9 13

9_12

9_14

(KnotPlot image)	See the full Rolfsen Knot Table. Visit 9 13's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 13 at Knotilus!

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]

Minimum Braid Representative	A Morse Link Presentation	An Arc Presentation
Length is 11, width is 4, Braid index is 4		[{3, 7}, {8, 6}, {7, 5}, {6, 4}, {5, 9}, {2, 8}, {10, 3}, {9, 11}, {1, 10}, {11, 2}, {4, 1}]

[edit Notes on presentations of 9 13]

Computer Talk

The above data is available with the Mathematica package KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.

(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 May 31, 2006, 14:15:20.091. Read more at http://katlas.org/wiki/KnotTheory.

In[3]:=

K = Knot["9 13"];

In[4]:=

PD[K]

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

Out[4]=

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

In[5]:=

GaussCode[K]

Out[5]=

1, -7, 8, -9, 2, -1, 3, -5, 4, -6, 9, -8, 7, -2, 6, -3, 5, -4

In[6]:=

DTCode[K]

Out[6]=

6 12 14 16 18 4 2 10 8

(The path below may be different on your system)

In[7]:=

AppendTo[$Path, "C:/bin/LinKnot/"];

In[8]:=

ConwayNotation[K]

Out[8]=

[3213]

In[9]:=

br = BR[K]

KnotTheory::credits: The minimum braids representing the knots with up to 10 crossings were provided by Thomas Gittings. See arXiv:math.GT/0401051.

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 \textrm{BR}(4,\{1,1,1,1,2,-1,2,2,3,-2,3\})}

In[10]:=

{First[br], Crossings[br], BraidIndex[K]}

KnotTheory::credits: The braid index data known to KnotTheory` is taken from Charles Livingston's http://www.indiana.edu/~knotinfo/.

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

Out[10]=

{ 4, 11, 4 }

In[11]:=

Show[BraidPlot[br]]

Out[11]=

-Graphics-

In[12]:=

Show[DrawMorseLink[K]]

KnotTheory::credits: "MorseLink was added to KnotTheory` by Siddarth Sankaran at the University of Toronto in the summer of 2005."

KnotTheory::credits: "DrawMorseLink was written by Siddarth Sankaran at the University of Toronto in the summer of 2005."

Out[12]=

-Graphics-

In[13]:=

ap = ArcPresentation[K]

Out[13]=

ArcPresentation[{3, 7}, {8, 6}, {7, 5}, {6, 4}, {5, 9}, {2, 8}, {10, 3}, {9, 11}, {1, 10}, {11, 2}, {4, 1}]

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

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 $\{4,6\}$ Nakanishi index 1 Maximal Thurston-Bennequin number [3][-14] Hyperbolic Volume 9.13509 A-Polynomial See Data:9 13/A-polynomial

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

Four dimensional invariants

Smooth 4 genus $2$ Topological 4 genus $2$ Concordance genus $2$ Rasmussen s-Invariant -4

