8 8

8_7

8_9

(KnotPlot image)	See the full Rolfsen Knot Table. Visit 8 8's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 8 8 at Knotilus!

Knot presentations

Planar diagram presentation	X1425 X3849 X11,15,12,14 X5,13,6,12 X13,7,14,6 X9,1,10,16 X15,11,16,10 X7283
Gauss code	-1, 8, -2, 1, -4, 5, -8, 2, -6, 7, -3, 4, -5, 3, -7, 6
Dowker-Thistlethwaite code	4 8 12 2 16 14 6 10
Conway Notation	[2312]

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

[edit Notes on presentations of 8 8]

Knot 8_8.

A graph, knot 8_8.

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["8 8"];

In[4]:=

PD[K]

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

Out[4]=

X1425 X3849 X11,15,12,14 X5,13,6,12 X13,7,14,6 X9,1,10,16 X15,11,16,10 X7283

In[5]:=

GaussCode[K]

Out[5]=

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

In[6]:=

DTCode[K]

Out[6]=

4 8 12 2 16 14 6 10

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[2312]

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]=

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

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, 9, 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[{10, 2}, {1, 8}, {4, 9}, {8, 10}, {3, 5}, {2, 4}, {6, 3}, {5, 7}, {9, 6}, {7, 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 [-4][-6] Hyperbolic Volume 7.80134 A-Polynomial See Data:8 8/A-polynomial

[edit Notes for 8 8's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus $0$ Topological 4 genus $0$ Concordance genus $0$ Rasmussen s-Invariant 0

[edit Notes for 8 8's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$2t^{2}-6t+9-6t^{-1}+2t^{-2}$
Conway polynomial	$2z^{4}+2z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 25, 0 }
Jones polynomial	$-q^{5}+2q^{4}-3q^{3}+4q^{2}-4q+5-3q^{-1}+2q^{-2}-q^{-3}$
HOMFLY-PT polynomial (db, data sources)	$z^{4}a^{-2}+z^{4}-a^{2}z^{2}+2z^{2}a^{-2}-z^{2}a^{-4}+2z^{2}-a^{2}+a^{-2}-a^{-4}+2$
Kauffman polynomial (db, data sources)	$z^{7}a^{-1}+z^{7}a^{-3}+4z^{6}a^{-2}+2z^{6}a^{-4}+2z^{6}+2az^{5}+z^{5}a^{-1}+z^{5}a^{-5}+2a^{2}z^{4}-9z^{4}a^{-2}-6z^{4}a^{-4}-z^{4}+a^{3}z^{3}-3z^{3}a^{-1}-5z^{3}a^{-3}-3z^{3}a^{-5}-2a^{2}z^{2}+5z^{2}a^{-2}+4z^{2}a^{-4}-z^{2}-a^{3}z-az+za^{-1}+3za^{-3}+2za^{-5}+a^{2}-a^{-2}-a^{-4}+2$
The A2 invariant	$-q^{10}-q^{4}+2q^{2}+1+2q^{-2}+q^{-4}+q^{-8}-q^{-10}-q^{-16}$
The G2 invariant	$q^{52}-q^{50}+2q^{48}-2q^{46}-3q^{40}+4q^{38}-5q^{36}+4q^{34}-5q^{32}+q^{30}+3q^{28}-6q^{26}+8q^{24}-9q^{22}+7q^{20}-4q^{18}-3q^{16}+7q^{14}-8q^{12}+9q^{10}-q^{8}-3q^{6}+6q^{4}-3q^{2}+1+6q^{-2}-10q^{-4}+12q^{-6}-5q^{-8}+10q^{-12}-14q^{-14}+17q^{-16}-11q^{-18}+3q^{-20}+4q^{-22}-9q^{-24}+14q^{-26}-10q^{-28}+6q^{-30}+q^{-32}-5q^{-34}+7q^{-36}-5q^{-38}+4q^{-42}-8q^{-44}+7q^{-46}-3q^{-48}-4q^{-50}+10q^{-52}-13q^{-54}+10q^{-56}-5q^{-58}-4q^{-60}+7q^{-62}-10q^{-64}+9q^{-66}-5q^{-68}+2q^{-72}-4q^{-74}+3q^{-76}-q^{-78}+q^{-80}$

Further Quantum Invariants

Further quantum knot invariants for 8_8.

A1 Invariants.

