9 13: Difference between revisions

Revision as of 21:07, 29 August 2005

9_12

9_14

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Visit 9 13's page at Knotilus!

Visit 9 13's page at the original Knot Atlas!

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]

Minimum Braid Representative:

Length is 11, width is 4.

Braid index is 4.

A Morse Link Presentation:

Three dimensional invariants

Symmetry type	Reversible
Unknotting number	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 \{2,3\}}
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	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	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 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)	$\{1\}$
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	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} }
The G2 invariant	$q^{-30}-q^{-32}+2q^{-34}-3q^{-36}+2q^{-38}-q^{-40}-2q^{-42}+7q^{-44}-8q^{-46}+11q^{-48}-9q^{-50}+4q^{-52}+4q^{-54}-12q^{-56}+19q^{-58}-20q^{-60}+17q^{-62}-8q^{-64}-4q^{-66}+18q^{-68}-22q^{-70}+23q^{-72}-13q^{-74}+q^{-76}+10q^{-78}-15q^{-80}+13q^{-82}-q^{-84}-8q^{-86}+22q^{-88}-18q^{-90}+6q^{-92}+13q^{-94}-27q^{-96}+36q^{-98}-32q^{-100}+16q^{-102}+4q^{-104}-21q^{-106}+34q^{-108}-36q^{-110}+24q^{-112}-9q^{-114}-11q^{-116}+18q^{-118}-22q^{-120}+14q^{-122}-2q^{-124}-10q^{-126}+15q^{-128}-14q^{-130}+2q^{-132}+12q^{-134}-24q^{-136}+24q^{-138}-17q^{-140}+q^{-142}+13q^{-144}-23q^{-146}+26q^{-148}-20q^{-150}+9q^{-152}+2q^{-154}-12q^{-156}+14q^{-158}-12q^{-160}+9q^{-162}-3q^{-164}-q^{-166}+3q^{-168}-4q^{-170}+3q^{-172}-q^{-174}+q^{-176}$

Further Quantum Invariants

Further quantum knot invariants for 9_13.

A1 Invariants.