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

Polynomial invariants

Alexander polynomial	$4t^{2}-9t+11-9t^{-1}+4t^{-2}$
Conway polynomial	$4z^{4}+7z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 37, 4 }
Jones polynomial	$-q^{11}+2q^{10}-4q^{9}+5q^{8}-6q^{7}+7q^{6}-5q^{5}+4q^{4}-2q^{3}+q^{2}$
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)	$z^{8}a^{-8}+z^{8}a^{-10}+2z^{7}a^{-7}+4z^{7}a^{-9}+2z^{7}a^{-11}+3z^{6}a^{-6}+z^{6}a^{-8}+2z^{6}a^{-12}+2z^{5}a^{-5}-2z^{5}a^{-7}-9z^{5}a^{-9}-4z^{5}a^{-11}+z^{5}a^{-13}+z^{4}a^{-4}-7z^{4}a^{-6}-4z^{4}a^{-8}-z^{4}a^{-10}-5z^{4}a^{-12}-3z^{3}a^{-5}+z^{3}a^{-7}+9z^{3}a^{-9}+2z^{3}a^{-11}-3z^{3}a^{-13}-2z^{2}a^{-4}+8z^{2}a^{-6}+6z^{2}a^{-8}-2z^{2}a^{-10}+2z^{2}a^{-12}+za^{-7}-3za^{-9}-2za^{-11}+2za^{-13}-3a^{-6}-a^{-8}+a^{-10}$
The A2 invariant	$q^{-6}-q^{-8}+q^{-10}+3q^{-16}+q^{-18}+2q^{-20}-q^{-24}-2q^{-28}-q^{-34}$
The G2 invariant	$q^{-30}-q^{-32}+2q^{-34}-3q^{-36}+2q^{-38}-q^{-40}-2q^{-42}+7q^{-44}-8q^{-46}+11q^{-48}-9q^{-50}+4q^{-52}+4q^{-54}-12q^{-56}+19q^{-58}-20q^{-60}+17q^{-62}-8q^{-64}-4q^{-66}+18q^{-68}-22q^{-70}+23q^{-72}-13q^{-74}+q^{-76}+10q^{-78}-15q^{-80}+13q^{-82}-q^{-84}-8q^{-86}+22q^{-88}-18q^{-90}+6q^{-92}+13q^{-94}-27q^{-96}+36q^{-98}-32q^{-100}+16q^{-102}+4q^{-104}-21q^{-106}+34q^{-108}-36q^{-110}+24q^{-112}-9q^{-114}-11q^{-116}+18q^{-118}-22q^{-120}+14q^{-122}-2q^{-124}-10q^{-126}+15q^{-128}-14q^{-130}+2q^{-132}+12q^{-134}-24q^{-136}+24q^{-138}-17q^{-140}+q^{-142}+13q^{-144}-23q^{-146}+26q^{-148}-20q^{-150}+9q^{-152}+2q^{-154}-12q^{-156}+14q^{-158}-12q^{-160}+9q^{-162}-3q^{-164}-q^{-166}+3q^{-168}-4q^{-170}+3q^{-172}-q^{-174}+q^{-176}$

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	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{-15} - q^{-17} +2 q^{-21} + q^{-27} - q^{-29} +3 q^{-33} + q^{-35} -3 q^{-37} +2 q^{-39} +3 q^{-41} +3 q^{-43} +2 q^{-45} -4 q^{-47} -11 q^{-49} -5 q^{-51} +8 q^{-53} +20 q^{-55} +19 q^{-57} -3 q^{-59} -31 q^{-61} -41 q^{-63} -17 q^{-65} +37 q^{-67} +76 q^{-69} +55 q^{-71} -22 q^{-73} -105 q^{-75} -120 q^{-77} -24 q^{-79} +126 q^{-81} +195 q^{-83} +100 q^{-85} -107 q^{-87} -264 q^{-89} -210 q^{-91} +54 q^{-93} +305 q^{-95} +313 q^{-97} +39 q^{-99} -299 q^{-101} -394 q^{-103} -148 q^{-105} +245 q^{-107} +429 q^{-109} +237 q^{-111} -158 q^{-113} -408 q^{-115} -294 q^{-117} +54 q^{-119} +337 q^{-121} +315 q^{-123} +28 q^{-125} -245 q^{-127} -282 q^{-129} -94 q^{-131} +139 q^{-133} +235 q^{-135} +129 q^{-137} -59 q^{-139} -174 q^{-141} -144 q^{-143} -9 q^{-145} +115 q^{-147} +159 q^{-149} +67 q^{-151} -91 q^{-153} -167 q^{-155} -104 q^{-157} +60 q^{-159} +199 q^{-161} +152 q^{-163} -59 q^{-165} -232 q^{-167} -192 q^{-169} +37 q^{-171} +268 q^{-173} +258 q^{-175} -16 q^{-177} -292 q^{-179} -309 q^{-181} -38 q^{-183} +295 q^{-185} +368 q^{-187} +106 q^{-189} -254 q^{-191} -395 q^{-193} -189 q^{-195} +181 q^{-197} +388 q^{-199} +258 q^{-201} -75 q^{-203} -334 q^{-205} -308 q^{-207} -38 q^{-209} +241 q^{-211} +304 q^{-213} +128 q^{-215} -122 q^{-217} -258 q^{-219} -188 q^{-221} +15 q^{-223} +178 q^{-225} +193 q^{-227} +66 q^{-229} -84 q^{-231} -159 q^{-233} -110 q^{-235} +14 q^{-237} +102 q^{-239} +103 q^{-241} +35 q^{-243} -44 q^{-245} -79 q^{-247} -49 q^{-249} +8 q^{-251} +44 q^{-253} +42 q^{-255} +10 q^{-257} -17 q^{-259} -26 q^{-261} -15 q^{-263} +5 q^{-265} +13 q^{-267} +8 q^{-269} -2 q^{-273} -5 q^{-275} -2 q^{-277} +3 q^{-279} + q^{-281} - q^{-283} - q^{-289} + q^{-291} + q^{-293} - q^{-295} }

