8 13

8_12

8_14

(KnotPlot image)	See the full Rolfsen Knot Table. Visit 8 13's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 8 13 at Knotilus!

Knot presentations

Planar diagram presentation	X1425 X3,10,4,11 X11,1,12,16 X5,13,6,12 X15,7,16,6 X7,15,8,14 X13,9,14,8 X9,2,10,3
Gauss code	-1, 8, -2, 1, -4, 5, -6, 7, -8, 2, -3, 4, -7, 6, -5, 3
Dowker-Thistlethwaite code	4 10 12 14 2 16 8 6
Conway Notation	[31112]

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

[edit Notes on presentations of 8 13]

Knot 8_13.

A graph, knot 8_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["8 13"];

In[4]:=

PD[K]

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

Out[4]=

X1425 X3,10,4,11 X11,1,12,16 X5,13,6,12 X15,7,16,6 X7,15,8,14 X13,9,14,8 X9,2,10,3

In[5]:=

GaussCode[K]

Out[5]=

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

In[6]:=

DTCode[K]

Out[6]=

4 10 12 14 2 16 8 6

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[31112]

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]=

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

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

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Three dimensional invariants

Symmetry type Reversible Unknotting number 1 3-genus 2 Bridge index 2 Super bridge index $\{4,5\}$ Nakanishi index 1 Maximal Thurston-Bennequin number [-4][-6] Hyperbolic Volume 8.53123 A-Polynomial See Data:8 13/A-polynomial

[edit Notes for 8 13's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus $1$ Topological 4 genus $1$ Concordance genus $2$ Rasmussen s-Invariant 0

[edit Notes for 8 13's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$2t^{2}-7t+11-7t^{-1}+2t^{-2}$
Conway polynomial	$2z^{4}+z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 29, 0 }
Jones polynomial	$-q^{5}+2q^{4}-3q^{3}+5q^{2}-5q+5-4q^{-1}+3q^{-2}-q^{-3}$
HOMFLY-PT polynomial (db, data sources)	$z^{4}a^{-2}+z^{4}-a^{2}z^{2}+2z^{2}a^{-2}-z^{2}a^{-4}+z^{2}+2a^{-2}-a^{-4}$
Kauffman polynomial (db, data sources)	$z^{7}a^{-1}+z^{7}a^{-3}+5z^{6}a^{-2}+2z^{6}a^{-4}+3z^{6}+4az^{5}+4z^{5}a^{-1}+z^{5}a^{-3}+z^{5}a^{-5}+3a^{2}z^{4}-11z^{4}a^{-2}-6z^{4}a^{-4}-2z^{4}+a^{3}z^{3}-4az^{3}-9z^{3}a^{-1}-7z^{3}a^{-3}-3z^{3}a^{-5}-2a^{2}z^{2}+7z^{2}a^{-2}+5z^{2}a^{-4}+az+3za^{-1}+4za^{-3}+2za^{-5}-2a^{-2}-a^{-4}$
The A2 invariant	$-q^{10}+q^{8}+q^{6}-q^{4}+q^{2}-1+q^{-2}+q^{-4}+q^{-6}+2q^{-8}-q^{-10}-q^{-16}$
The G2 invariant	$q^{52}-2q^{50}+3q^{48}-4q^{46}+q^{44}-3q^{40}+9q^{38}-11q^{36}+12q^{34}-8q^{32}+6q^{28}-13q^{26}+18q^{24}-16q^{22}+10q^{20}-8q^{16}+14q^{14}-10q^{12}+4q^{10}+2q^{8}-10q^{6}+9q^{4}-3q^{2}-7+17q^{-2}-22q^{-4}+19q^{-6}-6q^{-8}-9q^{-10}+19q^{-12}-26q^{-14}+26q^{-16}-14q^{-18}+3q^{-20}+11q^{-22}-16q^{-24}+21q^{-26}-10q^{-28}+q^{-30}+6q^{-32}-9q^{-34}+8q^{-36}-6q^{-40}+14q^{-42}-15q^{-44}+9q^{-46}+q^{-48}-13q^{-50}+18q^{-52}-19q^{-54}+11q^{-56}-4q^{-58}-6q^{-60}+11q^{-62}-13q^{-64}+10q^{-66}-4q^{-68}-q^{-70}+2q^{-72}-4q^{-74}+3q^{-76}-q^{-78}+q^{-80}$

Further Quantum Invariants

Further quantum knot invariants for 8_13.

