9 15: Difference between revisions

Revision as of 20:04, 29 August 2005

9_14

9_16

Visit 9 15's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 9 15's page at Knotilus!

Visit 9 15's page at the original Knot Atlas!

9 15 Quick Notes

9 15 Further Notes and Views

Knot presentations

Planar diagram presentation	X₁₄₂₅ X_7,10,8,11 X₃₉₄₈ X_9,3,10,2 X_13,17,14,16 X_5,15,6,14 X_15,7,16,6 X_11,1,12,18 X_17,13,18,12
Gauss code	-1, 4, -3, 1, -6, 7, -2, 3, -4, 2, -8, 9, -5, 6, -7, 5, -9, 8
Dowker-Thistlethwaite code	4 8 14 10 2 18 16 6 12
Conway Notation	[2322]

Minimum Braid Representative:

Length is 10, width is 5.

Braid index is 5.

A Morse Link Presentation:

Three dimensional invariants

Symmetry type	Reversible
Unknotting number	2
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 \{4,5\}}
Nakanishi index	1
Maximal Thurston-Bennequin number	[-1][-10]
Hyperbolic Volume	9.8855
A-Polynomial	See Data:9 15/A-polynomial

[edit Notes for 9 15'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 1}
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 1}
Concordance genus	$2$
Rasmussen s-Invariant	2

[edit Notes for 9 15'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 -2 t^2+10 t-15+10 t^{-1} -2 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 -2 z^4+2 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	{ 39, 2 }
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^8+2 q^7-4 q^6+6 q^5-6 q^4+7 q^3-6 q^2+4 q-2+ q^{-1} }
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^{-2} -z^4 a^{-4} -z^2 a^{-2} +2 z^2 a^{-6} +z^2- a^{-2} + a^{-4} + a^{-6} - a^{-8} +1}
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^{-4} +z^8 a^{-6} +2 z^7 a^{-3} +4 z^7 a^{-5} +2 z^7 a^{-7} +2 z^6 a^{-2} +z^6 a^{-4} +z^6 a^{-6} +2 z^6 a^{-8} +2 z^5 a^{-1} -z^5 a^{-3} -7 z^5 a^{-5} -3 z^5 a^{-7} +z^5 a^{-9} -4 z^4 a^{-6} -5 z^4 a^{-8} +z^4-3 z^3 a^{-1} +5 z^3 a^{-5} -z^3 a^{-7} -3 z^3 a^{-9} -3 z^2 a^{-2} -2 z^2 a^{-4} +2 z^2 a^{-6} +3 z^2 a^{-8} -2 z^2+z a^{-1} +z a^{-3} -z a^{-5} +z a^{-7} +2 z a^{-9} + a^{-2} + a^{-4} - a^{-6} - a^{-8} +1}
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^4+2 q^{-2} -2 q^{-4} +2 q^{-12} +2 q^{-16} - q^{-20} + q^{-22} - q^{-24} - q^{-26} }
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^{18}-q^{16}+3 q^{14}-3 q^{12}+2 q^{10}-3 q^6+8 q^4-9 q^2+11-9 q^{-2} +5 q^{-4} +4 q^{-6} -13 q^{-8} +23 q^{-10} -26 q^{-12} +21 q^{-14} -13 q^{-16} -5 q^{-18} +20 q^{-20} -31 q^{-22} +33 q^{-24} -20 q^{-26} +2 q^{-28} +14 q^{-30} -23 q^{-32} +15 q^{-34} -2 q^{-36} -13 q^{-38} +24 q^{-40} -23 q^{-42} +9 q^{-44} +17 q^{-46} -34 q^{-48} +46 q^{-50} -42 q^{-52} +21 q^{-54} +6 q^{-56} -28 q^{-58} +43 q^{-60} -44 q^{-62} +36 q^{-64} -11 q^{-66} -11 q^{-68} +26 q^{-70} -30 q^{-72} +20 q^{-74} - q^{-76} -15 q^{-78} +21 q^{-80} -15 q^{-82} +3 q^{-84} +18 q^{-86} -31 q^{-88} +33 q^{-90} -23 q^{-92} +18 q^{-96} -32 q^{-98} +33 q^{-100} -24 q^{-102} +10 q^{-104} +3 q^{-106} -15 q^{-108} +17 q^{-110} -15 q^{-112} +9 q^{-114} -3 q^{-116} -2 q^{-118} +3 q^{-120} -4 q^{-122} +3 q^{-124} - q^{-126} + q^{-128} }

Further Quantum Invariants

Further quantum knot invariants for 9_15.