A2 Invariants.

Weight	Invariant
1,0	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{-6} - q^{-8} + q^{-10} +3 q^{-16} + q^{-18} +2 q^{-20} - q^{-24} -2 q^{-28} - q^{-34} }
1,1	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{-12} -2 q^{-14} +4 q^{-16} -8 q^{-18} +17 q^{-20} -20 q^{-22} +32 q^{-24} -42 q^{-26} +56 q^{-28} -58 q^{-30} +70 q^{-32} -70 q^{-34} +69 q^{-36} -46 q^{-38} +30 q^{-40} -40 q^{-44} +74 q^{-46} -114 q^{-48} +130 q^{-50} -153 q^{-52} +148 q^{-54} -140 q^{-56} +120 q^{-58} -87 q^{-60} +52 q^{-62} -12 q^{-64} -18 q^{-66} +47 q^{-68} -72 q^{-70} +80 q^{-72} -78 q^{-74} +70 q^{-76} -62 q^{-78} +46 q^{-80} -30 q^{-82} +21 q^{-84} -12 q^{-86} +6 q^{-88} -2 q^{-90} + q^{-92} }
2,0	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{-12} - q^{-14} - q^{-16} +3 q^{-18} +2 q^{-20} -3 q^{-22} +5 q^{-26} +2 q^{-28} -3 q^{-30} + q^{-32} +5 q^{-34} - q^{-36} -2 q^{-38} +5 q^{-40} +3 q^{-42} + q^{-44} +2 q^{-46} +2 q^{-48} -3 q^{-50} -3 q^{-52} -3 q^{-56} -7 q^{-58} -2 q^{-60} +2 q^{-62} -2 q^{-64} -3 q^{-66} + q^{-68} +3 q^{-70} -2 q^{-74} + q^{-76} +2 q^{-78} + q^{-86} }

A3 Invariants.

Weight	Invariant
0,1,0	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{-12} - q^{-14} +2 q^{-18} -3 q^{-20} + q^{-22} +7 q^{-24} -3 q^{-26} + q^{-28} +9 q^{-30} -2 q^{-32} - q^{-34} +7 q^{-36} - q^{-38} - q^{-40} + q^{-44} -3 q^{-46} -6 q^{-48} +2 q^{-50} - q^{-52} -8 q^{-54} +3 q^{-56} +3 q^{-58} -6 q^{-60} +3 q^{-62} +2 q^{-64} -3 q^{-66} +2 q^{-68} + q^{-70} - q^{-72} + q^{-74} }
1,0,0	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{-9} - q^{-11} + q^{-13} - q^{-15} + q^{-17} +3 q^{-21} +2 q^{-23} +2 q^{-25} +2 q^{-27} -2 q^{-33} -2 q^{-37} - q^{-41} - q^{-45} }

A4 Invariants.