Weight	Invariant
1	$-q^{7}+q^{5}-q^{3}+2q+q^{-1}+q^{-5}-q^{-7}+q^{-9}-q^{-11}$
2	$q^{20}-q^{18}-q^{16}+2q^{14}-3q^{12}-q^{10}+5q^{8}-3q^{6}-2q^{4}+5q^{2}-q^{-2}+q^{-4}+3q^{-6}-2q^{-10}+3q^{-12}-5q^{-16}+2q^{-18}+2q^{-20}-4q^{-22}+q^{-24}+3q^{-26}-2q^{-28}-q^{-30}+q^{-32}$
3	$-q^{39}+q^{37}+q^{35}-q^{31}+q^{29}+q^{27}-4q^{25}-q^{23}+5q^{21}+2q^{19}-9q^{17}-4q^{15}+10q^{13}+7q^{11}-11q^{9}-7q^{7}+7q^{5}+11q^{3}-3q-7q^{-1}+q^{-3}+6q^{-5}+5q^{-7}-3q^{-9}-6q^{-11}+q^{-13}+9q^{-15}-q^{-17}-9q^{-19}-2q^{-21}+10q^{-23}+3q^{-25}-10q^{-27}-6q^{-29}+9q^{-31}+7q^{-33}-6q^{-35}-10q^{-37}+3q^{-39}+10q^{-41}-8q^{-45}-3q^{-47}+6q^{-49}+5q^{-51}-3q^{-53}-4q^{-55}+2q^{-59}+q^{-61}-q^{-63}$
4	$q^{64}-q^{62}-q^{60}-q^{56}+3q^{54}-q^{52}+q^{50}+2q^{48}-4q^{46}+q^{44}-4q^{42}+5q^{40}+10q^{38}-6q^{36}-8q^{34}-14q^{32}+9q^{30}+26q^{28}+q^{26}-17q^{24}-34q^{22}+3q^{20}+40q^{18}+18q^{16}-11q^{14}-43q^{12}-12q^{10}+30q^{8}+27q^{6}+10q^{4}-27q^{2}-21+6q^{-2}+18q^{-4}+18q^{-6}-3q^{-8}-16q^{-10}-15q^{-12}+2q^{-14}+19q^{-16}+14q^{-18}-10q^{-20}-25q^{-22}-3q^{-24}+20q^{-26}+24q^{-28}-8q^{-30}-33q^{-32}-8q^{-34}+19q^{-36}+31q^{-38}-3q^{-40}-35q^{-42}-17q^{-44}+9q^{-46}+37q^{-48}+12q^{-50}-26q^{-52}-24q^{-54}-9q^{-56}+29q^{-58}+24q^{-60}-4q^{-62}-19q^{-64}-26q^{-66}+8q^{-68}+21q^{-70}+13q^{-72}-q^{-74}-22q^{-76}-8q^{-78}+4q^{-80}+12q^{-82}+11q^{-84}-7q^{-86}-7q^{-88}-5q^{-90}+q^{-92}+6q^{-94}+q^{-96}-2q^{-100}-q^{-102}+q^{-104}$
5	$-q^{95}+q^{93}+q^{91}+q^{87}-q^{85}-3q^{83}-q^{81}+q^{79}+4q^{75}+3q^{73}-2q^{71}-6q^{69}-6q^{67}+9q^{63}+15q^{61}+7q^{59}-16q^{57}-27q^{55}-13q^{53}+18q^{51}+43q^{49}+34q^{47}-19q^{45}-68q^{43}-55q^{41}+10q^{39}+83q^{37}+87q^{35}+7q^{33}-94q^{31}-118q^{29}-36q^{27}+85q^{25}+139q^{23}+63q^{21}-64q^{19}-131q^{17}-93q^{15}+31q^{13}+120q^{11}+99q^{9}+3q^{7}-78q^{5}-93q^{3}-33q+47q^{-1}+77q^{-3}+48q^{-5}-9q^{-7}-50q^{-9}-53q^{-11}-20q^{-13}+30q^{-15}+53q^{-17}+35q^{-19}-10q^{-21}-54q^{-23}-52q^{-25}+3q^{-27}+59q^{-29}+57q^{-31}+2q^{-33}-62q^{-35}-71q^{-37}-3q^{-39}+74q^{-41}+77q^{-43}+9q^{-45}-78q^{-47}-92q^{-49}-16q^{-51}+80q^{-53}+103q^{-55}+31q^{-57}-73q^{-59}-114q^{-61}-50q^{-63}+57q^{-65}+111q^{-67}+73q^{-69}-30q^{-71}-106q^{-73}-89q^{-75}+82q^{-79}+96q^{-81}+36q^{-83}-52q^{-85}-89q^{-87}-58q^{-89}+15q^{-91}+68q^{-93}+67q^{-95}+21q^{-97}-38q^{-99}-63q^{-101}-41q^{-103}+7q^{-105}+42q^{-107}+45q^{-109}+19q^{-111}-19q^{-113}-37q^{-115}-28q^{-117}-q^{-119}+20q^{-121}+25q^{-123}+15q^{-125}-6q^{-127}-17q^{-129}-14q^{-131}-3q^{-133}+5q^{-135}+9q^{-137}+7q^{-139}-q^{-141}-4q^{-143}-3q^{-145}-q^{-147}+2q^{-151}+q^{-153}-q^{-155}$