A1 Invariants.

Weight	Invariant
1	$q^{3}-q+2q^{-1}-2q^{-3}+q^{-5}+q^{-7}+2q^{-11}-2q^{-13}+q^{-15}-q^{-17}$
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^{10}-q^8-q^6+3 q^4-3 q^2-2+9 q^{-2} -4 q^{-4} -6 q^{-6} +11 q^{-8} - q^{-10} -7 q^{-12} +5 q^{-14} +2 q^{-16} -3 q^{-18} -3 q^{-20} +5 q^{-22} +2 q^{-24} -8 q^{-26} +5 q^{-28} +6 q^{-30} -10 q^{-32} + q^{-34} +7 q^{-36} -6 q^{-38} -2 q^{-40} +4 q^{-42} - q^{-44} - q^{-46} + q^{-48} }
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^{21}-q^{19}-q^{17}+2 q^{13}-q^{11}-2 q^9+4 q^7+4 q^5-7 q^3-8 q+11 q^{-1} +16 q^{-3} -14 q^{-5} -25 q^{-7} +13 q^{-9} +33 q^{-11} -5 q^{-13} -38 q^{-15} +38 q^{-19} +8 q^{-21} -28 q^{-23} -14 q^{-25} +21 q^{-27} +16 q^{-29} -9 q^{-31} -19 q^{-33} -2 q^{-35} +17 q^{-37} +11 q^{-39} -19 q^{-41} -21 q^{-43} +19 q^{-45} +27 q^{-47} -14 q^{-49} -33 q^{-51} +10 q^{-53} +36 q^{-55} -37 q^{-59} -7 q^{-61} +28 q^{-63} +15 q^{-65} -21 q^{-67} -18 q^{-69} +12 q^{-71} +16 q^{-73} -3 q^{-75} -12 q^{-77} - q^{-79} +7 q^{-81} +2 q^{-83} -3 q^{-85} - q^{-87} + q^{-89} + q^{-91} - q^{-93} }
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^{36}-q^{34}-q^{32}-q^{28}+4 q^{26}-q^{24}+2 q^{20}-5 q^{18}+3 q^{16}-7 q^{14}+2 q^{12}+15 q^{10}-q^8-2 q^6-32 q^4-9 q^2+41+34 q^{-2} +13 q^{-4} -78 q^{-6} -62 q^{-8} +46 q^{-10} +92 q^{-12} +74 q^{-14} -95 q^{-16} -135 q^{-18} -7 q^{-20} +113 q^{-22} +151 q^{-24} -49 q^{-26} -159 q^{-28} -75 q^{-30} +68 q^{-32} +162 q^{-34} +20 q^{-36} -103 q^{-38} -97 q^{-40} -2 q^{-42} +107 q^{-44} +62 q^{-46} -25 q^{-48} -78 q^{-50} -50 q^{-52} +33 q^{-54} +76 q^{-56} +40 q^{-58} -56 q^{-60} -85 q^{-62} -25 q^{-64} +91 q^{-66} +95 q^{-68} -35 q^{-70} -115 q^{-72} -81 q^{-74} +96 q^{-76} +143 q^{-78} +5 q^{-80} -118 q^{-82} -136 q^{-84} +55 q^{-86} +152 q^{-88} +67 q^{-90} -67 q^{-92} -161 q^{-94} -20 q^{-96} +101 q^{-98} +99 q^{-100} +15 q^{-102} -113 q^{-104} -64 q^{-106} +17 q^{-108} +70 q^{-110} +61 q^{-112} -37 q^{-114} -47 q^{-116} -28 q^{-118} +16 q^{-120} +44 q^{-122} +4 q^{-124} -9 q^{-126} -21 q^{-128} -7 q^{-130} +14 q^{-132} +4 q^{-134} +3 q^{-136} -5 q^{-138} -4 q^{-140} +3 q^{-142} + q^{-146} - q^{-148} - q^{-150} + q^{-152} }
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^{55}-q^{53}-q^{51}-q^{47}+q^{45}+4 q^{43}+q^{41}-2 q^{39}-5 q^{35}-5 q^{33}+3 q^{31}+6 q^{29}+7 q^{27}+6 q^{25}-5 q^{23}-20 q^{21}-19 q^{19}+2 q^{17}+30 q^{15}+45 q^{13}+21 q^{11}-37 q^9-88 q^7-68 q^5+34 q^3+137 q+146 q^{-1} +8 q^{-3} -182 q^{-5} -260 q^{-7} -97 q^{-9} +206 q^{-11} +382 q^{-13} +235 q^{-15} -169 q^{-17} -487 q^{-19} -410 q^{-21} +67 q^{-23} +553 q^{-25} +581 q^{-27} +84 q^{-29} -529 q^{-31} -704 q^{-33} -266 q^{-35} +436 q^{-37} +765 q^{-39} +412 q^{-41} -287 q^{-43} -721 q^{-45} -510 q^{-47} +116 q^{-49} +604 q^{-51} +547 q^{-53} +28 q^{-55} -451 q^{-57} -501 q^{-59} -150 q^{-61} +277 q^{-63} +434 q^{-65} +222 q^{-67} -136 q^{-69} -336 q^{-71} -267 q^{-73} +5 q^{-75} +263 q^{-77} +298 q^{-79} +91 q^{-81} -198 q^{-83} -334 q^{-85} -181 q^{-87} +156 q^{-89} +385 q^{-91} +269 q^{-93} -125 q^{-95} -443 q^{-97} -366 q^{-99} +79 q^{-101} +496 q^{-103} +476 q^{-105} -16 q^{-107} -530 q^{-109} -576 q^{-111} -88 q^{-113} +522 q^{-115} +667 q^{-117} +208 q^{-119} -441 q^{-121} -711 q^{-123} -352 q^{-125} +317 q^{-127} +695 q^{-129} +462 q^{-131} -137 q^{-133} -595 q^{-135} -548 q^{-137} -43 q^{-139} +449 q^{-141} +534 q^{-143} +195 q^{-145} -252 q^{-147} -461 q^{-149} -292 q^{-151} +79 q^{-153} +328 q^{-155} +302 q^{-157} +61 q^{-159} -178 q^{-161} -252 q^{-163} -137 q^{-165} +57 q^{-167} +168 q^{-169} +141 q^{-171} +26 q^{-173} -80 q^{-175} -110 q^{-177} -59 q^{-179} +21 q^{-181} +63 q^{-183} +54 q^{-185} +9 q^{-187} -25 q^{-189} -34 q^{-191} -19 q^{-193} +7 q^{-195} +18 q^{-197} +11 q^{-199} -4 q^{-203} -7 q^{-205} -3 q^{-207} +4 q^{-209} +2 q^{-211} - q^{-213} - q^{-219} + q^{-221} + q^{-223} - q^{-225} }

