-58}-2q^{-60}+2q^{-62}-2q^{-64}-3q^{-66}+q^{-68}+3q^{-70}-2q^{-74}+q^{-76}+2q^{-78}+q^{-86}$

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} +7 q^{-24} -3 q^{-26} + q^{-28} +9 q^{-30} -2 q^{-32} - q^{-34} +7 q^{-36} - q^{-38} - q^{-40} + q^{-44} -3 q^{-46} -6 q^{-48} +2 q^{-50} - q^{-52} -8 q^{-54} +3 q^{-56} +3 q^{-58} -6 q^{-60} +3 q^{-62} +2 q^{-64} -3 q^{-66} +2 q^{-68} + q^{-70} - q^{-72} + 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} +3 q^{-21} +2 q^{-23} +2 q^{-25} +2 q^{-27} -2 q^{-33} -2 q^{-37} - q^{-41} - 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} +3 q^{-32} - q^{-34} + q^{-36} +6 q^{-38} +4 q^{-40} -2 q^{-42} +3 q^{-44} +10 q^{-46} +2 q^{-48} + q^{-50} +6 q^{-52} +5 q^{-54} -3 q^{-56} -2 q^{-58} -6 q^{-62} -9 q^{-64} -3 q^{-66} -3 q^{-68} -10 q^{-70} -3 q^{-72} +3 q^{-74} -2 q^{-76} -4 q^{-78} +3 q^{-80} +5 q^{-82} - q^{-86} +3 q^{-88} +2 q^{-90} - q^{-92} + 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} +3 q^{-26} +2 q^{-28} +3 q^{-30} +2 q^{-32} +2 q^{-34} - q^{-40} -2 q^{-42} -2 q^{-46} - q^{-50} - q^{-52} - 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} +7 q^{-24} -5 q^{-26} +7 q^{-28} -3 q^{-30} +2 q^{-32} +3 q^{-34} -5 q^{-36} +9 q^{-38} -11 q^{-40} +12 q^{-42} -13 q^{-44} +11 q^{-46} -10 q^{-48} +6 q^{-50} -3 q^{-52} +3 q^{-56} -5 q^{-58} +6 q^{-60} -7 q^{-62} +6 q^{-64} -5 q^{-66} +4 q^{-68} -3 q^{-70} + q^{-72} - 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} +7 q^{-38} -5 q^{-42} -3 q^{-44} +6 q^{-46} +7 q^{-48} - q^{-50} -6 q^{-52} + q^{-54} +6 q^{-56} +3 q^{-58} -4 q^{-60} -2 q^{-62} +4 q^{-64} +4 q^{-66} -3 q^{-68} -5 q^{-70} + q^{-72} +3 q^{-74} -2 q^{-76} -6 q^{-78} - q^{-80} +4 q^{-82} + q^{-84} -6 q^{-86} -6 q^{-88} +3 q^{-90} +7 q^{-92} -7 q^{-96} -4 q^{-98} +5 q^{-100} +5 q^{-102} - q^{-104} -4 q^{-106} - q^{-108} +3 q^{-110} +2 q^{-112} - q^{-114} - q^{-116} + q^{-120} }

D4 Invariants.

Weight

Invariant

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^{-18} - q^{-20} + q^{-22} -2 q^{-24} +3 q^{-26} -4 q^{-28} +4 q^{-30} -3 q^{-32} +7 q^{-34} -3 q^{-36} +6 q^{-38} - q^{-40} +8 q^{-42} + q^{-44} + q^{-46} +3 q^{-48} -2 q^{-50} +7 q^{-52} -7 q^{-54} +7 q^{-56} -10 q^{-58} +10 q^{-60} -10 q^{-62} +7 q^{-64} -11 q^{-66} +4 q^{-68} -6 q^{-70} + q^{-72} -4 q^{-74} -2 q^{-76} +2 q^{-78} -3 q^{-80} +4 q^{-82} -5 q^{-84} +6 q^{-86} -5 q^{-88} +4 q^{-90} -4 q^{-92} +4 q^{-94} -2 q^{-96} +2 q^{-98} - q^{-100} + 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} -8 q^{-46} +11 q^{-48} -9 q^{-50} +4 q^{-52} +4 q^{-54} -12 q^{-56} +19 q^{-58} -20 q^{-60} +17 q^{-62} -8 q^{-64} -4 q^{-66} +18 q^{-68} -22 q^{-70} +23 q^{-72} -13 q^{-74} + q^{-76} +10 q^{-78} -15 q^{-80} +13 q^{-82} - q^{-84} -8 q^{-86} +22 q^{-88} -18 q^{-90} +6 q^{-92} +13 q^{-94} -27 q^{-96} +36 q^{-98} -32 q^{-100} +16 q^{-102} +4 q^{-104} -21 q^{-106} +34 q^{-108} -36 q^{-110} +24 q^{-112} -9 q^{-114} -11 q^{-116} +18 q^{-118} -22 q^{-120} +14 q^{-122} -2 q^{-124} -10 q^{-126} +15 q^{-128} -14 q^{-130} +2 q^{-132} +12 q^{-134} -24 q^{-136} +24 q^{-138} -17 q^{-140} + q^{-142} +13 q^{-144} -23 q^{-146} +26 q^{-148} -20 q^{-150} +9 q^{-152} +2 q^{-154} -12 q^{-156} +14 q^{-158} -12 q^{-160} +9 q^{-162} -3 q^{-164} - q^{-166} +3 q^{-168} -4 q^{-170} +3 q^{-172} - q^{-174} + 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.