A2 Invariants.

Weight	Invariant
1,0	$-q^{10}-q^{4}+2q^{2}+1+2q^{-2}+q^{-4}+q^{-8}-q^{-10}-q^{-16}$
1,1	$q^{28}-2q^{26}+4q^{24}-6q^{22}+9q^{20}-12q^{18}+14q^{16}-20q^{14}+19q^{12}-24q^{10}+22q^{8}-22q^{6}+18q^{4}-4q^{2}+20q^{-2}-27q^{-4}+44q^{-6}-48q^{-8}+56q^{-10}-52q^{-12}+48q^{-14}-40q^{-16}+24q^{-18}-15q^{-20}-4q^{-22}+14q^{-24}-24q^{-26}+31q^{-28}-32q^{-30}+30q^{-32}-24q^{-34}+17q^{-36}-12q^{-38}+6q^{-40}-2q^{-42}+q^{-44}$
2,0	$q^{26}-q^{22}+q^{18}-q^{16}-3q^{14}+2q^{10}-3q^{8}-3q^{6}+2q^{4}+2q^{2}+2+2q^{-2}+5q^{-4}+2q^{-6}+2q^{-8}+3q^{-10}+q^{-12}-2q^{-14}-4q^{-20}-3q^{-22}+q^{-26}-q^{-28}-q^{-30}+2q^{-32}+q^{-34}-q^{-36}-q^{-38}+q^{-42}$
3,0	$-q^{48}+q^{44}+q^{42}-2q^{38}+2q^{36}+3q^{34}+q^{32}-4q^{30}-5q^{28}+4q^{26}+6q^{24}-q^{22}-10q^{20}-8q^{18}+8q^{16}+9q^{14}-3q^{12}-13q^{10}-8q^{8}+8q^{6}+6q^{4}-4+4q^{-2}+10q^{-4}+8q^{-6}+3q^{-8}+3q^{-10}+7q^{-12}+2q^{-14}-3q^{-16}-2q^{-18}+3q^{-20}+2q^{-22}-8q^{-24}-9q^{-26}+6q^{-30}+q^{-32}-10q^{-34}-7q^{-36}+4q^{-38}+8q^{-40}+2q^{-42}-8q^{-44}-5q^{-46}+3q^{-48}+7q^{-50}+4q^{-52}-5q^{-54}-4q^{-56}+5q^{-60}+4q^{-62}-2q^{-64}-3q^{-66}-3q^{-68}+q^{-70}+2q^{-72}+q^{-74}-q^{-78}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{22}-q^{20}+2q^{16}-3q^{14}-3q^{12}+2q^{10}-3q^{8}-3q^{6}+5q^{4}+q^{2}+2+4q^{-2}+3q^{-4}+q^{-6}+3q^{-10}+2q^{-12}-3q^{-14}+q^{-16}+q^{-18}-5q^{-20}-q^{-22}+q^{-24}-3q^{-26}+q^{-28}+q^{-30}-q^{-32}+q^{-34}$
1,0,0	$-q^{13}-q^{9}-q^{5}+2q^{3}+q+2q^{-1}+2q^{-3}+q^{-5}+q^{-7}+q^{-11}-q^{-13}-q^{-17}-q^{-21}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{28}+q^{22}+q^{20}-2q^{18}-4q^{16}-2q^{14}-q^{12}-4q^{10}-4q^{8}+3q^{6}+2q^{4}+q^{2}+4+7q^{-2}+4q^{-4}+3q^{-6}+5q^{-8}+4q^{-10}+2q^{-14}+4q^{-16}-2q^{-18}-3q^{-20}+q^{-22}-2q^{-24}-6q^{-26}-4q^{-28}-q^{-32}-2q^{-34}+q^{-36}+2q^{-38}+q^{-44}$
1,0,0,0	$-q^{16}-q^{12}-q^{10}-q^{6}+2q^{4}+q^{2}+2+2q^{-2}+2q^{-4}+q^{-6}+q^{-8}+q^{-10}+q^{-14}-q^{-16}-q^{-20}-q^{-22}-q^{-26}$