A1 Invariants.

Weight	Invariant
1	$-q^{7}+2q^{5}-q^{3}+q+2q^{-5}-q^{-7}+q^{-9}-q^{-11}$
2	$q^{20}-2q^{18}-q^{16}+5q^{14}-4q^{12}-3q^{10}+7q^{8}-3q^{6}-3q^{4}+6q^{2}+1-2q^{-2}+3q^{-6}-q^{-8}-5q^{-10}+4q^{-12}+3q^{-14}-6q^{-16}+3q^{-18}+4q^{-20}-5q^{-22}+3q^{-26}-2q^{-28}-q^{-30}+q^{-32}$
3	$-q^{39}+2q^{37}+q^{35}-3q^{33}-2q^{31}+4q^{29}+6q^{27}-9q^{25}-8q^{23}+10q^{21}+11q^{19}-12q^{17}-15q^{15}+12q^{13}+19q^{11}-8q^{9}-18q^{7}+5q^{5}+16q^{3}+q-11q^{-1}-6q^{-3}+7q^{-5}+10q^{-7}-q^{-9}-12q^{-11}-3q^{-13}+14q^{-15}+8q^{-17}-15q^{-19}-11q^{-21}+13q^{-23}+13q^{-25}-10q^{-27}-17q^{-29}+8q^{-31}+17q^{-33}-2q^{-35}-17q^{-37}-q^{-39}+14q^{-41}+5q^{-43}-10q^{-45}-7q^{-47}+5q^{-49}+6q^{-51}-2q^{-53}-4q^{-55}+2q^{-59}+q^{-61}-q^{-63}$
4	$q^{64}-2q^{62}-q^{60}+3q^{58}+2q^{54}-7q^{52}-q^{50}+10q^{48}+q^{46}+3q^{44}-21q^{42}-6q^{40}+26q^{38}+13q^{36}-47q^{32}-19q^{30}+43q^{28}+40q^{26}+11q^{24}-67q^{22}-47q^{20}+36q^{18}+60q^{16}+35q^{14}-55q^{12}-60q^{10}+6q^{8}+46q^{6}+49q^{4}-18q^{2}-43-22q^{-2}+11q^{-4}+39q^{-6}+19q^{-8}-13q^{-10}-39q^{-12}-19q^{-14}+24q^{-16}+43q^{-18}+9q^{-20}-44q^{-22}-35q^{-24}+9q^{-26}+59q^{-28}+26q^{-30}-46q^{-32}-47q^{-34}-7q^{-36}+64q^{-38}+41q^{-40}-34q^{-42}-52q^{-44}-32q^{-46}+51q^{-48}+56q^{-50}-4q^{-52}-42q^{-54}-54q^{-56}+18q^{-58}+50q^{-60}+26q^{-62}-12q^{-64}-51q^{-66}-12q^{-68}+21q^{-70}+29q^{-72}+14q^{-74}-25q^{-76}-18q^{-78}-4q^{-80}+12q^{-82}+16q^{-84}-4q^{-86}-6q^{-88}-6q^{-90}+6q^{-94}+q^{-96}-2q^{-100}-q^{-102}+q^{-104}$
5	$-q^{95}+2q^{93}+q^{91}-3q^{89}+q^{83}+2q^{81}-6q^{77}-4q^{75}+7q^{73}+12q^{71}+4q^{69}-13q^{67}-20q^{65}-14q^{63}+20q^{61}+52q^{59}+24q^{57}-36q^{55}-78q^{53}-56q^{51}+38q^{49}+124q^{47}+100q^{45}-36q^{43}-161q^{41}-154q^{39}+5q^{37}+184q^{35}+211q^{33}+41q^{31}-183q^{29}-251q^{27}-94q^{25}+149q^{23}+266q^{21}+146q^{19}-95q^{17}-245q^{15}-175q^{13}+34q^{11}+195q^{9}+183q^{7}+27q^{5}-131q^{3}-166q-71q^{-1}+63q^{-3}+132q^{-5}+100q^{-7}+q^{-9}-98q^{-11}-115q^{-13}-48q^{-15}+65q^{-17}+128q^{-19}+82q^{-21}-39q^{-23}-136q^{-25}-111q^{-27}+27q^{-29}+148q^{-31}+130q^{-33}-18q^{-35}-157q^{-37}-154q^{-39}+6q^{-41}+171q^{-43}+177q^{-45}+11q^{-47}-176q^{-49}-200q^{-51}-41q^{-53}+166q^{-55}+225q^{-57}+77q^{-59}-142q^{-61}-231q^{-63}-119q^{-65}+90q^{-67}+226q^{-69}+164q^{-71}-37q^{-73}-191q^{-75}-185q^{-77}-36q^{-79}+139q^{-81}+190q^{-83}+86q^{-85}-73q^{-87}-162q^{-89}-121q^{-91}+8q^{-93}+117q^{-95}+127q^{-97}+42q^{-99}-62q^{-101}-107q^{-103}-66q^{-105}+13q^{-107}+70q^{-109}+69q^{-111}+18q^{-113}-35q^{-115}-53q^{-117}-29q^{-119}+7q^{-121}+30q^{-123}+28q^{-125}+6q^{-127}-14q^{-129}-17q^{-131}-7q^{-133}+2q^{-135}+8q^{-137}+8q^{-139}-4q^{-143}-3q^{-145}-q^{-147}+2q^{-151}+q^{-153}-q^{-155}$