Weight	Invariant
1	$q^{-3}-q^{-5}+2q^{-7}-q^{-9}+2q^{-11}+q^{-13}-q^{-15}+q^{-17}-2q^{-19}+q^{-21}-q^{-23}$
2	$q^{-6}-q^{-8}+4q^{-12}-2q^{-14}-2q^{-16}+7q^{-18}-3q^{-20}-5q^{-22}+9q^{-24}-q^{-26}-6q^{-28}+5q^{-30}+2q^{-32}-2q^{-34}-3q^{-36}+4q^{-38}+q^{-40}-8q^{-42}+3q^{-44}+4q^{-46}-8q^{-48}+q^{-50}+6q^{-52}-5q^{-54}-q^{-56}+4q^{-58}-q^{-60}-q^{-62}+q^{-64}$
3	$q^{-9}-q^{-11}+2q^{-15}+2q^{-17}-2q^{-19}-2q^{-21}+4q^{-23}+5q^{-25}-5q^{-27}-6q^{-29}+7q^{-31}+12q^{-33}-8q^{-35}-17q^{-37}+7q^{-39}+24q^{-41}-3q^{-43}-27q^{-45}-q^{-47}+27q^{-49}+5q^{-51}-22q^{-53}-11q^{-55}+16q^{-57}+12q^{-59}-7q^{-61}-12q^{-63}-4q^{-65}+14q^{-67}+8q^{-69}-14q^{-71}-18q^{-73}+13q^{-75}+19q^{-77}-10q^{-79}-24q^{-81}+7q^{-83}+25q^{-85}-q^{-87}-24q^{-89}-4q^{-91}+21q^{-93}+11q^{-95}-15q^{-97}-13q^{-99}+10q^{-101}+12q^{-103}-3q^{-105}-10q^{-107}+6q^{-111}+q^{-113}-3q^{-115}-q^{-117}+q^{-119}+q^{-121}-q^{-123}$
4	$q^{-12}-q^{-14}+2q^{-18}+2q^{-22}-3q^{-24}+5q^{-28}-2q^{-30}+4q^{-32}-6q^{-34}+11q^{-38}-2q^{-40}+2q^{-42}-19q^{-44}-4q^{-46}+25q^{-48}+14q^{-50}+9q^{-52}-45q^{-54}-34q^{-56}+28q^{-58}+50q^{-60}+47q^{-62}-57q^{-64}-82q^{-66}-5q^{-68}+66q^{-70}+97q^{-72}-31q^{-74}-100q^{-76}-48q^{-78}+39q^{-80}+106q^{-82}+12q^{-84}-66q^{-86}-63q^{-88}-5q^{-90}+70q^{-92}+38q^{-94}-15q^{-96}-47q^{-98}-33q^{-100}+16q^{-102}+47q^{-104}+29q^{-106}-32q^{-108}-54q^{-110}-22q^{-112}+57q^{-114}+62q^{-116}-21q^{-118}-69q^{-120}-53q^{-122}+60q^{-124}+89q^{-126}-71q^{-130}-84q^{-132}+38q^{-134}+95q^{-136}+39q^{-138}-40q^{-140}-100q^{-142}-10q^{-144}+62q^{-146}+61q^{-148}+14q^{-150}-73q^{-152}-43q^{-154}+7q^{-156}+45q^{-158}+45q^{-160}-25q^{-162}-32q^{-164}-21q^{-166}+10q^{-168}+33q^{-170}+2q^{-172}-7q^{-174}-16q^{-176}-5q^{-178}+12q^{-180}+2q^{-182}+2q^{-184}-4q^{-186}-3q^{-188}+3q^{-190}+q^{-194}-q^{-196}-q^{-198}+q^{-200}$
5	$q^{-15}-q^{-17}+2q^{-21}+q^{-27}-q^{-29}+3q^{-33}+q^{-35}-3q^{-37}+2q^{-39}+3q^{-41}+3q^{-43}+2q^{-45}-4q^{-47}-11q^{-49}-5q^{-51}+8q^{-53}+20q^{-55}+19q^{-57}-3q^{-59}-31q^{-61}-41q^{-63}-17q^{-65}+37q^{-67}+76q^{-69}+55q^{-71}-22q^{-73}-105q^{-75}-120q^{-77}-24q^{-79}+126q^{-81}+195q^{-83}+100q^{-85}-107q^{-87}-264q^{-89}-210q^{-91}+54q^{-93}+305q^{-95}+313q^{-97}+39q^{-99}-299q^{-101}-394q^{-103}-148q^{-105}+245q^{-107}+429q^{-109}+237q^{-111}-158q^{-113}-408q^{-115}-294q^{-117}+54q^{-119}+337q^{-121}+315q^{-123}+28q^{-125}-245q^{-127}-282q^{-129}-94q^{-131}+139q^{-133}+235q^{-135}+129q^{-137}-59q^{-139}-174q^{-141}-144q^{-143}-9q^{-145}+115q^{-147}+159q^{-149}+67q^{-151}-91q^{-153}-167q^{-155}-104q^{-157}+60q^{-159}+199q^{-161}+152q^{-163}-59q^{-165}-232q^{-167}-192q^{-169}+37q^{-171}+268q^{-173}+258q^{-175}-16q^{-177}-292q^{-179}-309q^{-181}-38q^{-183}+295q^{-185}+368q^{-187}+106q^{-189}-254q^{-191}-395q^{-193}-189q^{-195}+181q^{-197}+388q^{-199}+258q^{-201}-75q^{-203}-334q^{-205}-308q^{-207}-38q^{-209}+241q^{-211}+304q^{-213}+128q^{-215}-122q^{-217}-258q^{-219}-188q^{-221}+15q^{-223}+178q^{-225}+193q^{-227}+66q^{-229}-84q^{-231}-159q^{-233}-110q^{-235}+14q^{-237}+102q^{-239}+103q^{-241}+35q^{-243}-44q^{-245}-79q^{-247}-49q^{-249}+8q^{-251}+44q^{-253}+42q^{-255}+10q^{-257}-17q^{-259}-26q^{-261}-15q^{-263}+5q^{-265}+13q^{-267}+8q^{-269}-2q^{-273}-5q^{-275}-2q^{-277}+3q^{-279}+q^{-281}-q^{-283}-q^{-289}+q^{-291}+q^{-293}-q^{-295}$

A2 Invariants.