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^4+2 q^{-2} -2 q^{-4} +2 q^{-12} +2 q^{-16} - q^{-20} + q^{-22} - q^{-24} - q^{-26} }
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^{10}+6 q^8-10 q^6+17 q^4-26 q^2+36-52 q^{-2} +68 q^{-4} -82 q^{-6} +98 q^{-8} -102 q^{-10} +106 q^{-12} -82 q^{-14} +52 q^{-16} -8 q^{-18} -47 q^{-20} +102 q^{-22} -156 q^{-24} +194 q^{-26} -217 q^{-28} +220 q^{-30} -204 q^{-32} +174 q^{-34} -126 q^{-36} +72 q^{-38} -12 q^{-40} -38 q^{-42} +78 q^{-44} -108 q^{-46} +118 q^{-48} -116 q^{-50} +98 q^{-52} -80 q^{-54} +58 q^{-56} -38 q^{-58} +23 q^{-60} -12 q^{-62} +6 q^{-64} -2 q^{-66} + q^{-68} }
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^8+2 q^4-4+7 q^{-4} - q^{-6} -6 q^{-8} +3 q^{-10} +7 q^{-12} + q^{-14} -4 q^{-16} +3 q^{-18} +3 q^{-20} -3 q^{-22} - q^{-24} -3 q^{-28} - q^{-30} +5 q^{-32} - q^{-34} -2 q^{-36} +3 q^{-38} +6 q^{-40} -2 q^{-42} -6 q^{-44} +2 q^{-46} +3 q^{-48} -3 q^{-50} -5 q^{-52} +3 q^{-56} -2 q^{-60} + q^{-64} + q^{-66} }