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

B2 Invariants.

Weight	Invariant
0,1	$q^{-12}-q^{-14}+2q^{-16}-4q^{-18}+5q^{-20}-5q^{-22}+7q^{-24}-5q^{-26}+7q^{-28}-3q^{-30}+2q^{-32}+3q^{-34}-5q^{-36}+9q^{-38}-11q^{-40}+12q^{-42}-13q^{-44}+11q^{-46}-10q^{-48}+6q^{-50}-3q^{-52}+3q^{-56}-5q^{-58}+6q^{-60}-7q^{-62}+6q^{-64}-5q^{-66}+4q^{-68}-3q^{-70}+q^{-72}-q^{-74}$
1,0	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{-18} - q^{-22} - q^{-24} + q^{-26} +3 q^{-28} -4 q^{-32} -2 q^{-34} +4 q^{-36} +7 q^{-38} -5 q^{-42} -3 q^{-44} +6 q^{-46} +7 q^{-48} - q^{-50} -6 q^{-52} + q^{-54} +6 q^{-56} +3 q^{-58} -4 q^{-60} -2 q^{-62} +4 q^{-64} +4 q^{-66} -3 q^{-68} -5 q^{-70} + q^{-72} +3 q^{-74} -2 q^{-76} -6 q^{-78} - q^{-80} +4 q^{-82} + q^{-84} -6 q^{-86} -6 q^{-88} +3 q^{-90} +7 q^{-92} -7 q^{-96} -4 q^{-98} +5 q^{-100} +5 q^{-102} - q^{-104} -4 q^{-106} - q^{-108} +3 q^{-110} +2 q^{-112} - q^{-114} - q^{-116} + q^{-120} }

D4 Invariants.

Weight

Invariant

1,0,0,0

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{-18} - q^{-20} + q^{-22} -2 q^{-24} +3 q^{-26} -4 q^{-28} +4 q^{-30} -3 q^{-32} +7 q^{-34} -3 q^{-36} +6 q^{-38} - q^{-40} +8 q^{-42} + q^{-44} + q^{-46} +3 q^{-48} -2 q^{-50} +7 q^{-52} -7 q^{-54} +7 q^{-56} -10 q^{-58} +10 q^{-60} -10 q^{-62} +7 q^{-64} -11 q^{-66} +4 q^{-68} -6 q^{-70} + q^{-72} -4 q^{-74} -2 q^{-76} +2 q^{-78} -3 q^{-80} +4 q^{-82} -5 q^{-84} +6 q^{-86} -5 q^{-88} +4 q^{-90} -4 q^{-92} +4 q^{-94} -2 q^{-96} +2 q^{-98} - q^{-100} + q^{-102} }

G2 Invariants.

Weight

Invariant

1,0

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{-30} - q^{-32} +2 q^{-34} -3 q^{-36} +2 q^{-38} - q^{-40} -2 q^{-42} +7 q^{-44} -8 q^{-46} +11 q^{-48} -9 q^{-50} +4 q^{-52} +4 q^{-54} -12 q^{-56} +19 q^{-58} -20 q^{-60} +17 q^{-62} -8 q^{-64} -4 q^{-66} +18 q^{-68} -22 q^{-70} +23 q^{-72} -13 q^{-74} + q^{-76} +10 q^{-78} -15 q^{-80} +13 q^{-82} - q^{-84} -8 q^{-86} +22 q^{-88} -18 q^{-90} +6 q^{-92} +13 q^{-94} -27 q^{-96} +36 q^{-98} -32 q^{-100} +16 q^{-102} +4 q^{-104} -21 q^{-106} +34 q^{-108} -36 q^{-110} +24 q^{-112} -9 q^{-114} -11 q^{-116} +18 q^{-118} -22 q^{-120} +14 q^{-122} -2 q^{-124} -10 q^{-126} +15 q^{-128} -14 q^{-130} +2 q^{-132} +12 q^{-134} -24 q^{-136} +24 q^{-138} -17 q^{-140} + q^{-142} +13 q^{-144} -23 q^{-146} +26 q^{-148} -20 q^{-150} +9 q^{-152} +2 q^{-154} -12 q^{-156} +14 q^{-158} -12 q^{-160} +9 q^{-162} -3 q^{-164} - q^{-166} +3 q^{-168} -4 q^{-170} +3 q^{-172} - q^{-174} + q^{-176} }

.