B2 Invariants.

Weight	Invariant
0,1	$-q^{22}+q^{20}-2q^{18}+2q^{16}-3q^{14}+3q^{12}-4q^{10}+3q^{8}-q^{6}+q^{4}+3q^{2}-2+6q^{-2}-5q^{-4}+7q^{-6}-6q^{-8}+5q^{-10}-4q^{-12}+3q^{-14}-q^{-16}-q^{-18}+3q^{-20}-3q^{-22}+3q^{-24}-3q^{-26}+3q^{-28}-3q^{-30}+q^{-32}-q^{-34}$
1,0	$q^{36}-q^{32}-q^{30}+q^{28}+2q^{26}-3q^{22}-3q^{20}+3q^{16}-4q^{12}-2q^{10}+2q^{8}+4q^{6}-q^{4}-q^{2}+2+5q^{-2}+q^{-4}-q^{-6}+3q^{-10}+2q^{-12}-q^{-14}-q^{-16}+2q^{-18}+2q^{-20}-q^{-22}-3q^{-24}+3q^{-28}+q^{-30}-4q^{-32}-4q^{-34}+q^{-36}+3q^{-38}-3q^{-42}-2q^{-44}+2q^{-46}+2q^{-48}-q^{-50}-q^{-52}+q^{-56}$

D4 Invariants.

Weight	Invariant
1,0,0,0	$q^{30}-q^{28}+q^{26}-q^{24}+2q^{22}-3q^{20}-4q^{16}+2q^{14}-4q^{12}-2q^{8}+2q^{6}+3q^{4}+q^{2}+5-q^{-2}+7q^{-4}-2q^{-6}+6q^{-8}-4q^{-10}+6q^{-12}-2q^{-14}+5q^{-16}-2q^{-18}+2q^{-20}-q^{-22}-q^{-24}-q^{-26}-4q^{-28}+q^{-30}-4q^{-32}+q^{-34}-3q^{-36}+3q^{-38}-2q^{-40}+2q^{-42}-q^{-44}+q^{-46}$

G2 Invariants.

Weight

Invariant

1,0

$q^{52}-q^{50}+2q^{48}-2q^{46}-3q^{40}+4q^{38}-5q^{36}+4q^{34}-5q^{32}+q^{30}+3q^{28}-6q^{26}+8q^{24}-9q^{22}+7q^{20}-4q^{18}-3q^{16}+7q^{14}-8q^{12}+9q^{10}-q^{8}-3q^{6}+6q^{4}-3q^{2}+1+6q^{-2}-10q^{-4}+12q^{-6}-5q^{-8}+10q^{-12}-14q^{-14}+17q^{-16}-11q^{-18}+3q^{-20}+4q^{-22}-9q^{-24}+14q^{-26}-10q^{-28}+6q^{-30}+q^{-32}-5q^{-34}+7q^{-36}-5q^{-38}+4q^{-42}-8q^{-44}+7q^{-46}-3q^{-48}-4q^{-50}+10q^{-52}-13q^{-54}+10q^{-56}-5q^{-58}-4q^{-60}+7q^{-62}-10q^{-64}+9q^{-66}-5q^{-68}+2q^{-72}-4q^{-74}+3q^{-76}-q^{-78}+q^{-80}$

.