A2 Invariants.

Weight	Invariant
1,0	$-q^{10}+q^{8}+q^{6}-q^{4}+q^{2}-1+q^{-2}+q^{-4}+q^{-6}+2q^{-8}-q^{-10}-q^{-16}$
1,1	$q^{28}-4q^{26}+8q^{24}-12q^{22}+20q^{20}-32q^{18}+38q^{16}-44q^{14}+51q^{12}-52q^{10}+44q^{8}-30q^{6}+13q^{4}+12q^{2}-36+66q^{-2}-82q^{-4}+100q^{-6}-102q^{-8}+98q^{-10}-86q^{-12}+66q^{-14}-42q^{-16}+12q^{-18}+10q^{-20}-28q^{-22}+46q^{-24}-54q^{-26}+55q^{-28}-48q^{-30}+40q^{-32}-32q^{-34}+19q^{-36}-12q^{-38}+6q^{-40}-2q^{-42}+q^{-44}$
2,0	$q^{26}-q^{24}-2q^{22}+q^{20}+3q^{18}-5q^{14}+3q^{10}-2q^{8}-2q^{6}+4q^{4}+4q^{2}+1+q^{-2}+2q^{-4}-q^{-6}-2q^{-8}+q^{-10}-q^{-12}-3q^{-14}+3q^{-16}+4q^{-18}-q^{-20}-2q^{-22}+2q^{-24}+2q^{-26}-2q^{-28}-3q^{-30}+q^{-32}+q^{-34}-q^{-36}-q^{-38}+q^{-42}$

A3 Invariants.

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

A4 Invariants.

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

B2 Invariants.

Weight	Invariant
0,1	$-q^{22}+2q^{20}-3q^{18}+4q^{16}-5q^{14}+5q^{12}-4q^{10}+3q^{8}-q^{6}+4q^{2}-6+8q^{-2}-8q^{-4}+9q^{-6}-8q^{-8}+7q^{-10}-4q^{-12}+2q^{-14}+q^{-16}-2q^{-18}+4q^{-20}-4q^{-22}+5q^{-24}-4q^{-26}+3q^{-28}-3q^{-30}+q^{-32}-q^{-34}$
1,0	$q^{36}-2q^{32}-2q^{30}+q^{28}+4q^{26}+q^{24}-4q^{22}-3q^{20}+3q^{18}+5q^{16}-5q^{12}-2q^{10}+3q^{8}+3q^{6}-3q^{4}-3q^{2}+2+4q^{-2}-3q^{-6}+4q^{-10}+2q^{-12}-2q^{-14}-q^{-16}+4q^{-18}+3q^{-20}-2q^{-22}-4q^{-24}+2q^{-26}+5q^{-28}+q^{-30}-5q^{-32}-4q^{-34}+2q^{-36}+4q^{-38}-q^{-40}-4q^{-42}-2q^{-44}+2q^{-46}+2q^{-48}-q^{-50}-q^{-52}+q^{-56}$

D4 Invariants.

