9 15

9_14

9_16

(KnotPlot image)	See the full Rolfsen Knot Table. Visit 9 15's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 15 at Knotilus!

Knot presentations

Planar diagram presentation	X1425 X7,10,8,11 X3948 X9,3,10,2 X13,17,14,16 X5,15,6,14 X15,7,16,6 X11,1,12,18 X17,13,18,12
Gauss code	-1, 4, -3, 1, -6, 7, -2, 3, -4, 2, -8, 9, -5, 6, -7, 5, -9, 8
Dowker-Thistlethwaite code	4 8 14 10 2 18 16 6 12
Conway Notation	[2322]

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

[edit Notes on presentations of 9 15]

Computer Talk[show]

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 15"];

In[4]:=

PD[K]

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

Out[4]=

X1425 X7,10,8,11 X3948 X9,3,10,2 X13,17,14,16 X5,15,6,14 X15,7,16,6 X11,1,12,18 X17,13,18,12

In[5]:=

GaussCode[K]

Out[5]=

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

In[6]:=

DTCode[K]

Out[6]=

4 8 14 10 2 18 16 6 12

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[2322]

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]=

${\textrm {BR}}(5,\{1,1,1,2,-1,-3,2,4,-3,4\})$

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]=

{ 5, 10, 5 }

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

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 2 Bridge index 2 Super bridge index $\{4,5\}$ Nakanishi index 1 Maximal Thurston-Bennequin number [-1][-10] Hyperbolic Volume 9.8855 A-Polynomial See Data:9 15/A-polynomial

[edit Notes for 9 15's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus $1$ Topological 4 genus $1$ Concordance genus $2$ Rasmussen s-Invariant 2

[edit Notes for 9 15's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-2t^{2}+10t-15+10t^{-1}-2t^{-2}$
Conway polynomial	$-2z^{4}+2z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 39, 2 }
Jones polynomial	$-q^{8}+2q^{7}-4q^{6}+6q^{5}-6q^{4}+7q^{3}-6q^{2}+4q-2+q^{-1}$
HOMFLY-PT polynomial (db, data sources)	$-z^{4}a^{-2}-z^{4}a^{-4}-z^{2}a^{-2}+2z^{2}a^{-6}+z^{2}-a^{-2}+a^{-4}+a^{-6}-a^{-8}+1$
Kauffman polynomial (db, data sources)	$z^{8}a^{-4}+z^{8}a^{-6}+2z^{7}a^{-3}+4z^{7}a^{-5}+2z^{7}a^{-7}+2z^{6}a^{-2}+z^{6}a^{-4}+z^{6}a^{-6}+2z^{6}a^{-8}+2z^{5}a^{-1}-z^{5}a^{-3}-7z^{5}a^{-5}-3z^{5}a^{-7}+z^{5}a^{-9}-4z^{4}a^{-6}-5z^{4}a^{-8}+z^{4}-3z^{3}a^{-1}+5z^{3}a^{-5}-z^{3}a^{-7}-3z^{3}a^{-9}-3z^{2}a^{-2}-2z^{2}a^{-4}+2z^{2}a^{-6}+3z^{2}a^{-8}-2z^{2}+za^{-1}+za^{-3}-za^{-5}+za^{-7}+2za^{-9}+a^{-2}+a^{-4}-a^{-6}-a^{-8}+1$
The A2 invariant	$q^{4}+2q^{-2}-2q^{-4}+2q^{-12}+2q^{-16}-q^{-20}+q^{-22}-q^{-24}-q^{-26}$
The G2 invariant	$q^{18}-q^{16}+3q^{14}-3q^{12}+2q^{10}-3q^{6}+8q^{4}-9q^{2}+11-9q^{-2}+5q^{-4}+4q^{-6}-13q^{-8}+23q^{-10}-26q^{-12}+21q^{-14}-13q^{-16}-5q^{-18}+20q^{-20}-31q^{-22}+33q^{-24}-20q^{-26}+2q^{-28}+14q^{-30}-23q^{-32}+15q^{-34}-2q^{-36}-13q^{-38}+24q^{-40}-23q^{-42}+9q^{-44}+17q^{-46}-34q^{-48}+46q^{-50}-42q^{-52}+21q^{-54}+6q^{-56}-28q^{-58}+43q^{-60}-44q^{-62}+36q^{-64}-11q^{-66}-11q^{-68}+26q^{-70}-30q^{-72}+20q^{-74}-q^{-76}-15q^{-78}+21q^{-80}-15q^{-82}+3q^{-84}+18q^{-86}-31q^{-88}+33q^{-90}-23q^{-92}+18q^{-96}-32q^{-98}+33q^{-100}-24q^{-102}+10q^{-104}+3q^{-106}-15q^{-108}+17q^{-110}-15q^{-112}+9q^{-114}-3q^{-116}-2q^{-118}+3q^{-120}-4q^{-122}+3q^{-124}-q^{-126}+q^{-128}$

Further Quantum Invariants[show]

Further quantum knot invariants for 9_15.