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^8-q^6+q^4+3 q^2-3+6 q^{-4} -6 q^{-6} -2 q^{-8} +9 q^{-10} -5 q^{-12} -2 q^{-14} +6 q^{-16} -2 q^{-18} -2 q^{-20} + q^{-22} +4 q^{-24} -2 q^{-28} +6 q^{-30} +3 q^{-32} -8 q^{-34} +4 q^{-36} +2 q^{-38} -9 q^{-40} +2 q^{-42} + q^{-44} -4 q^{-46} +2 q^{-48} + q^{-50} - q^{-52} + q^{-54} }

1,0,0

q^{5}+q+2q^{-3}-2q^{-5}-q^{-9}+q^{-15}+2q^{-17}+2q^{-21}+q^{-25}-q^{-27}+q^{-29}-q^{-31}-q^{-33}-q^{-35}

B2 Invariants.

Weight	Invariant
0,1	$q^{8}-q^{6}+3q^{4}-3q^{2}+5-6q^{-2}+8q^{-4}-8q^{-6}+6q^{-8}-5q^{-10}+q^{-12}+2q^{-14}-6q^{-16}+10q^{-18}-12q^{-20}+15q^{-22}-14q^{-24}+14q^{-26}-10q^{-28}+8q^{-30}-3q^{-32}+4q^{-36}-6q^{-38}+7q^{-40}-8q^{-42}+7q^{-44}-6q^{-46}+4q^{-48}-3q^{-50}+q^{-52}-q^{-54}$
1,0	$q^{14}-q^{10}-q^{8}+2q^{6}+3q^{4}-4-3q^{-2}+3q^{-4}+7q^{-6}-8q^{-10}-4q^{-12}+6q^{-14}+8q^{-16}-2q^{-18}-7q^{-20}+7q^{-24}+2q^{-26}-6q^{-28}-3q^{-30}+4q^{-32}+4q^{-34}-3q^{-36}-4q^{-38}+2q^{-40}+6q^{-42}-4q^{-46}+7q^{-50}+3q^{-52}-6q^{-54}-7q^{-56}+4q^{-58}+8q^{-60}-q^{-62}-9q^{-64}-5q^{-66}+5q^{-68}+5q^{-70}-2q^{-72}-5q^{-74}-q^{-76}+3q^{-78}+2q^{-80}-q^{-82}-q^{-84}+q^{-88}$

G2 Invariants.

Weight

Invariant

1,0

q^{18}-q^{16}+3q^{14}-3q^{12}+2q^{10}-3q^{6}+8q^{4}-9q^{2}+11-9q^{-2}+5q^{-4}+4q^{-6}-13q^{-8}+23q^{-10}-26q^{-12}+21q^{-14}-13q^{-16}-5q^{-18}+20q^{-20}-31q^{-22}+33q^{-24}-20q^{-26}+2q^{-28}+14q^{-30}-23q^{-32}+15q^{-34}-2q^{-36}-13q^{-38}+24q^{-40}-23q^{-42}+9q^{-44}+17q^{-46}-34q^{-48}+46q^{-50}-42q^{-52}+21q^{-54}+6q^{-56}-28q^{-58}+43q^{-60}-44q^{-62}+36q^{-64}-11q^{-66}-11q^{-68}+26q^{-70}-30q^{-72}+20q^{-74}-q^{-76}-15q^{-78}+21q^{-80}-15q^{-82}+3q^{-84}+18q^{-86}-31q^{-88}+33q^{-90}-23q^{-92}+18q^{-96}-32q^{-98}+33q^{-100}-24q^{-102}+10q^{-104}+3q^{-106}-15q^{-108}+17q^{-110}-15q^{-112}+9q^{-114}-3q^{-116}-2q^{-118}+3q^{-120}-4q^{-122}+3q^{-124}-q^{-126}+q^{-128}

.