Weight	Invariant
1,0	$q^{-6}-q^{-8}+q^{-10}+3q^{-16}+q^{-18}+2q^{-20}-q^{-24}-2q^{-28}-q^{-34}$
1,1	$q^{-12}-2q^{-14}+4q^{-16}-8q^{-18}+17q^{-20}-20q^{-22}+32q^{-24}-42q^{-26}+56q^{-28}-58q^{-30}+70q^{-32}-70q^{-34}+69q^{-36}-46q^{-38}+30q^{-40}-40q^{-44}+74q^{-46}-114q^{-48}+130q^{-50}-153q^{-52}+148q^{-54}-140q^{-56}+120q^{-58}-87q^{-60}+52q^{-62}-12q^{-64}-18q^{-66}+47q^{-68}-72q^{-70}+80q^{-72}-78q^{-74}+70q^{-76}-62q^{-78}+46q^{-80}-30q^{-82}+21q^{-84}-12q^{-86}+6q^{-88}-2q^{-90}+q^{-92}$
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	$q^{-12}-q^{-14}+2q^{-18}-3q^{-20}+q^{-22}+7q^{-24}-3q^{-26}+q^{-28}+9q^{-30}-2q^{-32}-q^{-34}+7q^{-36}-q^{-38}-q^{-40}+q^{-44}-3q^{-46}-6q^{-48}+2q^{-50}-q^{-52}-8q^{-54}+3q^{-56}+3q^{-58}-6q^{-60}+3q^{-62}+2q^{-64}-3q^{-66}+2q^{-68}+q^{-70}-q^{-72}+q^{-74}$
1,0,0	$q^{-9}-q^{-11}+q^{-13}-q^{-15}+q^{-17}+3q^{-21}+2q^{-23}+2q^{-25}+2q^{-27}-2q^{-33}-2q^{-37}-q^{-41}-q^{-45}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{-18}-q^{-20}+q^{-24}-q^{-26}-q^{-28}+3q^{-30}+3q^{-32}-q^{-34}+q^{-36}+6q^{-38}+4q^{-40}-2q^{-42}+3q^{-44}+10q^{-46}+2q^{-48}+q^{-50}+6q^{-52}+5q^{-54}-3q^{-56}-2q^{-58}-6q^{-62}-9q^{-64}-3q^{-66}-3q^{-68}-10q^{-70}-3q^{-72}+3q^{-74}-2q^{-76}-4q^{-78}+3q^{-80}+5q^{-82}-q^{-86}+3q^{-88}+2q^{-90}-q^{-92}+q^{-96}$
1,0,0,0	$q^{-12}-q^{-14}+q^{-16}-q^{-18}+q^{-22}+3q^{-26}+2q^{-28}+3q^{-30}+2q^{-32}+2q^{-34}-q^{-40}-2q^{-42}-2q^{-46}-q^{-50}-q^{-52}-q^{-56}$

B2 Invariants.

Weight

Invariant

0,1

q^{-12}-q^{-14}+2q^{-16}-4q^{-18}+5q^{-20}-5q^{-22}+7q^{-24}-5q^{-26}+7q^{-28}-3q^{-30}+2q^{-32}+3q^{-34}-5q^{-36}+9q^{-38}-11q^{-40}+12q^{-42}-13q^{-44}+11q^{-46}-10q^{-48}+6q^{-50}-3q^{-52}+3q^{-56}-5q^{-58}+6q^{-60}-7q^{-62}+6q^{-64}-5q^{-66}+4q^{-68}-3q^{-70}+q^{-72}-q^{-74}

1,0

q^{-18}-q^{-22}-q^{-24}+q^{-26}+3q^{-28}-4q^{-32}-2q^{-34}+4q^{-36}+7q^{-38}-5q^{-42}-3q^{-44}+6q^{-46}+7q^{-48}-q^{-50}-6q^{-52}+q^{-54}+6q^{-56}+3q^{-58}-4q^{-60}-2q^{-62}+4q^{-64}+4q^{-66}-3q^{-68}-5q^{-70}+q^{-72}+3q^{-74}-2q^{-76}-6q^{-78}-q^{-80}+4q^{-82}+q^{-84}-6q^{-86}-6q^{-88}+3q^{-90}+7q^{-92}-7q^{-96}-4q^{-98}+5q^{-100}+5q^{-102}-q^{-104}-4q^{-106}-q^{-108}+3q^{-110}+2q^{-112}-q^{-114}-q^{-116}+q^{-120}

D4 Invariants.