Computer Talk

The above data is available with the Mathematica package KnotTheory`, as shown in the (simulated) Mathematica session below. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting. This Mathematica session is also available (albeit only for the knot 5_2) as the notebook PolynomialInvariantsSession.nb.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of August 31, 2006, 11:25:27.5625. 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]=

$4t^{2}-9t+11-9t^{-1}+4t^{-2}$

In[5]:=

Conway[K][z]

Out[5]=

$4z^{4}+7z^{2}+1$

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]=

$-q^{11}+2q^{10}-4q^{9}+5q^{8}-6q^{7}+7q^{6}-5q^{5}+4q^{4}-2q^{3}+q^{2}$

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]=

$z^{8}a^{-8}+z^{8}a^{-10}+2z^{7}a^{-7}+4z^{7}a^{-9}+2z^{7}a^{-11}+3z^{6}a^{-6}+z^{6}a^{-8}+2z^{6}a^{-12}+2z^{5}a^{-5}-2z^{5}a^{-7}-9z^{5}a^{-9}-4z^{5}a^{-11}+z^{5}a^{-13}+z^{4}a^{-4}-7z^{4}a^{-6}-4z^{4}a^{-8}-z^{4}a^{-10}-5z^{4}a^{-12}-3z^{3}a^{-5}+z^{3}a^{-7}+9z^{3}a^{-9}+2z^{3}a^{-11}-3z^{3}a^{-13}-2z^{2}a^{-4}+8z^{2}a^{-6}+6z^{2}a^{-8}-2z^{2}a^{-10}+2z^{2}a^{-12}+za^{-7}-3za^{-9}-2za^{-11}+2za^{-13}-3a^{-6}-a^{-8}+a^{-10}$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, Failed to parse (SVG (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\leftrightarrow q^{-1}} ): {}

Computer Talk

The above data is available with the Mathematica package KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.

(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 May 31, 2006, 14:15:20.091. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["9 13"];

In[4]:=

{A = Alexander[K][t], J = Jones[K][q]}

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

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

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} } , Failed to parse (SVG (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[5]:=

DeleteCases[Select[AllKnots[], (A === Alexander[#][t]) &], K]

KnotTheory::loading: Loading precomputed data in DTCode4KnotsTo11`.

KnotTheory::credits: The GaussCode to PD conversion was written by Siddarth Sankaran at the University of Toronto in the summer of 2005.

Out[5]=

{}

In[6]:=

DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]

KnotTheory::loading: Loading precomputed data in Jones4Knots11`.

Out[6]=

{}

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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 28} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 144} $392$ 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}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 10368} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{82040}{3}} ${\frac {13384}{3}}$ 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j} , alternation over Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r} ). The squares with yellow highlighting are those on the "critical diagonals", where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j-2r=s+1} or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j-2r=s-1} , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s=} 4 is the signature of 9 13. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ 0 1 2 3 4 5 6 7 8 9 χ 23                   1 -1 21                 1   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)

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

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	Failed to parse (SVG (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{NotAvailable}(q)}
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 \textrm{NotAvailable}(q)}
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 \textrm{NotAvailable}(q)}

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session, or any of the Computer Talk sections above.

Modifying This Page

Read me first: Modifying Knot Pages See/edit the Rolfsen Knot Page master template (intermediate). See/edit the Rolfsen_Splice_Base (expert). Back to the top.

9_12 9_14