Weight	Invariant
1,0,0,0	$q^{30}-2q^{28}+q^{26}-2q^{24}+4q^{22}-4q^{20}+3q^{18}-3q^{16}+5q^{14}-2q^{12}+q^{10}-2q^{8}+2q^{4}-4q^{2}+3-6q^{-2}+7q^{-4}-5q^{-6}+8q^{-8}-5q^{-10}+9q^{-12}-q^{-14}+7q^{-16}-q^{-18}+2q^{-20}+q^{-22}-2q^{-24}-5q^{-28}+2q^{-30}-5q^{-32}+2q^{-34}-4q^{-36}+3q^{-38}-2q^{-40}+2q^{-42}-q^{-44}+q^{-46}$

G2 Invariants.

Weight

Invariant

1,0

$q^{52}-2q^{50}+3q^{48}-4q^{46}+q^{44}-3q^{40}+9q^{38}-11q^{36}+12q^{34}-8q^{32}+6q^{28}-13q^{26}+18q^{24}-16q^{22}+10q^{20}-8q^{16}+14q^{14}-10q^{12}+4q^{10}+2q^{8}-10q^{6}+9q^{4}-3q^{2}-7+17q^{-2}-22q^{-4}+19q^{-6}-6q^{-8}-9q^{-10}+19q^{-12}-26q^{-14}+26q^{-16}-14q^{-18}+3q^{-20}+11q^{-22}-16q^{-24}+21q^{-26}-10q^{-28}+q^{-30}+6q^{-32}-9q^{-34}+8q^{-36}-6q^{-40}+14q^{-42}-15q^{-44}+9q^{-46}+q^{-48}-13q^{-50}+18q^{-52}-19q^{-54}+11q^{-56}-4q^{-58}-6q^{-60}+11q^{-62}-13q^{-64}+10q^{-66}-4q^{-68}-q^{-70}+2q^{-72}-4q^{-74}+3q^{-76}-q^{-78}+q^{-80}$

.

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 13"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$2t^{2}-7t+11-7t^{-1}+2t^{-2}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 29, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{7}a^{-1}+z^{7}a^{-3}+5z^{6}a^{-2}+2z^{6}a^{-4}+3z^{6}+4az^{5}+4z^{5}a^{-1}+z^{5}a^{-3}+z^{5}a^{-5}+3a^{2}z^{4}-11z^{4}a^{-2}-6z^{4}a^{-4}-2z^{4}+a^{3}z^{3}-4az^{3}-9z^{3}a^{-1}-7z^{3}a^{-3}-3z^{3}a^{-5}-2a^{2}z^{2}+7z^{2}a^{-2}+5z^{2}a^{-4}+az+3za^{-1}+4za^{-3}+2za^{-5}-2a^{-2}-a^{-4}$

"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["8 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]=

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

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]=

{}

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:

(1, 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 $4$ $8$ $8$ ${\frac {14}{3}}$ $-{\frac {38}{3}}$ $32$ ${\frac {80}{3}}$ ${\frac {32}{3}}$ $-24$ ${\frac {32}{3}}$ $32$ ${\frac {56}{3}}$ $-{\frac {152}{3}}$ ${\frac {1951}{30}}$ ${\frac {526}{5}}$ $-{\frac {6058}{45}}$ $-{\frac {31}{18}}$ $-{\frac {1409}{30}}$

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

Khovanov Homology

The coefficients of the monomials $t^{r}q^{j}$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$). The squares with yellow highlighting are those on the "critical diagonals", where $j-2r=s+1$ or $j-2r=s-1$, where $s=$0 is the signature of 8 13. 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           3 1     2 3         2 2       0 1       3 3         0 -1     2 3           1 -3   1 2             -1 -5   2               2 -7 1                 -1

Integral Khovanov Homology (db, data source)