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 15"];

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 -2 t^2+10 t-15+10 t^{-1} -2 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 -2 z^4+2 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]= { 39, 2 }

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^8+2 q^7-4 q^6+6 q^5-6 q^4+7 q^3-6 q^2+4 q-2+ q^{-1} }

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^{-2} -z^4 a^{-4} -z^2 a^{-2} +2 z^2 a^{-6} +z^2- a^{-2} + a^{-4} + a^{-6} - a^{-8} +1}

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^{-4} +z^8 a^{-6} +2 z^7 a^{-3} +4 z^7 a^{-5} +2 z^7 a^{-7} +2 z^6 a^{-2} +z^6 a^{-4} +z^6 a^{-6} +2 z^6 a^{-8} +2 z^5 a^{-1} -z^5 a^{-3} -7 z^5 a^{-5} -3 z^5 a^{-7} +z^5 a^{-9} -4 z^4 a^{-6} -5 z^4 a^{-8} +z^4-3 z^3 a^{-1} +5 z^3 a^{-5} -z^3 a^{-7} -3 z^3 a^{-9} -3 z^2 a^{-2} -2 z^2 a^{-4} +2 z^2 a^{-6} +3 z^2 a^{-8} -2 z^2+z a^{-1} +z a^{-3} -z a^{-5} +z a^{-7} +2 z a^{-9} + a^{-2} + a^{-4} - a^{-6} - a^{-8} +1}

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_165, K11n63, K11n101, ...}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$ ): {...}

Vassiliev invariants

V₂ and V₃:

(2, 5)

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

8

40

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 \frac{508}{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{92}{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 320}

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{2416}{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{352}{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 168}

{\frac {256}{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 800}

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{4064}{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{736}{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{59071}{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{1628}{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 \frac{92164}{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{881}{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{4351}{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

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=} 2 is the signature of 9 15. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\		r
	\
j		\

-2

-1

0

1

2

3

4

5

6

7

χ

17

1

-1

15

1

13

3

1

-2

11

3

1

2

9

3

0

7

4

3

1

5

2

3

1

3

2

4

-2

1

3

2

-1

1

-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}}	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=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 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 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}}
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}\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}}
$r=0$	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}^{3}\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}^{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=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}^{4}\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=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}^{3}\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=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^{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=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}\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}\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=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}	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}}