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["8 8"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$2t^{2}-6t+9-6t^{-1}+2t^{-2}$

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]=

{ 25, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

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

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_129, K11n39, K11n45, K11n50, K11n132,}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {10_129,}

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["8 8"];

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]=

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

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_129, K11n39, K11n45, K11n50, K11n132,}

In[6]:=

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

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

Out[6]=

{10_129,}

Vassiliev invariants

V2 and V3:

(2, 1)

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 $8$ $8$ $32$ ${\frac {124}{3}}$ $-{\frac {4}{3}}$ $64$ ${\frac {272}{3}}$ ${\frac {32}{3}}$ $8$ ${\frac {256}{3}}$ $32$ ${\frac {992}{3}}$ $-{\frac {32}{3}}$ ${\frac {6271}{15}}$ ${\frac {1516}{15}}$ ${\frac {484}{45}}$ ${\frac {113}{9}}$ $-{\frac {449}{15}}$

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=$0 is the signature of 8 8. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -3 -2 -1 0 1 2 3 4 5 χ 11                 1 -1 9               1   1 7             2 1   -1 5           2 1     1 3         2 2       0 1       3 2         1 -1     1 3           2 -3   1 2             -1 -5   1               1 -7 1                 -1

Integral Khovanov Homology (db, data source)