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

The Coloured Jones Polynomials

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

$n$	$J_{n}$
2	$q^{15}-2q^{14}-q^{13}+6q^{12}-5q^{11}-6q^{10}+15q^{9}-6q^{8}-15q^{7}+24q^{6}-5q^{5}-24q^{4}+28q^{3}-q^{2}-27q+26+2q^{-1}-22q^{-2}+17q^{-3}+2q^{-4}-12q^{-5}+7q^{-6}+q^{-7}-3q^{-8}+q^{-9}$
3	$-q^{30}+2q^{29}+q^{28}-2q^{27}-5q^{26}+4q^{25}+9q^{24}-3q^{23}-17q^{22}+q^{21}+24q^{20}+6q^{19}-32q^{18}-15q^{17}+39q^{16}+25q^{15}-41q^{14}-40q^{13}+46q^{12}+48q^{11}-41q^{10}-64q^{9}+42q^{8}+71q^{7}-35q^{6}-81q^{5}+33q^{4}+82q^{3}-24q^{2}-84q+20+77q^{-1}-12q^{-2}-69q^{-3}+9q^{-4}+54q^{-5}-2q^{-6}-42q^{-7}+2q^{-8}+27q^{-9}+q^{-10}-19q^{-11}+q^{-12}+9q^{-13}-4q^{-15}-q^{-16}+3q^{-17}-q^{-18}$
4	$q^{50}-2q^{49}-q^{48}+2q^{47}+q^{46}+6q^{45}-8q^{44}-7q^{43}+2q^{42}+3q^{41}+26q^{40}-12q^{39}-23q^{38}-12q^{37}-4q^{36}+65q^{35}+3q^{34}-31q^{33}-45q^{32}-43q^{31}+104q^{30}+41q^{29}-7q^{28}-77q^{27}-115q^{26}+116q^{25}+79q^{24}+53q^{23}-82q^{22}-198q^{21}+96q^{20}+97q^{19}+128q^{18}-59q^{17}-269q^{16}+56q^{15}+98q^{14}+200q^{13}-26q^{12}-319q^{11}+12q^{10}+89q^{9}+253q^{8}+8q^{7}-338q^{6}-31q^{5}+69q^{4}+279q^{3}+40q^{2}-318q-59+36q^{-1}+258q^{-2}+65q^{-3}-251q^{-4}-62q^{-5}-4q^{-6}+192q^{-7}+70q^{-8}-161q^{-9}-37q^{-10}-28q^{-11}+109q^{-12}+50q^{-13}-83q^{-14}-8q^{-15}-25q^{-16}+47q^{-17}+22q^{-18}-36q^{-19}+5q^{-20}-12q^{-21}+15q^{-22}+7q^{-23}-12q^{-24}+3q^{-25}-3q^{-26}+4q^{-27}+q^{-28}-3q^{-29}+q^{-30}$
5	$-q^{75}+2q^{74}+q^{73}-2q^{72}-q^{71}-2q^{70}-2q^{69}+6q^{68}+9q^{67}-2q^{66}-7q^{65}-11q^{64}-12q^{63}+9q^{62}+29q^{61}+20q^{60}-5q^{59}-34q^{58}-48q^{57}-15q^{56}+47q^{55}+73q^{54}+46q^{53}-33q^{52}-105q^{51}-94q^{50}+6q^{49}+118q^{48}+150q^{47}+52q^{46}-115q^{45}-203q^{44}-123q^{43}+77q^{42}+239q^{41}+211q^{40}-11q^{39}-254q^{38}-298q^{37}-72q^{36}+233q^{35}+365q^{34}+190q^{33}-192q^{32}-434q^{31}-281q^{30}+121q^{29}+454q^{28}+409q^{27}-44q^{26}-493q^{25}-488q^{24}-38q^{23}+478q^{22}+596q^{21}+122q^{20}-499q^{19}-653q^{18}-198q^{17}+475q^{16}+735q^{15}+270q^{14}-481q^{13}-774q^{12}-336q^{11}+450q^{10}+832q^{9}+391q^{8}-435q^{7}-837q^{6}-449q^{5}+383q^{4}+849q^{3}+490q^{2}-336q-805-518q^{-1}+249q^{-2}+754q^{-3}+525q^{-4}-178q^{-5}-649