Weight	Invariant
1,0,0,0	$q^{-18}-q^{-20}+q^{-22}-2q^{-24}+3q^{-26}-4q^{-28}+4q^{-30}-3q^{-32}+7q^{-34}-3q^{-36}+6q^{-38}-q^{-40}+8q^{-42}+q^{-44}+q^{-46}+3q^{-48}-2q^{-50}+7q^{-52}-7q^{-54}+7q^{-56}-10q^{-58}+10q^{-60}-10q^{-62}+7q^{-64}-11q^{-66}+4q^{-68}-6q^{-70}+q^{-72}-4q^{-74}-2q^{-76}+2q^{-78}-3q^{-80}+4q^{-82}-5q^{-84}+6q^{-86}-5q^{-88}+4q^{-90}-4q^{-92}+4q^{-94}-2q^{-96}+2q^{-98}-q^{-100}+q^{-102}$

G2 Invariants.

Weight

Invariant

1,0

q^{-30}-q^{-32}+2q^{-34}-3q^{-36}+2q^{-38}-q^{-40}-2q^{-42}+7q^{-44}-8q^{-46}+11q^{-48}-9q^{-50}+4q^{-52}+4q^{-54}-12q^{-56}+19q^{-58}-20q^{-60}+17q^{-62}-8q^{-64}-4q^{-66}+18q^{-68}-22q^{-70}+23q^{-72}-13q^{-74}+q^{-76}+10q^{-78}-15q^{-80}+13q^{-82}-q^{-84}-8q^{-86}+22q^{-88}-18q^{-90}+6q^{-92}+13q^{-94}-27q^{-96}+36q^{-98}-32q^{-100}+16q^{-102}+4q^{-104}-21q^{-106}+34q^{-108}-36q^{-110}+24q^{-112}-9q^{-114}-11q^{-116}+18q^{-118}-22q^{-120}+14q^{-122}-2q^{-124}-10q^{-126}+15q^{-128}-14q^{-130}+2q^{-132}+12q^{-134}-24q^{-136}+24q^{-138}-17q^{-140}+q^{-142}+13q^{-144}-23q^{-146}+26q^{-148}-20q^{-150}+9q^{-152}+2q^{-154}-12q^{-156}+14q^{-158}-12q^{-160}+9q^{-162}-3q^{-164}-q^{-166}+3q^{-168}-4q^{-170}+3q^{-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]= $\{1\}$

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} }

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {...}

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

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

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}

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{10976}{3}}

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 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

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}}	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}	${\mathbb {Z} }$	${\mathbb {Z} }$
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}}