A1 Invariants.

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

A2 Invariants.

Weight	Invariant
1,0	$q^{4}+2q^{-2}-2q^{-4}+2q^{-12}+2q^{-16}-q^{-20}+q^{-22}-q^{-24}-q^{-26}$
1,1	$q^{12}-2q^{10}+6q^{8}-10q^{6}+17q^{4}-26q^{2}+36-52q^{-2}+68q^{-4}-82q^{-6}+98q^{-8}-102q^{-10}+106q^{-12}-82q^{-14}+52q^{-16}-8q^{-18}-47q^{-20}+102q^{-22}-156q^{-24}+194q^{-26}-217q^{-28}+220q^{-30}-204q^{-32}+174q^{-34}-126q^{-36}+72q^{-38}-12q^{-40}-38q^{-42}+78q^{-44}-108q^{-46}+118q^{-48}-116q^{-50}+98q^{-52}-80q^{-54}+58q^{-56}-38q^{-58}+23q^{-60}-12q^{-62}+6q^{-64}-2q^{-66}+q^{-68}$
2,0	$q^{12}-q^{8}+2q^{4}-4+7q^{-4}-q^{-6}-6q^{-8}+3q^{-10}+7q^{-12}+q^{-14}-4q^{-16}+3q^{-18}+3q^{-20}-3q^{-22}-q^{-24}-3q^{-28}-q^{-30}+5q^{-32}-q^{-34}-2q^{-36}+3q^{-38}+6q^{-40}-2q^{-42}-6q^{-44}+2q^{-46}+3q^{-48}-3q^{-50}-5q^{-52}+3q^{-56}-2q^{-60}+q^{-64}+q^{-66}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{8}-q^{6}+q^{4}+3q^{2}-3+6q^{-4}-6q^{-6}-2q^{-8}+9q^{-10}-5q^{-12}-2q^{-14}+6q^{-16}-2q^{-18}-2q^{-20}+q^{-22}+4q^{-24}-2q^{-28}+6q^{-30}+3q^{-32}-8q^{-34}+4q^{-36}+2q^{-38}-9q^{-40}+2q^{-42}+q^{-44}-4q^{-46}+2q^{-48}+q^{-50}-q^{-52}+q^{-54}$
1,0,0	$q^{5}+q+2q^{-3}-2q^{-5}-q^{-9}+q^{-15}+2q^{-17}+2q^{-21}+q^{-25}-q^{-27}+q^{-29}-q^{-31}-q^{-33}-q^{-35}$

B2 Invariants.

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

G2 Invariants.

Weight

Invariant

1,0

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

.

Computer Talk[show]

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 15"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-2t^{2}+10t-15+10t^{-1}-2t^{-2}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 39, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

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

"Similar" Knots (within the Atlas)

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

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

Computer Talk[show]

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 15"];

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]=

{ $-2t^{2}+10t-15+10t^{-1}-2t^{-2}$, $-q^{8}+2q^{7}-4q^{6}+6q^{5}-6q^{4}+7q^{3}-6q^{2}+4q-2+q^{-1}$ }

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]=

{10_165, K11n63, K11n101,}

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:

(2, 5)

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 $8$ $40$ $32$ ${\frac {508}{3}}$ ${\frac {92}{3}}$ $320$ ${\frac {2416}{3}}$ ${\frac {352}{3}}$ $168$ ${\frac {256}{3}}$ $800$ ${\frac {4064}{3}}$ ${\frac {736}{3}}$ ${\frac {59071}{15}}$ $-{\frac {1628}{5}}$ ${\frac {92164}{45}}$ ${\frac {881}{9}}$ ${\frac {4351}{15}}$

V_2,1 through V_6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V₂ and V₃.