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

The Coloured Jones Polynomials

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

$n$	$J_{n}$
2	$q^{15}-2q^{14}-q^{13}+6q^{12}-4q^{11}-6q^{10}+12q^{9}-4q^{8}-13q^{7}+17q^{6}-q^{5}-18q^{4}+19q^{3}+2q^{2}-20q+17+3q^{-1}-15q^{-2}+10q^{-3}+2q^{-4}-7q^{-5}+4q^{-6}-2q^{-8}+q^{-9}$
3	$-q^{30}+2q^{29}+q^{28}-2q^{27}-5q^{26}+3q^{25}+9q^{24}-q^{23}-14q^{22}-2q^{21}+17q^{20}+9q^{19}-21q^{18}-15q^{17}+21q^{16}+22q^{15}-19q^{14}-30q^{13}+17q^{12}+35q^{11}-12q^{10}-42q^{9}+10q^{8}+43q^{7}-2q^{6}-50q^{5}+3q^{4}+46q^{3}+6q^{2}-49q-2+38q^{-1}+10q^{-2}-35q^{-3}-6q^{-4}+24q^{-5}+6q^{-6}-17q^{-7}-3q^{-8}+10q^{-9}+q^{-10}-6q^{-11}+4q^{-13}-2q^{-14}-q^{-15}+2q^{-17}-q^{-18}$
4	$q^{50}-2q^{49}-q^{48}+2q^{47}+q^{46}+6q^{45}-7q^{44}-7q^{43}+q^{41}+24q^{40}-6q^{39}-15q^{38}-12q^{37}-13q^{36}+45q^{35}+8q^{34}-7q^{33}-25q^{32}-47q^{31}+52q^{30}+23q^{29}+21q^{28}-20q^{27}-85q^{26}+37q^{25}+21q^{24}+59q^{23}+5q^{22}-113q^{21}+11q^{20}+3q^{19}+91q^{18}+39q^{17}-125q^{16}-16q^{15}-22q^{14}+116q^{13}+71q^{12}-129q^{11}-39q^{10}-44q^{9}+131q^{8}+95q^{7}-124q^{6}-56q^{5}-61q^{4}+130q^{3}+108q^{2}-103q-56-73q^{-1}+103q^{-2}+102q^{-3}-66q^{-4}-39q^{-5}-70q^{-6}+61q^{-7}+71q^{-8}-34q^{-9}-10q^{-10}-48q^{-11}+24q^{-12}+34q^{-13}-17q^{-14}+8q^{-15}-23q^{-16}+7q^{-17}+11q^{-18}-11q^{-19}+10q^{-20}-7q^{-21}+2q^{-22}+2q^{-23}-6q^{-24}+5q^{-25}-q^{-26}+q^{-27}-2q^{-29}+q^{-30}$
5	$-q^{75}+2q^{74}+q^{73}-2q^{72}-q^{71}-2q^{70}-2q^{69}+5q^{68}+9q^{67}-5q^{65}-10q^{64}-13q^{63}+2q^{62}+20q^{61}+21q^{60}+5q^{59}-15q^{58}-34q^{57}-25q^{56}+11q^{55}+39q^{54}+43q^{53}+11q^{52}-37q^{51}-60q^{50}-37q^{49}+17q^{48}+68q^{47}+70q^{46}+9q^{45}-59q^{44}-90q^{43}-56q^{42}+37q^{41}+107q^{40}+97q^{39}+q^{38}-104q^{37}-138q^{36}-52q^{35}+90q^{34}+173q^{33}+104q^{32}-66q^{31}-192q^{30}-159q^{29}+26q^{28}+214q^{27}+208q^{26}+6q^{25}-215q^{24}-255q^{23}-50q^{22}+228q^{21}+295q^{20}+74q^{19}-218q^{18}-332q^{17}-118q^{16}+237q^{15}+359q^{14}+129q^{13}-216q^{12}-388q^{11}-173q^{10}+235q^{9}+403q^{8}+174q^{7}-198q^{6}-411q^{5}-223q^{4}+202q^{3}+406q^{2}+215q-141-382q^{-1}-253q^{-2}+122q^{-3}+346q^{-4}+230q^{-5}-60q^{-6}-286q^{-7}-232q^{-8}+33q^{-9}+222q^{-10}+192q^{-11}+7q^{-12}-159q^{-13}-156q^{-14}-21q^{-15}+101q^{-16}+110q^{-17}+31q^{-18}-58q^{-19}-76q^{-20}-25q^{-21}+28q^{-22}+45q^{-23}+18q^{-24}-9q^{-25}-23q^{-26}-16q^{-27}+3q^{-28}+14q^{-29}+4q^{-30}+2q^{-31}-8q^{-33}-3q^{-34}+5q^{-35}-2q^{-36}+2q^{-37}+4q^{-38}-3q^{-39}-2q^{-40}+q^{-41}-q^{-42}+2q^{-44}-q^{-45}$
6	$q^{105}-2q^{104}-q^{103}+2q^{102}+q^{101}+2q^{100}-2q^{99}+4q^{98}-7q^{97}-9q^{96}+3q^{95}+4q^{94}+11q^{93}+2q^{92}+18q^{91}-14q^{90}-27q^{89}-13q^{88}-7q^{87}+16q^{86}+9q^{85}+65q^{84}+6q^{83}-30q^{82}-37q^{81}-47q^{80}-21q^{79}-24q^{78}+111q^{77}+60q^{76}+30q^{75}-10q^{74}-58q^{73}-90q^{72}-139q^{71}+78q^{70}+64q^{69}+116q^{68}+100q^{67}+49q^{66}-82q^{65}-267q^{64}-54q^{63}-70q^{62}+100q^{61}+200q^{60}+269q^{59}+82q^{58}-267q^{57}-171q^{56}-313q^{55}-89q^{54}+154q^{53}+472q^{52}+354q^{51}-85q^{50}-146q^{49}-531q^{48}-389q^{47}-69q^{46}+540q^{45}+606q^{44}+210q^{43}+35q^{42}-628q^{41}-682q^{40}-390q^{39}+472q^{38}+759q^{37}+508q^{36}+292q^{35}-615q^{34}-899q^{33}-705q^{32}+340q^{31}+827q^{30}+750q^{29}+534q^{28}-557