In[3]:= K = Knot["9 13"];

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^2-9 t+11-9 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+7 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]= { 37, 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}+2 q^{10}-4 q^9+5 q^8-6 q^7+7 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]= 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^4 a^{-4} +2 z^4 a^{-6} +z^4 a^{-8} +2 z^2 a^{-4} +5 z^2 a^{-6} +z^2 a^{-8} -z^2 a^{-10} +3 a^{-6} - a^{-8} - a^{-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} +4 z^7 a^{-9} +2 z^7 a^{-11} +3 z^6 a^{-6} +z^6 a^{-8} +2 z^6 a^{-12} +2 z^5 a^{-5} -2 z^5 a^{-7} -9 z^5 a^{-9} -4 z^5 a^{-11} +z^5 a^{-13} +z^4 a^{-4} -7 z^4 a^{-6} -4 z^4 a^{-8} -z^4 a^{-10} -5 z^4 a^{-12} -3 z^3 a^{-5} +z^3 a^{-7} +9 z^3 a^{-9} +2 z^3 a^{-11} -3 z^3 a^{-13} -2 z^2 a^{-4} +8 z^2 a^{-6} +6 z^2 a^{-8} -2 z^2 a^{-10} +2 z^2 a^{-12} +z a^{-7} -3 z a^{-9} -2 z a^{-11} +2 z a^{-13} -3 a^{-6} - a^{-8} + a^{-10} }

Vassiliev invariants

V₂ and V₃:

(7, 18)

V_2,1 through V_6,9:

V_2,1

V_3,1

V_4,1

V_4,2

V_4,3

V_5,1

V_5,2

V_5,3

V_5,4

V_6,1

V_6,2

V_6,3

V_6,4

V_6,5

V_6,6

V_6,7

V_6,8

V_6,9

28

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 144}

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 392}

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{2930}{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{478}{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 4032}

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 7264}

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 1280}

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 1040}

{\frac {10976}{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 10368}

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{82040}{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{13384}{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{1644937}{30}}

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{9566}{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{1018754}{45}}

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{7223}{18}}

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{91657}{30}}

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

t^{r}q^{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 13. 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