The Coloured Jones Polynomials

The Coloured Jones Polynomials (in the 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 n+1} -dimensional representation of 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 sl(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 n}	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_n}
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^{23}-2 q^{22}+6 q^{20}-8 q^{19}-4 q^{18}+19 q^{17}-14 q^{16}-15 q^{15}+35 q^{14}-15 q^{13}-28 q^{12}+45 q^{11}-12 q^{10}-36 q^9+45 q^8-7 q^7-33 q^6+33 q^5-q^4-21 q^3+16 q^2+q-8+5 q^{-1} -2 q^{-3} + q^{-4} }
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^{45}+2 q^{44}-2 q^{42}-3 q^{41}+7 q^{40}+5 q^{39}-10 q^{38}-14 q^{37}+16 q^{36}+24 q^{35}-14 q^{34}-44 q^{33}+13 q^{32}+60 q^{31}-q^{30}-79 q^{29}-17 q^{28}+97 q^{27}+35 q^{26}-105 q^{25}-60 q^{24}+116 q^{23}+76 q^{22}-113 q^{21}-100 q^{20}+118 q^{19}+106 q^{18}-107 q^{17}-119 q^{16}+101 q^{15}+116 q^{14}-82 q^{13}-114 q^{12}+66 q^{11}+102 q^{10}-46 q^9-84 q^8+28 q^7+64 q^6-13 q^5-46 q^4+8 q^3+26 q^2-2 q-16+3 q^{-1} +7 q^{-2} - q^{-3} -5 q^{-4} +3 q^{-5} + q^{-6} -2 q^{-8} + q^{-9} }
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^{74}-2 q^{73}+2 q^{71}-q^{70}+4 q^{69}-9 q^{68}-q^{67}+10 q^{66}+14 q^{64}-30 q^{63}-15 q^{62}+22 q^{61}+13 q^{60}+54 q^{59}-58 q^{58}-59 q^{57}+3 q^{56}+23 q^{55}+152 q^{54}-49 q^{53}-112 q^{52}-78 q^{51}-26 q^{50}+280 q^{49}+35 q^{48}-110 q^{47}-199 q^{46}-167 q^{45}+374 q^{44}+169 q^{43}-25 q^{42}-296 q^{41}-358 q^{40}+392 q^{39}+292 q^{38}+113 q^{37}-343 q^{36}-535 q^{35}+358 q^{34}+372 q^{33}+243 q^{32}-347 q^{31}-651 q^{30}+298 q^{29}+401 q^{28}+339 q^{27}-311 q^{26}-694 q^{25}+215 q^{24}+373 q^{23}+392 q^{22}-224 q^{21}-649 q^{20}+106 q^{19}+278 q^{18}+386 q^{17}-101 q^{16}-507 q^{15}+12 q^{14}+135 q^{13}+302 q^{12}+9 q^{11}-307 q^{10}-26 q^9+15 q^8+174 q^7+49 q^6-138 q^5-8 q^4-31 q^3+66 q^2+33 q-47+13 q^{-1} -24 q^{-2} +16 q^{-3} +10 q^{-4} -17 q^{-5} +14 q^{-6} -8 q^{-7} +3 q^{-8} + q^{-9} -7 q^{-10} +6 q^{-11} - q^{-12} + q^{-13} -2 q^{-15} + q^{-16} }
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^{110}+2 q^{109}-2 q^{107}+q^{106}-2 q^{104}+5 q^{103}+2 q^{102}-9 q^{101}-3 q^{100}+3 q^{99}+2 q^{98}+16 q^{97}+9 q^{96}-20 q^{95}-29 q^{94}-12 q^{93}+11 q^{92}+50 q^{91}+54 q^{90}-11 q^{89}-71 q^{88}-92 q^{87}-40 q^{86}+80 q^{85}+160 q^{84}+104 q^{83}-44 q^{82}-203 q^{81}-234 q^{80}-35 q^{79}+234 q^{78}+343 q^{77}+197 q^{76}-177 q^{75}-483 q^{74}-406 q^{73}+65 q^{72}+552 q^{71}+644 q^{70}+162 q^{69}-568 q^{68}-898 q^{67}-440 q^{66}+505 q^{65}+1102 q^{64}+761 q^{63}-335 q^{62}-1276 q^{61}-1109 q^{60}+146 q^{59}+1372 q^{58}+1410 q^{57}+124 q^{56}-1421 q^{55}-1719 q^{54}-342 q^{53}+1418 q^{52}+1924 q^{51}+616 q^{50}-1401 q^{49}-2136 q^{48}-787 q^{47}+1341 q^{46}+2242 q^{45}+1010 q^{44}-1285 q^{43}-2365 q^{42}-1124 q^{41}+1188 q^{40}+2378 