q^{27}-1050q^{26}-957q^{25}+220q^{24}+859q^{23}+934q^{22}+715q^{21}-508q^{20}-1162q^{19}-1146q^{18}+133q^{17}+887q^{16}+1075q^{15}+844q^{14}-464q^{13}-1239q^{12}-1287q^{11}+41q^{10}+880q^{9}+1171q^{8}+956q^{7}-367q^{6}-1237q^{5}-1374q^{4}-105q^{3}+768q^{2}+1170q+1047-174q^{-1}-1085q^{-2}-1342q^{-3}-283q^{-4}+519q^{-5}+1000q^{-6}+1029q^{-7}+62q^{-8}-766q^{-9}-1118q^{-10}-382q^{-11}+213q^{-12}+667q^{-13}+839q^{-14}+210q^{-15}-397q^{-16}-743q^{-17}-328q^{-18}+309q^{-20}+534q^{-21}+212q^{-22}-137q^{-23}-382q^{-24}-181q^{-25}-65q^{-26}+79q^{-27}+264q^{-28}+128q^{-29}-25q^{-30}-157q^{-31}-59q^{-32}-46q^{-33}-11q^{-34}+111q^{-35}+54q^{-36}+q^{-37}-56q^{-38}-7q^{-39}-19q^{-40}-25q^{-41}+43q^{-42}+18q^{-43}+4q^{-44}-19q^{-45}+5q^{-46}-7q^{-47}-17q^{-48}+16q^{-49}+4q^{-50}+4q^{-51}-6q^{-52}+4q^{-53}-2q^{-54}-8q^{-55}+5q^{-56}+2q^{-58}-q^{-59}+q^{-60}-2q^{-62}+q^{-63}$
7	$-q^{140}+2q^{139}+q^{138}-2q^{137}-q^{136}-2q^{135}+2q^{134}-2q^{132}+7q^{131}+6q^{130}-2q^{129}-4q^{128}-13q^{127}-4q^{126}-9q^{124}+18q^{123}+23q^{122}+16q^{121}+9q^{120}-27q^{119}-24q^{118}-21q^{117}-43q^{116}+4q^{115}+33q^{114}+49q^{113}+77q^{112}+7q^{111}-12q^{110}-32q^{109}-109q^{108}-69q^{107}-40q^{106}+15q^{105}+130q^{104}+97q^{103}+97q^{102}+76q^{101}-89q^{100}-116q^{99}-177q^{98}-180q^{97}+12q^{96}+60q^{95}+195q^{94}+296q^{93}+139q^{92}+67q^{91}-148q^{90}-373q^{89}-293q^{88}-273q^{87}-3q^{86}+366q^{85}+411q^{84}+514q^{83}+262q^{82}-228q^{81}-460q^{80}-750q^{79}-577q^{78}-18q^{77}+361q^{76}+896q^{75}+931q^{74}+387q^{73}-134q^{72}-950q^{71}-1232q^{70}-794q^{69}-228q^{68}+840q^{67}+1448q^{66}+1226q^{65}+685q^{64}-599q^{63}-1559q^{62}-1605q^{61}-1183q^{60}+250q^{59}+1527q^{58}+1903q^{57}+1700q^{56}+190q^{55}-1405q^{54}-2132q^{53}-2160q^{52}-646q^{51}+1183q^{50}+2250q^{49}+2583q^{48}+1126q^{47}-933q^{46}-2318q^{45}-2934q^{44}-1548q^{43}+654q^{42}+2317q^{41}+3230q^{40}+1952q^{39}-396q^{38}-2323q^{37}-3464q^{36}-2271q^{35}+160q^{34}+2287q^{33}+3675q^{32}+2578q^{31}+14q^{30}-2306q^{29}-3832q^{28}-2785q^{27}-181q^{26}+2282q^{25}+4005q^{24}+3018q^{23}+281q^{22}-2337q^{21}-4118q^{20}-3156q^{19}-411q^{18}+2302q^{17}+4257q^{16}+3370q^{15}+515q^{14}-2338q^{13}-4324q^{12}-3478q^{11}-681q^{10}+2199q^{9}+4373q^{8}+3687q^{7}+877q^{6}-2110q^{5}-4318q^{4}-3738q^{3}-1121q^{2}+1787q+4157+3852q^{-1}+1387q^{-2}-1500q^{-3}-3863q^{-4}-3738q^{-5}-1611q^{-6}+1019q^{-7}+3423q^{-8}+3585q^{-9}+1790q^{-10}-617q^{-11}-2879q^{-12}-3209q^{-13}-1837q^{-14}+175q^{-15}+2267q^{-16}+2766q^{-17}+1751q^{-18}+142q^{-19}-1658q^{-20}-2213q^{-21}-1558q^{-22}-363q^{-23}+1133q^{-24}+1662q^{-25}+1269q^{-26}+446q^{-27}-695q^{-28}-1157q^{-29}-964q^{-30}-445q^{-31}+396q^{-32}+763q^{-33}+664q^{-34}+369q^{-35}-210q^{-36}-452q^{-37}-424q^{-38}-286q^{-39}+103q^{-40}+261q^{-41}+256q^{-42}+195q^{-43}-61q^{-44}-142q^{-45}-133q^{-46}-121q^{-47}+29q^{-48}+67q^{-49}+74q^{-50}+90q^{-51}-30q^{-52}-47q^{-53}-32q^{-54}-40q^{-55}+18q^{-56}+8q^{-57}+17q^{-58}+44q^{-59}-14q^{-60}-21q^{-61}-8q^{-62}-13q^{-63}+11q^{-64}-3q^{-65}+2q^{-66}+22q^{-67}-5q^{-68}-8q^{-69}-2q^{-70}-6q^{-71}+5q^{-72}-3q^{-73}+8q^{-75}-q^{-76}-2q^{-77}-2q^{-79}+q^{-80}-q^{-81}+2q^{-83}-q^{-84}$

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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