q^{-6}-506q^{-7}+88q^{-8}+545q^{-9}+455q^{-10}-28q^{-11}-408q^{-12}-386q^{-13}-29q^{-14}+302q^{-15}+298q^{-16}+40q^{-17}-184q^{-18}-216q^{-19}-56q^{-20}+123q^{-21}+139q^{-22}+33q^{-23}-59q^{-24}-80q^{-25}-32q^{-26}+37q^{-27}+45q^{-28}+11q^{-29}-17q^{-30}-20q^{-31}-4q^{-32}+5q^{-33}+11q^{-34}+5q^{-35}-10q^{-36}-3q^{-37}+4q^{-38}+3q^{-41}-4q^{-42}-q^{-43}+3q^{-44}-q^{-45}$
6	$q^{105}-2q^{104}-q^{103}+2q^{102}+q^{101}+2q^{100}-2q^{99}+4q^{98}-8q^{97}-9q^{96}+5q^{95}+6q^{94}+12q^{93}+16q^{91}-22q^{90}-35q^{89}-9q^{88}+5q^{87}+35q^{86}+22q^{85}+72q^{84}-21q^{83}-80q^{82}-71q^{81}-50q^{80}+25q^{79}+48q^{78}+207q^{77}+70q^{76}-60q^{75}-148q^{74}-188q^{73}-118q^{72}-42q^{71}+346q^{70}+265q^{69}+146q^{68}-77q^{67}-283q^{66}-387q^{65}-373q^{64}+283q^{63}+390q^{62}+491q^{61}+264q^{60}-98q^{59}-565q^{58}-852q^{57}-94q^{56}+195q^{55}+718q^{54}+750q^{53}+454q^{52}-405q^{51}-1201q^{50}-632q^{49}-372q^{48}+593q^{47}+1106q^{46}+1193q^{45}+118q^{44}-1213q^{43}-1073q^{42}-1118q^{41}+124q^{40}+1165q^{39}+1866q^{38}+808q^{37}-921q^{36}-1286q^{35}-1807q^{34}-489q^{33}+981q^{32}+2349q^{31}+1453q^{30}-504q^{29}-1325q^{28}-2337q^{27}-1057q^{26}+720q^{25}+2671q^{24}+1961q^{23}-114q^{22}-1313q^{21}-2725q^{20}-1506q^{19}+492q^{18}+2897q^{17}+2343q^{16}+215q^{15}-1294q^{14}-3006q^{13}-1861q^{12}+281q^{11}+3008q^{10}+2627q^{9}+542q^{8}-1193q^{7}-3132q^{6}-2158q^{5}-11q^{4}+2891q^{3}+2759q^{2}+906q-888-2973q^{-1}-2331q^{-2}-420q^{-3}+2429q^{-4}+2600q^{-5}+1207q^{-6}-376q^{-7}-2423q^{-8}-2213q^{-9}-802q^{-10}+1667q^{-11}+2056q^{-12}+1246q^{-13}+147q^{-14}-1585q^{-15}-1725q^{-16}-928q^{-17}+876q^{-18}+1271q^{-19}+950q^{-20}+421q^{-21}-775q^{-22}-1040q^{-23}-739q^{-24}+343q^{-25}+570q^{-26}+506q^{-27}+391q^{-28}-266q^{-29}-463q^{-30}-418q^{-31}+119q^{-32}+167q^{-33}+168q^{-34}+228q^{-35}-63q^{-36}-148q^{-37}-172q^{-38}+57q^{-39}+25q^{-40}+19q^{-41}+94q^{-42}-12q^{-43}-35q^{-44}-54q^{-45}+34q^{-46}-2q^{-47}-10q^{-48}+28q^{-49}-5q^{-50}-5q^{-51}-15q^{-52}+17q^{-53}-2q^{-54}-8q^{-55}+8q^{-56}-3q^{-57}-3q^{-59}+4q^{-60}+q^{-61}-3q^{-62}+q^{-63}$