q^{39}+1299 q^{38}-1078 q^{37}-2400 q^{36}-1382 q^{35}+916 q^{34}+2309 q^{33}+1499 q^{32}-720 q^{31}-2188 q^{30}-1539 q^{29}+489 q^{28}+1958 q^{27}+1549 q^{26}-241 q^{25}-1669 q^{24}-1482 q^{23}+q^{22}+1332 q^{21}+1338 q^{20}+193 q^{19}-970 q^{18}-1129 q^{17}-328 q^{16}+630 q^{15}+900 q^{14}+368 q^{13}-357 q^{12}-632 q^{11}-356 q^{10}+144 q^9+423 q^8+291 q^7-39 q^6-228 q^5-209 q^4-32 q^3+120 q^2+128 q+39-38 q^{-1} -71 q^{-2} -41 q^{-3} +17 q^{-4} +26 q^{-5} +19 q^{-6} +13 q^{-7} -13 q^{-8} -17 q^{-9} +2 q^{-10} -2 q^{-11} -2 q^{-12} +12 q^{-13} +2 q^{-14} -6 q^{-15} +3 q^{-16} -3 q^{-17} -5 q^{-18} +4 q^{-19} +2 q^{-20} - q^{-21} + q^{-22} -2 q^{-24} + q^{-25} }
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 q^{153}-2 q^{152}+2 q^{150}-q^{149}-2 q^{147}+6 q^{146}-6 q^{145}-3 q^{144}+11 q^{143}-q^{142}-2 q^{141}-13 q^{140}+12 q^{139}-14 q^{138}-8 q^{137}+36 q^{136}+14 q^{135}+4 q^{134}-42 q^{133}+9 q^{132}-56 q^{131}-40 q^{130}+78 q^{129}+73 q^{128}+74 q^{127}-47 q^{126}+18 q^{125}-169 q^{124}-189 q^{123}+32 q^{122}+127 q^{121}+245 q^{120}+106 q^{119}+225 q^{118}-239 q^{117}-465 q^{116}-296 q^{115}-103 q^{114}+287 q^{113}+379 q^{112}+870 q^{111}+139 q^{110}-498 q^{109}-797 q^{108}-869 q^{107}-338 q^{106}+251 q^{105}+1722 q^{104}+1220 q^{103}+356 q^{102}-797 q^{101}-1808 q^{100}-1854 q^{99}-973 q^{98}+1961 q^{97}+2517 q^{96}+2241 q^{95}+419 q^{94}-2012 q^{93}-3651 q^{92}-3328 q^{91}+884 q^{90}+3095 q^{89}+4451 q^{88}+2782 q^{87}-882 q^{86}-4793 q^{85}-6017 q^{84}-1336 q^{83}+2457 q^{82}+6087 q^{81}+5471 q^{80}+1273 q^{79}-4876 q^{78}-8170 q^{77}-3863 q^{76}+959 q^{75}+6803 q^{74}+7686 q^{73}+3621 q^{72}-4237 q^{71}-9459 q^{70}-5964 q^{69}-679 q^{68}+6860 q^{67}+9134 q^{66}+5537 q^{65}-3414 q^{64}-10060 q^{63}-7406 q^{62}-2014 q^{61}+6618 q^{60}+9940 q^{59}+6889 q^{58}-2632 q^{57}-10192 q^{56}-8327 q^{55}-3070 q^{54}+6131 q^{53}+10254 q^{52}+7867 q^{51}-1723 q^{50}-9805 q^{49}-8855 q^{48}-4092 q^{47}+5144 q^{46}+9955 q^{45}+8556 q^{44}-408 q^{43}-8580 q^{42}-8795 q^{41}-5116 q^{40}+3405 q^{39}+8681 q^{38}+8653 q^{37}+1237 q^{36}-6307 q^{35}-7713 q^{34}-5715 q^{33}+1143 q^{32}+6283 q^{31}+7664 q^{30}+2591 q^{29}-3415 q^{28}-5504 q^{27}-5290 q^{26}-813 q^{25}+3352 q^{24}+5548 q^{23}+2923 q^{22}-933 q^{21}-2863 q^{20}-3801 q^{19}-1647 q^{18}+976 q^{17}+3078 q^{16}+2182 q^{15}+326 q^{14}-831 q^{13}-2006 q^{12}-1377 q^{11}-174 q^{10}+1227 q^9+1096 q^8+480 q^7+99 q^6-724 q^5-722 q^4-356 q^3+333 q^2+346 q+231+252 q^{-1} -159 q^{-2} -254 q^{-3} -206 q^{-4} +67 q^{-5} +50 q^{-6} +41 q^{-7} +159 q^{-8} -11 q^{-9} -62 q^{-10} -79 q^{-11} +18 q^{-12} -10 q^{-13} -17 q^{-14} +69 q^{-15} +7 q^{-16} -8 q^{-17} -25 q^{-18} +10 q^{-19} -10 q^{-20} -18 q^{-21} +24 q^{-22} +3 q^{-23} +3 q^{-24} -7 q^{-25} +5 q^{-26} -3 q^{-27} -9 q^{-28} +6 q^{-29} +2 q^{-31} - q^{-32} + q^{-33} -2 q^{-35} + q^{-36} }
7	Not Available

