9 13

9_12

9_14

Visit 9 13's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) 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 X14,6,15,5 X16,8,17,7 X18,10,1,9 X8,18,9,17 X10,16,11,15 X2,14,3,13 X12,4,13,3 X4,12,5,11
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 Failed to parse (SVG (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)	Failed to parse (SVG (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)	$z^{4}a^{-4}+2z^{4}a^{-6}+z^{4}a^{-8}+2z^{2}a^{-4}+5z^{2}a^{-6}+z^{2}a^{-8}-z^{2}a^{-10}+3a^{-6}-a^{-8}-a^{-10}$
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	Failed to parse (SVG (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	$q^{-15}-q^{-17}+2q^{-21}+q^{-27}-q^{-29}+3q^{-33}+q^{-35}-3q^{-37}+2q^{-39}+3q^{-41}+3q^{-43}+2q^{-45}-4q^{-47}-11q^{-49}-5q^{-51}+8q^{-53}+20q^{-55}+19q^{-57}-3q^{-59}-31q^{-61}-41q^{-63}-17q^{-65}+37q^{-67}+76q^{-69}+55q^{-71}-22q^{-73}-105q^{-75}-120q^{-77}-24q^{-79}+126q^{-81}+195q^{-83}+100q^{-85}-107q^{-87}-264q^{-89}-210q^{-91}+54q^{-93}+305q^{-95}+313q^{-97}+39q^{-99}-299q^{-101}-394q^{-103}-148q^{-105}+245q^{-107}+429q^{-109}+237q^{-111}-158q^{-113}-408q^{-115}-294q^{-117}+54q^{-119}+337q^{-121}+315q^{-123}+28q^{-125}-245q^{-127}-282q^{-129}-94q^{-131}+139q^{-133}+235q^{-135}+129q^{-137}-59q^{-139}-174q^{-141}-144q^{-143}-9q^{-145}+115q^{-147}+159q^{-149}+67q^{-151}-91q^{-153}-167q^{-155}-104q^{-157}+60q^{-159}+199q^{-161}+152q^{-163}-59q^{-165}-232q^{-167}-192q^{-169}+37q^{-171}+268q^{-173}+258q^{-175}-16q^{-177}-292q^{-179}-309q^{-181}-38q^{-183}+295q^{-185}+368q^{-187}+106q^{-189}-254q^{-191}-395q^{-193}-189q^{-195}+181q^{-197}+388q^{-199}+258q^{-201}-75q^{-203}-334q^{-205}-308q^{-207}-38q^{-209}+241q^{-211}+304q^{-213}+128q^{-215}-122q^{-217}-258q^{-219}-188q^{-221}+15q^{-223}+178q^{-225}+193q^{-227}+66q^{-229}-84q^{-231}-159q^{-233}-110q^{-235}+14q^{-237}+102q^{-239}+103q^{-241}+35q^{-243}-44q^{-245}-79q^{-247}-49q^{-249}+8q^{-251}+44q^{-253}+42q^{-255}+10q^{-257}-17q^{-259}-26q^{-261}-15q^{-263}+5q^{-265}+13q^{-267}+8q^{-269}-2q^{-273}-5q^{-275}-2q^{-277}+3q^{-279}+q^{-281}-q^{-283}-q^{-289}+q^{-291}+q^{-293}-q^{-295}$

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. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

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]=

$z^{4}a^{-4}+2z^{4}a^{-6}+z^{4}a^{-8}+2z^{2}a^{-4}+5z^{2}a^{-6}+z^{2}a^{-8}-z^{2}a^{-10}+3a^{-6}-a^{-8}-a^{-10}$

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)

V2,1 through V6,9:

V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9 $28$ $144$ $392$ ${\frac {2930}{3}}$ ${\frac {478}{3}}$ $4032$ $7264$ $1280$ $1040$ ${\frac {10976}{3}}$ $10368$ ${\frac {82040}{3}}$ ${\frac {13384}{3}}$ ${\frac {1644937}{30}}$ ${\frac {9566}{15}}$ ${\frac {1018754}{45}}$ ${\frac {7223}{18}}$ ${\frac {91657}{30}}$

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 $\chi$ (fixed $j$, alternation over $r$). The squares with yellow highlighting are those on the "critical diagonals", where $j-2r=s+1$ or $j-2r=s-1$, where $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   1 19               3 1   -2 17             2 1     1 15           4 3       -1 13         3 2         1 11       2 4           2 9     2 3             -1 7     2               2 5 1 2                 -1 3 1                   1

Integral Khovanov Homology (db, data source)

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