7	$-q^{140}+2q^{139}+q^{138}-2q^{137}-q^{136}-2q^{135}+2q^{134}-2q^{132}+8q^{131}+6q^{130}-4q^{129}-6q^{128}-14q^{127}-2q^{126}+3q^{125}-6q^{124}+25q^{123}+28q^{122}+10q^{121}-5q^{120}-48q^{119}-35q^{118}-22q^{117}-32q^{116}+45q^{115}+85q^{114}+82q^{113}+69q^{112}-57q^{111}-101q^{110}-121q^{109}-170q^{108}-25q^{107}+102q^{106}+211q^{105}+300q^{104}+122q^{103}-33q^{102}-201q^{101}-442q^{100}-335q^{99}-149q^{98}+141q^{97}+551q^{96}+538q^{95}+423q^{94}+101q^{93}-521q^{92}-731q^{91}-781q^{90}-482q^{89}+315q^{88}+768q^{87}+1112q^{86}+1011q^{85}+115q^{84}-594q^{83}-1312q^{82}-1559q^{81}-751q^{80}+118q^{79}+1284q^{78}+2043q^{77}+1489q^{76}+608q^{75}-924q^{74}-2293q^{73}-2229q^{72}-1563q^{71}+231q^{70}+2253q^{69}+2838q^{68}+2603q^{67}+735q^{66}-1864q^{65}-3166q^{64}-3609q^{63}-1943q^{62}+1125q^{61}+3246q^{60}+4506q^{59}+3170q^{58}-168q^{57}-2947q^{56}-5125q^{55}-4448q^{54}-1027q^{53}+2453q^{52}+5588q^{51}+5550q^{50}+2183q^{49}-1703q^{48}-5715q^{47}-6547q^{46}-3436q^{45}+894q^{44}+5790q^{43}+7334q^{42}+4473q^{41}-40q^{40}-5607q^{39}-7992q^{38}-5517q^{37}-752q^{36}+5502q^{35}+8509q^{34}+6296q^{33}+1474q^{32}-5273q^{31}-8946q^{30}-7071q^{29}-2070q^{28}+5177q^{27}+9302q^{26}+7641q^{25}+2595q^{24}-5016q^{23}-9657q^{22}-8219q^{21}-3010q^{20}+4948q^{19}+9926q^{18}+8672q^{17}+3456q^{16}-4798q^{15}-10186q^{14}-9151q^{13}-3854q^{12}+4617q^{11}+10293q^{10}+9531q^{9}+4354q^{8}-4240q^{7}-10292q^{6}-9891q^{5}-4866q^{4}+3744q^{3}+10026q^{2}+10041q+5442-2978q^{-1}-9500q^{-2}-10054q^{-3}-5946q^{-4}+2107q^{-5}+8639q^{-6}+9702q^{-7}+6341q^{-8}-1071q^{-9}-7507q^{-10}-9069q^{-11}-6480q^{-12}+92q^{-13}+6140q^{-14}+8055q^{-15}+6347q^{-16}+783q^{-17}-4706q^{-18}-6812q^{-19}-5852q^{-20}-1392q^{-21}+3288q^{-22}+5393q^{-23}+5128q^{-24}+1742q^{-25}-2107q^{-26}-4026q^{-27}-4165q^{-28}-1758q^{-29}+1145q^{-30}+2734q^{-31}+3215q^{-32}+1621q^{-33}-534q^{-34}-1768q^{-35}-2280q^{-36}-1257q^{-37}+140q^{-38}+973q^{-39}+1532q^{-40}+966q^{-41}+22q^{-42}-532q^{-43}-954q^{-44}-606q^{-45}-73q^{-46}+203q^{-47}+554q^{-48}+390q^{-49}+75q^{-50}-72q^{-51}-317q^{-52}-209q^{-53}-37q^{-54}+8q^{-55}+153q^{-56}+98q^{-57}+27q^{-58}+30q^{-59}-88q^{-60}-61q^{-61}+4q^{-62}-9q^{-63}+35q^{-64}+5q^{-65}+29q^{-67}-22q^{-68}-16q^{-69}+6q^{-70}-q^{-71}+9q^{-72}-7q^{-73}-5q^{-74}+13q^{-75}-4q^{-76}-5q^{-77}+3q^{-78}+3q^{-80}-4q^{-81}-q^{-82}+3q^{-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.

8_12 8_14