Computer Talk

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

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 29, 2005, 15:27:48)...
In[2]:=	PD[Knot[9, 15]]
Out[2]=	PD[X[1, 4, 2, 5], X[7, 10, 8, 11], X[3, 9, 4, 8], X[9, 3, 10, 2], X[13, 17, 14, 16], X[5, 15, 6, 14], X[15, 7, 16, 6], X[11, 1, 12, 18], X[17, 13, 18, 12]]
In[3]:=	GaussCode[Knot[9, 15]]
Out[3]=	GaussCode[-1, 4, -3, 1, -6, 7, -2, 3, -4, 2, -8, 9, -5, 6, -7, 5, -9, 8]
In[4]:=	DTCode[Knot[9, 15]]
Out[4]=	DTCode[4, 8, 14, 10, 2, 18, 16, 6, 12]
In[5]:=	br = BR[Knot[9, 15]]
Out[5]=	BR[5, {1, 1, 1, 2, -1, -3, 2, 4, -3, 4}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{5, 10}
In[7]:=	BraidIndex[Knot[9, 15]]
Out[7]=	5
In[8]:=	Show[DrawMorseLink[Knot[9, 15]]]

`Out[8]=`	`-Graphics-`
In[9]:=	(#[Knot[9, 15]]&) /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 2, 2, 2, {4, 5}, 1}
In[10]:=	alex = Alexander[Knot[9, 15]][t]
Out[10]=	2 10 2 -15 - -- + -- + 10 t - 2 t 2 t t
In[11]:=	Conway[Knot[9, 15]][z]
Out[11]=	2 4 1 + 2 z - 2 z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[9, 15], Knot[10, 165], Knot[11, NonAlternating, 63], Knot[11, NonAlternating, 101]}
In[13]:=	{KnotDet[Knot[9, 15]], KnotSignature[Knot[9, 15]]}
Out[13]=	{39, 2}
In[14]:=	Jones[Knot[9, 15]][q]
Out[14]=	1 2 3 4 5 6 7 8 -2 + - + 4 q - 6 q + 7 q - 6 q + 6 q - 4 q + 2 q - q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] \|\| (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[9, 15]}
In[16]:=	A2Invariant[Knot[9, 15]][q]
Out[16]=	-4 2 4 12 16 20 22 24 26 q + 2 q - 2 q + 2 q + 2 q - q + q - q - q
In[17]:=	HOMFLYPT[Knot[9, 15]][a, z]
Out[17]=	2 2 4 4 -8 -6 -4 -2 2 2 z z z z 1 - a + a + a - a + z + ---- - -- - -- - -- 6 2 4 2 a a a a
In[18]:=	Kauffman[Knot[9, 15]][a, z]
Out[18]=	2 -8 -6 -4 -2 2 z z z z z 2 3 z 1 - a - a + a + a + --- + -- - -- + -- + - - 2 z + ---- + 9 7 5 3 a 8 a a a a a 2 2 2 3 3 3 3 4 4 2 z 2 z 3 z 3 z z 5 z 3 z 4 5 z 4 z ---- - ---- - ---- - ---- - -- + ---- - ---- + z - ---- - ---- + 6 4 2 9 7 5 a 8 6 a a a a a a a a 5 5 5 5 5 6 6 6 6 7 7 z 3 z 7 z z 2 z 2 z z z 2 z 2 z 4 z -- - ---- - ---- - -- + ---- + ---- + -- + -- + ---- + ---- + ---- + 9 7 5 3 a 8 6 4 2 7 5 a a a a a a a a a a 7 8 8 2 z z z ---- + -- + -- 3 6 4 a a a
In[19]:=	{Vassiliev[2][Knot[9, 15]], Vassiliev[3][Knot[9, 15]]}
Out[19]=	{2, 5}
In[20]:=	Kh[Knot[9, 15]][q, t]
Out[20]=	3 1 1 q 3 5 5 2 7 2 3 q + 2 q + ----- + --- + - + 4 q t + 2 q t + 3 q t + 4 q t + 3 2 q t t q t 7 3 9 3 9 4 11 4 11 5 13 5 13 6 3 q t + 3 q t + 3 q t + 3 q t + q t + 3 q t + q t + 15 6 17 7 q t + q t
In[21]:=	ColouredJones[Knot[9, 15], 2][q]
Out[21]=	-4 2 5 2 3 4 5 6 7 -8 + q - -- + - + q + 16 q - 21 q - q + 33 q - 33 q - 7 q + 3 q q 8 9 10 11 12 13 14 15 45 q - 36 q - 12 q + 45 q - 28 q - 15 q + 35 q - 15 q - 16 17 18 19 20 22 23 14 q + 19 q - 4 q - 8 q + 6 q - 2 q + q

See/edit the Rolfsen_Splice_Template.

Back to the top.

9_14

9_16

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+<!-- This page was generated from the splice template "Rolfsen_Splice_Template". Please do not edit! -->
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 </table>
+{| width=100%
-See/edit the [[Rolfsen_Splice_Template]].
+|align=left|See/edit the [[Rolfsen_Splice_Template]].
+Back to the [[#top|top]].
+|align=right|{{Knot Navigation Links|ext=gif}}
+|}
  [[Category:Knot Page]]

9 15: Difference between revisions

Revision as of 20:04, 29 August 2005

Contents

Knot presentations

Three dimensional invariants

Four dimensional invariants

Polynomial invariants

"Similar" Knots (within the Atlas)

Vassiliev invariants

Khovanov Homology

The Coloured Jones Polynomials

Computer Talk

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