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]=

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

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

Further Quantum Invariants

Further quantum knot invariants for 9_13.

A1 Invariants.

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

A2 Invariants.

Weight	Invariant
1,0	$q^{-6} - q^{-8} + q^{-10} +3 q^{-16} + q^{-18} +2 q^{-20} - q^{-24} -2 q^{-28} - q^{-34}$
1,1	$q^{-12} -2 q^{-14} +4 q^{-16} -8 q^{-18} +17 q^{-20} -20 q^{-22} +32 q^{-24} -42 q^{-26} +56 q^{-28} -58 q^{-30} +70 q^{-32} -70 q^{-34} +69 q^{-36} -46 q^{-38} +30 q^{-40} -40 q^{-44} +74 q^{-46} -114 q^{-48} +130 q^{-50} -153 q^{-52} +148 q^{-54} -140 q^{-56} +120 q^{-58} -87 q^{-60} +52 q^{-62} -12 q^{-64} -18 q^{-66} +47 q^{-68} -72 q^{-70} +80 q^{-72} -78 q^{-74} +70 q^{-76} -62 q^{-78} +46 q^{-80} -30 q^{-82} +21 q^{-84} -12 q^{-86} +6 q^{-88} -2 q^{-90} + q^{-92}$
2,0	$q^{-12} - q^{-14} - q^{-16} +3 q^{-18} +2 q^{-20} -3 q^{-22} +5 q^{-26} +2 q^{-28} -3 q^{-30} + q^{-32} +5 q^{-34} - q^{-36} -2 q^{-38} +5 q^{-40} +3 q^{-42} + q^{-44} +2 q^{-46} +2 q^{-48} -3 q^{-50} -3 q^{-52} -3 q^{-56} -7 q^{-58} -2 q^{-60} +2 q^{-62} -2 q^{-64} -3 q^{-66} + q^{-68} +3 q^{-70} -2 q^{-74} + q^{-76} +2 q^{-78} + q^{-86}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{-12} - q^{-14} +2 q^{-18} -3 q^{-20} + q^{-22} +7 q^{-24} -3 q^{-26} + q^{-28} +9 q^{-30} -2 q^{-32} - q^{-34} +7 q^{-36} - q^{-38} - q^{-40} + q^{-44} -3 q^{-46} -6 q^{-48} +2 q^{-50} - q^{-52} -8 q^{-54} +3 q^{-56} +3 q^{-58} -6 q^{-60} +3 q^{-62} +2 q^{-64} -3 q^{-66} +2 q^{-68} + q^{-70} - q^{-72} + q^{-74}$
1,0,0	$q^{-9} - q^{-11} + q^{-13} - q^{-15} + q^{-17} +3 q^{-21} +2 q^{-23} +2 q^{-25} +2 q^{-27} -2 q^{-33} -2 q^{-37} - q^{-41} - q^{-45}$

A4 Invariants.

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

B2 Invariants.

Weight	Invariant
0,1	$q^{-12} - q^{-14} +2 q^{-16} -4 q^{-18} +5 q^{-20} -5 q^{-22} +7 q^{-24} -5 q^{-26} +7 q^{-28} -3 q^{-30} +2 q^{-32} +3 q^{-34} -5 q^{-36} +9 q^{-38} -11 q^{-40} +12 q^{-42} -13 q^{-44} +11 q^{-46} -10 q^{-48} +6 q^{-50} -3 q^{-52} +3 q^{-56} -5 q^{-58} +6 q^{-60} -7 q^{-62} +6 q^{-64} -5 q^{-66} +4 q^{-68} -3 q^{-70} + q^{-72} - q^{-74}$
1,0	$q^{-18} - q^{-22} - q^{-24} + q^{-26} +3 q^{-28} -4 q^{-32} -2 q^{-34} +4 q^{-36} +7 q^{-38} -5 q^{-42} -3 q^{-44} +6 q^{-46} +7 q^{-48} - q^{-50} -6 q^{-52} + q^{-54} +6 q^{-56} +3 q^{-58} -4 q^{-60} -2 q^{-62} +4 q^{-64} +4 q^{-66} -3 q^{-68} -5 q^{-70} + q^{-72} +3 q^{-74} -2 q^{-76} -6 q^{-78} - q^{-80} +4 q^{-82} + q^{-84} -6 q^{-86} -6 q^{-88} +3 q^{-90} +7 q^{-92} -7 q^{-96} -4 q^{-98} +5 q^{-100} +5 q^{-102} - q^{-104} -4 q^{-106} - q^{-108} +3 q^{-110} +2 q^{-112} - q^{-114} - q^{-116} + q^{-120}$

D4 Invariants.

Weight

Invariant

1,0,0,0

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

G2 Invariants.