Khovanov Homology

The coefficients of the monomials $t^{r}q^{j}$ are shown, along with their alternating sums $\chi$ (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=$2 is the signature of 9 15. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -2 -1 0 1 2 3 4 5 6 7 χ 17                   1 -1 15                 1   1 13               3 1   -2 11             3 1     2 9           3 3       0 7         4 3         1 5       2 3           1 3     2 4             -2 1   1 3               2 -1   1                 -1 -3 1                   1

Integral Khovanov Homology (db, data source)

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

The Coloured Jones Polynomials

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

$n$	$J_{n}$
2	$q^{23}-2q^{22}+6q^{20}-8q^{19}-4q^{18}+19q^{17}-14q^{16}-15q^{15}+35q^{14}-15q^{13}-28q^{12}+45q^{11}-12q^{10}-36q^{9}+45q^{8}-7q^{7}-33q^{6}+33q^{5}-q^{4}-21q^{3}+16q^{2}+q-8+5q^{-1}-2q^{-3}+q^{-4}$
3	$-q^{45}+2q^{44}-2q^{42}-3q^{41}+7q^{40}+5q^{39}-10q^{38}-14q^{37}+16q^{36}+24q^{35}-14q^{34}-44q^{33}+13q^{32}+60q^{31}-q^{30}-79q^{29}-17q^{28}+97q^{27}+35q^{26}-105q^{25}-60q^{24}+116q^{23}+76q^{22}-113q^{21}-100q^{20}+118q^{19}+106q^{18}-107q^{17}-119q^{16}+101q^{15}+116q^{14}-82q^{13}-114q^{12}+66q^{11}+102q^{10}-46q^{9}-84q^{8}+28q^{7}+64q^{6}-13q^{5}-46q^{4}+8q^{3}+26q^{2}-2q-16+3q^{-1}+7q^{-2}-q^{-3}-5q^{-4}+3q^{-5}+q^{-6}-2q^{-8}+q^{-9}$
4	$q^{74}-2q^{73}+2q^{71}-q^{70}+4q^{69}-9q^{68}-q^{67}+10q^{66}+14q^{64}-30q^{63}-15q^{62}+22q^{61}+13q^{60}+54q^{59}-58q^{58}-59q^{57}+3q^{56}+23q^{55}+152q^{54}-49q^{53}-112q^{52}-78q^{51}-26q^{50}+280q^{49}+35q^{48}-110q^{47}-199q^{46}-167q^{45}+374q^{44}+169q^{43}-25q^{42}-296q^{41}-358q^{40}+392q^{39}+292q^{38}+113q^{37}-343q^{36}-535q^{35}+358q^{34}+372q^{33}+243q^{32}-347q^{31}-651q^{30}+298q^{29}+401q^{28}+339q^{27}-311q^{26}-694q^{25}+215q^{24}+373q^{23}+392q^{22}-224q^{21}-649q^{20}+106q^{19}+278q^{18}+386q^{17}-101q^{16}-507q^{15}+12q^{14}+135q^{13}+302q^{12}+9q^{11}-307q^{10}-26q^{9}+15q^{8}+174q^{7}+49q^{6}-138q^{5}-8q^{4}-31q^{3}+66q^{2}+33q-47+13q^{-1}-24q^{-2}+16q^{-3}+10q^{-4}-17q^{-5}+14q^{-6}-8q^{-7}+3q^{-8}+q^{-9}-7q^{-10}+6q^{-11}-q^{-12}+q^{-13}-2q^{-15}+q^{-16}$
5	$-q^{110}+2q^{109}-2q^{107}+q^{106}-2q^{104}+5q^{103}+2q^{102}-9q^{101}-3q^{100}+3q^{99}+2q^{98}+16q^{97}+9q^{96}-20q^{95}-29q^{94}-12q^{93}+11q^{92}+50q^{91}+54q^{90}-11q^{89}-71q^{88}-92q^{87}-40q^{86}+80q^{85}+160q^{84}+104q^{83}-44q^{82}-203q^{81}-234q^{80}-35q^{79}+234q^{78}+343q^{77}+197q^{76}-177q^{75}-483q^{74}-406q^{73}+65q^{72}+552q^{71}+644q^{70}+162q^{69}-568q^{68}-898q^{67}-440q^{66}+505q^{65}+1102q^{64}+761q^{63}-335q^{62}-1276q^{61}-1109q^{60}+146q^{59}+1372q^{58}+1410q^{57}+124q^{56}-1421q^{55}-1719q^{54}-342q^{53}+1418q^{52}+1924