1

19

3

1

-2

17

2

1

15

4

3

-1

13

3

2

1

11

2

4

2

9

2

3

-1

7

2

5

1

2

-1

3

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}}	$i=3$	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}}
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=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}^{4}\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^{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}^{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 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}\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=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}\oplus{\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}}
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, 13]]
Out[2]=	9
In[3]:=	PD[Knot[9, 13]]
Out[3]=	PD[X[6, 2, 7, 1], X[14, 6, 15, 5], X[16, 8, 17, 7], X[18, 10, 1, 9], X[8, 18, 9, 17], X[10, 16, 11, 15], X[2, 14, 3, 13], X[12, 4, 13, 3], X[4, 12, 5, 11]]
In[4]:=	GaussCode[Knot[9, 13]]
Out[4]=	GaussCode[1, -7, 8, -9, 2, -1, 3, -5, 4, -6, 9, -8, 7, -2, 6, -3, 5, -4]
In[5]:=	BR[Knot[9, 13]]
Out[5]=	BR[4, {1, 1, 1, 1, 2, -1, 2, 2, 3, -2, 3}]
In[6]:=	alex = Alexander[Knot[9, 13]][t]
Out[6]=	4 9 2 11 + -- - - - 9 t + 4 t 2 t t
In[7]:=	Conway[Knot[9, 13]][z]
Out[7]=	2 4 1 + 7 z + 4 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[9, 13]}
In[9]:=	{KnotDet[Knot[9, 13]], KnotSignature[Knot[9, 13]]}
Out[9]=	{37, 4}
In[10]:=	J=Jones[Knot[9, 13]][q]
Out[10]=	2 3 4 5 6 7 8 9 10 11 q - 2 q + 4 q - 5 q + 7 q - 6 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, 13]}
In[12]:=	A2Invariant[Knot[9, 13]][q]
Out[12]=	6 8 10 16 18 20 24 28 34 q - q + q + 3 q + q + 2 q - q - 2 q - q
In[13]:=	Kauffman[Knot[9, 13]][a, z]
Out[13]=	2 2 2 2 -10 -8 3 2 z 2 z 3 z z 2 z 2 z 6 z 8 z a - a - -- + --- - --- - --- + -- + ---- - ---- + ---- + ---- - 6 13 11 9 7 12 10 8 6 a a a a a a a a a 2 3 3 3 3 3 4 4 4 4 2 z 3 z 2 z 9 z z 3 z 5 z z 4 z 7 z ---- - ---- + ---- + ---- + -- - ---- - ---- - --- - ---- - ---- + 4 13 11 9 7 5 12 10 8 6 a a a a a a a a a a 4 5 5 5 5 5 6 6 6 7 z z 4 z 9 z 2 z 2 z 2 z z 3 z 2 z -- + --- - ---- - ---- - ---- + ---- + ---- + -- + ---- + ---- + 4 13 11 9 7 5 12 8 6 11 a a a a a a a a a a 7 7 8 8 4 z 2 z z z ---- + ---- + --- + -- 9 7 10 8 a a a a
In[14]:=	{Vassiliev[2][Knot[9, 13]], Vassiliev[3][Knot[9, 13]]}
Out[14]=	{0, 18}
In[15]:=	Kh[Knot[9, 13]][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 + 4 q t + 13 4 13 5 15 5 15 6 17 6 17 7 3 q t + 2 q t + 4 q t + 3 q t + 2 q t + q t + 19 7 19 8 21 8 23 9 3 q t + q t + q t + q t

@@ Line 1: / Line 1: @@
 <!--  -->
+<!-- -->
+<!--  -->
+<!-- -->
 <!-- provide an anchor so we can return to the top of the page -->
 <span id="top"></span>
+<!-- -->
 <!-- this relies on transclusion for next and previous links -->
 {{Knot Navigation Links|ext=gif}}
+{{Rolfsen Knot Page Header|n=9|k=13|KnotilusURL=http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/1,-7,8,-9,2,-1,3,-5,4,-6,9,-8,7,-2,6,-3,5,-4/goTop.html}}
-{| align=left
-|- valign=top
-|[[Image:{{PAGENAME}}.gif]]
-|{{Rolfsen Knot Site Links|n=9|k=13|KnotilusURL=http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/1,-7,8,-9,2,-1,3,-5,4,-6,9,-8,7,-2,6,-3,5,-4/goTop.html}}
-|{{:{{PAGENAME}} Quick Notes}}
-|}
 <br style="clear:both" />
@@ Line 24: / Line 21: @@
 {{Vassiliev Invariants}}
-===[[Khovanov Homology]]===
+{{Khovanov Homology|table=<table border=1>
-The coefficients of the monomials <math>t^rq^j</math> are shown, along with their alternating sums <math>\chi</math> (fixed <math>j</math>, alternation over <math>r</math>). The squares with <font class=HLYellow>yellow</font> highlighting are those on the "critical diagonals", where <math>j-2r=s+1</math> or <math>j-2r=s+1</math>, where <math>s=</math>{{Data:{{PAGENAME}}/Signature}} is the signature of {{PAGENAME}}. Nonzero entries off the critical diagonals (if any exist) are highlighted in <font class=HLRed>red</font>.
-<center><table border=1>
 <tr align=center>
 <td width=14.2857%><table cellpadding=0 cellspacing=0>
@@ Line 47: / Line 40: @@
 <tr align=center><td>5</td><td bgcolor=yellow>1</td><td bgcolor=yellow>2</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>-1</td></tr>
 <tr align=center><td>3</td><td bgcolor=yellow>1</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>1</td></tr>
-</table></center>
+</table>}}
 {{Computer Talk Header}}
@@ Line 126: / Line 118: @@
 q   t  + q   t  + q   t  + q   t</nowiki></pre></td></tr>
 </table>
+ [[Category:Knot Page]]

9 13: Difference between revisions

Revision as of 20:13, 28 August 2005

Contents

Knot presentations

Three dimensional invariants

Four dimensional invariants

Polynomial invariants

Vassiliev invariants

Khovanov Homology

Computer Talk

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