Weight

Invariant

1,0

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

. </div></div>

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]=

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

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

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

In[9]:=

HOMFLYPT[K][a, z]

KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.

Out[10]=

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

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

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

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]=

{ $4 t^2-9 t+11-9 t^{-1} +4 t^{-2}$, $-q^{11}+2 q^{10}-4 q^9+5 q^8-6 q^7+7 q^6-5 q^5+4 q^4-2 q^3+q^2$ }

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 $28$ $144$ $392$ $\frac{2930}{3}$ $\frac{478}{3}$ $4032$ $7264$ $1280$ $1040$ $\frac{10976}{3}$ $10368$ $\frac{82040}{3}$ $\frac{13384}{3}$ $\frac{1644937}{30}$ $\frac{9566}{15}$ $\frac{1018754}{45}$ $\frac{7223}{18}$ $\frac{91657}{30}$

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 $t^rq^j$ 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)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=3$ $i=5$ $r=0$ ${\mathbb Z}$ ${\mathbb Z}$ $r=1$ ${\mathbb Z}^{2}$ $r=2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=3$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=4$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=5$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=6$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=7$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=8$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=9$ ${\mathbb Z}_2$ ${\mathbb Z}$

The Coloured Jones Polynomials

The Coloured Jones Polynomials (in the $n+1$-dimensional representation of $sl(2)$)

<p>

$n$	$J_n$
2	$q^{31}-2 q^{30}+6 q^{28}-7 q^{27}-4 q^{26}+17 q^{25}-12 q^{24}-13 q^{23}+29 q^{22}-13 q^{21}-24 q^{20}+38 q^{19}-10 q^{18}-31 q^{17}+39 q^{16}-6 q^{15}-28 q^{14}+28 q^{13}-q^{12}-18 q^{11}+14 q^{10}+q^9-8 q^8+5 q^7+q^6-2 q^5+q^4$
3	$-q^{60}+2 q^{59}-2 q^{57}-3 q^{56}+6 q^{55}+5 q^{54}-8 q^{53}-13 q^{52}+13 q^{51}+20 q^{50}-10 q^{49}-36 q^{48}+11 q^{47}+46 q^{46}-61 q^{44}-9 q^{43}+69 q^{42}+26 q^{41}-79 q^{40}-40 q^{39}+83 q^{38}+55 q^{37}-85 q^{36}-71 q^{35}+87 q^{34}+77 q^{33}-79 q^{32}-89 q^{31}+79 q^{30}+82 q^{29}-60 q^{28}-85 q^{27}+52 q^{26}+71 q^{25}-33 q^{24}-63 q^{23}+24 q^{22}+45 q^{21}-9 q^{20}-36 q^{19}+7 q^{18}+21 q^{17}-16 q^{15}+2 q^{14}+8 q^{13}+q^{12}-6 q^{11}+q^{10}+2 q^9+q^8-2 q^7+q^6$
4	$q^{98}-2 q^{97}+2 q^{95}-q^{94}+4 q^{93}-8 q^{92}-q^{91}+8 q^{90}-q^{89}+14 q^{88}-25 q^{87}-12 q^{86}+17 q^{85}+8 q^{84}+45 q^{83}-48 q^{82}-43 q^{81}+6 q^{80}+15 q^{79}+115 q^{78}-48 q^{77}-81 q^{76}-44 q^{75}-15 q^{74}+202 q^{73}-q^{72}-80 q^{71}-116 q^{70}-105 q^{69}+262 q^{68}+78 q^{67}-24 q^{66}-173 q^{65}-227 q^{64}+275 q^{63}+149 q^{62}+65 q^{61}-202 q^{60}-340 q^{59}+259 q^{58}+197 q^{57}+148 q^{56}-207 q^{55}-419 q^{54}+227 q^{53}+219 q^{52}+209 q^{51}-189 q^{50}-450 q^{49}+178 q^{48}+205 q^{47}+241 q^{46}-136 q^{45}-418 q^{44}+103 q^{43}+147 q^{42}+238 q^{41}-58 q^{40}-324 q^{39}+36 q^{38}+60 q^{37}+186 q^{36}+11 q^{35}-196 q^{34}+5 q^{33}-11 q^{32}+109 q^{31}+36 q^{30}-92 q^{29}+8 q^{28}-33 q^{27}+47 q^{26}+25 q^{25}-38 q^{24}+13 q^{23}-22 q^{22}+18 q^{21}+10 q^{20}-17 q^{19}+9 q^{18}-9 q^{17}+7 q^{16}+4 q^{15}-7 q^{14}+3 q^{13}-2 q^{12}+2 q^{11}+q^{10}-2 q^9+q^8$

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.

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