q^{51}+616q^{50}-1401q^{49}-2136q^{48}-787q^{47}+1341q^{46}+2242q^{45}+1010q^{44}-1285q^{43}-2365q^{42}-1124q^{41}+1188q^{40}+2378q^{39}+1299q^{38}-1078q^{37}-2400q^{36}-1382q^{35}+916q^{34}+2309q^{33}+1499q^{32}-720q^{31}-2188q^{30}-1539q^{29}+489q^{28}+1958q^{27}+1549q^{26}-241q^{25}-1669q^{24}-1482q^{23}+q^{22}+1332q^{21}+1338q^{20}+193q^{19}-970q^{18}-1129q^{17}-328q^{16}+630q^{15}+900q^{14}+368q^{13}-357q^{12}-632q^{11}-356q^{10}+144q^{9}+423q^{8}+291q^{7}-39q^{6}-228q^{5}-209q^{4}-32q^{3}+120q^{2}+128q+39-38q^{-1}-71q^{-2}-41q^{-3}+17q^{-4}+26q^{-5}+19q^{-6}+13q^{-7}-13q^{-8}-17q^{-9}+2q^{-10}-2q^{-11}-2q^{-12}+12q^{-13}+2q^{-14}-6q^{-15}+3q^{-16}-3q^{-17}-5q^{-18}+4q^{-19}+2q^{-20}-q^{-21}+q^{-22}-2q^{-24}+q^{-25}$
6	$q^{153}-2q^{152}+2q^{150}-q^{149}-2q^{147}+6q^{146}-6q^{145}-3q^{144}+11q^{143}-q^{142}-2q^{141}-13q^{140}+12q^{139}-14q^{138}-8q^{137}+36q^{136}+14q^{135}+4q^{134}-42q^{133}+9q^{132}-56q^{131}-40q^{130}+78q^{129}+73q^{128}+74q^{127}-47q^{126}+18q^{125}-169q^{124}-189q^{123}+32q^{122}+127q^{121}+245q^{120}+106q^{119}+225q^{118}-239q^{117}-465q^{116}-296q^{115}-103q^{114}+287q^{113}+379q^{112}+870q^{111}+139q^{110}-498q^{109}-797q^{108}-869q^{107}-338q^{106}+251q^{105}+1722q^{104}+1220q^{103}+356q^{102}-797q^{101}-1808q^{100}-1854q^{99}-973q^{98}+1961q^{97}+2517q^{96}+2241q^{95}+419q^{94}-2012q^{93}-3651q^{92}-3328q^{91}+884q^{90}+3095q^{89}+4451q^{88}+2782q^{87}-882q^{86}-4793q^{85}-6017q^{84}-1336q^{83}+2457q^{82}+6087q^{81}+5471q^{80}+1273q^{79}-4876q^{78}-8170q^{77}-3863q^{76}+959q^{75}+6803q^{74}+7686q^{73}+3621q^{72}-4237q^{71}-9459q^{70}-5964q^{69}-679q^{68}+6860q^{67}+9134q^{66}+5537q^{65}-3414q^{64}-10060q^{63}-7406q^{62}-2014q^{61}+6618q^{60}+9940q^{59}+6889q^{58}-2632q^{57}-10192q^{56}-8327q^{55}-3070q^{54}+6131q^{53}+10254q^{52}+7867q^{51}-1723q^{50}-9805q^{49}-8855q^{48}-4092q^{47}+5144q^{46}+9955q^{45}+8556q^{44}-408q^{43}-8580q^{42}-8795q^{41}-5116q^{40}+3405q^{39}+8681q^{38}+8653q^{37}+1237q^{36}-6307q^{35}-7713q^{34}-5715q^{33}+1143q^{32}+6283q^{31}+7664q^{30}+2591q^{29}-3415q^{28}-5504q^{27}-5290q^{26}-813q^{25}+3352q^{24}+5548q^{23}+2923q^{22}-933q^{21}-2863q^{20}-3801q^{19}-1647q^{18}+976q^{17}+3078q^{16}+2182q^{15}+326q^{14}-831q^{13}-2006q^{12}-1377q^{11}-174q^{10}+1227q^{9}+1096q^{8}+480q^{7}+99q^{6}-724q^{5}-722q^{4}-356q^{3}+333q^{2}+346q+231+252q^{-1}-159q^{-2}-254q^{-3}-206q^{-4}+67q^{-5}+50q^{-6}+41q^{-7}+159q^{-8}-11q^{-9}-62q^{-10}-79q^{-11}+18q^{-12}-10q^{-13}-17q^{-14}+69q^{-15}+7q^{-16}-8q^{-17}-25q^{-18}+10q^{-19}-10q^{-20}-18q^{-21}+24q^{-22}+3q^{-23}+3q^{-24}-7q^{-25}+5q^{-26}-3q^{-27}-9q^{-28}+6q^{-29}+2q^{-31}-q^{-32}+q^{-33}-2q^{-35}+q^{-36}$

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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