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^{31}-2 q^{30}+6 q^{28}-7 q^{27}-4 q^{26}+17 q^{25}-12 q^{24}-13 q^{23}+29 q^{22}-13 q^{21}-24 q^{20}+38 q^{19}-10 q^{18}-31 q^{17}+39 q^{16}-6 q^{15}-28 q^{14}+28 q^{13}-q^{12}-18 q^{11}+14 q^{10}+q^9-8 q^8+5 q^7+q^6-2 q^5+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^{60}+2 q^{59}-2 q^{57}-3 q^{56}+6 q^{55}+5 q^{54}-8 q^{53}-13 q^{52}+13 q^{51}+20 q^{50}-10 q^{49}-36 q^{48}+11 q^{47}+46 q^{46}-61 q^{44}-9 q^{43}+69 q^{42}+26 q^{41}-79 q^{40}-40 q^{39}+83 q^{38}+55 q^{37}-85 q^{36}-71 q^{35}+87 q^{34}+77 q^{33}-79 q^{32}-89 q^{31}+79 q^{30}+82 q^{29}-60 q^{28}-85 q^{27}+52 q^{26}+71 q^{25}-33 q^{24}-63 q^{23}+24 q^{22}+45 q^{21}-9 q^{20}-36 q^{19}+7 q^{18}+21 q^{17}-16 q^{15}+2 q^{14}+8 q^{13}+q^{12}-6 q^{11}+q^{10}+2 q^9+q^8-2 q^7+q^6}
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^{98}-2 q^{97}+2 q^{95}-q^{94}+4 q^{93}-8 q^{92}-q^{91}+8 q^{90}-q^{89}+14 q^{88}-25 q^{87}-12 q^{86}+17 q^{85}+8 q^{84}+45 q^{83}-48 q^{82}-43 q^{81}+6 q^{80}+15 q^{79}+115 q^{78}-48 q^{77}-81 q^{76}-44 q^{75}-15 q^{74}+202 q^{73}-q^{72}-80 q^{71}-116 q^{70}-105 q^{69}+262 q^{68}+78 q^{67}-24 q^{66}-173 q^{65}-227 q^{64}+275 q^{63}+149 q^{62}+65 q^{61}-202 q^{60}-340 q^{59}+259 q^{58}+197 q^{57}+148 q^{56}-207 q^{55}-419 q^{54}+227 q^{53}+219 q^{52}+209 q^{51}-189 q^{50}-450 q^{49}+178 q^{48}+205 q^{47}+241 q^{46}-136 q^{45}-418 q^{44}+103 q^{43}+147 q^{42}+238 q^{41}-58 q^{40}-324 q^{39}+36 q^{38}+60 q^{37}+186 q^{36}+11 q^{35}-196 q^{34}+5 q^{33}-11 q^{32}+109 q^{31}+36 q^{30}-92 q^{29}+8 q^{28}-33 q^{27}+47 q^{26}+25 q^{25}-38 q^{24}+13 q^{23}-22 q^{22}+18 q^{21}+10 q^{20}-17 q^{19}+9 q^{18}-9 q^{17}+7 q^{16}+4 q^{15}-7 q^{14}+3 q^{13}-2 q^{12}+2 q^{11}+q^{10}-2 q^9+q^8}
5	Not Available
6	Not Available
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, 13]]
Out[2]=	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[3]:=	GaussCode[Knot[9, 13]]
Out[3]=	GaussCode[1, -7, 8, -9, 2, -1, 3, -5, 4, -6, 9, -8, 7, -2, 6, -3, 5, -4]
In[4]:=	DTCode[Knot[9, 13]]
Out[4]=	DTCode[6, 12, 14, 16, 18, 4, 2, 10, 8]
In[5]:=	br = BR[Knot[9, 13]]
Out[5]=	BR[4, {1, 1, 1, 1, 2, -1, 2, 2, 3, -2, 3}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{4, 11}
In[7]:=	BraidIndex[Knot[9, 13]]
Out[7]=	4
In[8]:=	Show[DrawMorseLink[Knot[9, 13]]]

`Out[8]=`	`-Graphics-`
In[9]:=	(#[Knot[9, 13]]&) /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, {2, 3}, 2, 2, {4, 6}, 1}
In[10]:=	alex = Alexander[Knot[9, 13]][t]
Out[10]=	4 9 2 11 + -- - - - 9 t + 4 t 2 t t
In[11]:=	Conway[Knot[9, 13]][z]
Out[11]=	2 4 1 + 7 z + 4 z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[9, 13]}
In[13]:=	{KnotDet[Knot[9, 13]], KnotSignature[Knot[9, 13]]}
Out[13]=	{37, 4}
In[14]:=	Jones[Knot[9, 13]][q]
Out[14]=	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[15]:=	Select[AllKnots[], (J === Jones[#][q] \|\| (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[9, 13]}
In[16]:=	A2Invariant[Knot[9, 13]][q]
Out[16]=	6 8 10 16 18 20 24 28 34 q - q + q + 3 q + q + 2 q - q - 2 q - q
In[17]:=	HOMFLYPT[Knot[9, 13]][a, z]
Out[17]=	2 2 2 2 4 4 4 -10 -8 3 z z 5 z 2 z z 2 z z -a - a + -- - --- + -- + ---- + ---- + -- + ---- + -- 6 10 8 6 4 8 6 4 a a a a a a a a
In[18]:=	Kauffman[Knot[9, 13]][a, z]
Out[18]=	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[19]:=	{Vassiliev[2][Knot[9, 13]], Vassiliev[3][Knot[9, 13]]}
Out[19]=	{7, 18}
In[20]:=	Kh[Knot[9, 13]][q, t]
Out[20]=	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
In[21]:=	ColouredJones[Knot[9, 13], 2][q]
Out[21]=	4 5 6 7 8 9 10 11 12 13 q - 2 q + q + 5 q - 8 q + q + 14 q - 18 q - q + 28 q - 14 15 16 17 18 19 20 28 q - 6 q + 39 q - 31 q - 10 q + 38 q - 24 q - 21 22 23 24 25 26 27 28 13 q + 29 q - 13 q - 12 q + 17 q - 4 q - 7 q + 6 q - 30 31 2 q + q

See/edit the Rolfsen_Splice_Template.

Back to the top.

9_12

9_14

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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 13: Difference between revisions

Revision as of 21:07, 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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