8 15

8_14

8_16

(KnotPlot image)	See the full Rolfsen Knot Table. Visit 8 15's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 8 15 at Knotilus!
	Two trefoil knots along a closed loop, mutually interlinked. (See also 10 120.)

Symmetrical depiction.

Knot presentations

Planar diagram presentation	X1425 X3849 X5,12,6,13 X13,16,14,1 X9,14,10,15 X15,10,16,11 X11,6,12,7 X7283
Gauss code	-1, 8, -2, 1, -3, 7, -8, 2, -5, 6, -7, 3, -4, 5, -6, 4
Dowker-Thistlethwaite code	4 8 12 2 14 6 16 10
Conway Notation	[21,21,2]

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

[edit Notes on presentations of 8 15]

Knot 8_15.

A graph, knot 8_15.

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 15"];

In[4]:=

PD[K]

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

Out[4]=

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

In[5]:=

GaussCode[K]

Out[5]=

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

In[6]:=

DTCode[K]

Out[6]=

4 8 12 2 14 6 16 10

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[21,21,2]

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]=

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

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

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 2 Bridge index 3 Super bridge index $\{4,6\}$ Nakanishi index 1 Maximal Thurston-Bennequin number [-13][3] Hyperbolic Volume 9.93065 A-Polynomial See Data:8 15/A-polynomial

[edit Notes for 8 15'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 8 15's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$3t^{2}-8t+11-8t^{-1}+3t^{-2}$
Conway polynomial	$3z^{4}+4z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 33, -4 }
Jones polynomial	$q^{-2}-2q^{-3}+5q^{-4}-5q^{-5}+6q^{-6}-6q^{-7}+4q^{-8}-3q^{-9}+q^{-10}$
HOMFLY-PT polynomial (db, data sources)	$a^{10}-3z^{2}a^{8}-4a^{8}+2z^{4}a^{6}+5z^{2}a^{6}+3a^{6}+z^{4}a^{4}+2z^{2}a^{4}+a^{4}$
Kauffman polynomial (db, data sources)	$z^{4}a^{12}-z^{2}a^{12}+3z^{5}a^{11}-5z^{3}a^{11}+2za^{11}+3z^{6}a^{10}-3z^{4}a^{10}-a^{10}+z^{7}a^{9}+6z^{5}a^{9}-14z^{3}a^{9}+8za^{9}+6z^{6}a^{8}-10z^{4}a^{8}+8z^{2}a^{8}-4a^{8}+z^{7}a^{7}+5z^{5}a^{7}-11z^{3}a^{7}+6za^{7}+3z^{6}a^{6}-5z^{4}a^{6}+5z^{2}a^{6}-3a^{6}+2z^{5}a^{5}-2z^{3}a^{5}+z^{4}a^{4}-2z^{2}a^{4}+a^{4}$
The A2 invariant	$q^{32}+q^{30}-2q^{28}-q^{26}-2q^{24}-2q^{22}+q^{20}+3q^{16}+q^{14}+q^{12}+2q^{10}-q^{8}+q^{6}$
The G2 invariant	$q^{162}-2q^{160}+4q^{158}-6q^{156}+3q^{154}-q^{152}-6q^{150}+14q^{148}-18q^{146}+20q^{144}-12q^{142}-q^{140}+17q^{138}-27q^{136}+34q^{134}-24q^{132}+7q^{130}+10q^{128}-21q^{126}+25q^{124}-16q^{122}+q^{120}+14q^{118}-20q^{116}+12q^{114}-24q^{110}+34q^{108}-36q^{106}+18q^{104}-2q^{102}-24q^{100}+40q^{98}-47q^{96}+34q^{94}-18q^{92}-7q^{90}+25q^{88}-33q^{86}+26q^{84}-9q^{82}-2q^{80}+16q^{78}-17q^{76}+9q^{74}+10q^{72}-21q^{70}+29q^{68}-21q^{66}+6q^{64}+17q^{62}-27q^{60}+33q^{58}-23q^{56}+12q^{54}+2q^{52}-14q^{50}+17q^{48}-14q^{46}+10q^{44}-2q^{42}-q^{40}+3q^{38}-3q^{36}+3q^{34}-q^{32}+q^{30}$

Further Quantum Invariants

Further quantum knot invariants for 8_15.

A1 Invariants.

Weight	Invariant
1	$q^{21}-2q^{19}+q^{17}-2q^{15}+q^{11}+3q^{7}-q^{5}+q^{3}$
2	$q^{58}-2q^{56}-2q^{54}+6q^{52}-2q^{50}-6q^{48}+9q^{46}+q^{44}-9q^{42}+6q^{40}+3q^{38}-6q^{36}-q^{34}+2q^{32}-7q^{28}+2q^{26}+7q^{24}-8q^{22}+10q^{18}-5q^{16}-2q^{14}+6q^{12}-q^{8}+q^{6}$
3	$q^{111}-2q^{109}-2q^{107}+3q^{105}+6q^{103}-2q^{101}-13q^{99}+2q^{97}+18q^{95}+4q^{93}-25q^{91}-13q^{89}+27q^{87}+22q^{85}-27q^{83}-29q^{81}+20q^{79}+35q^{77}-12q^{75}-32q^{73}+7q^{71}+27q^{69}+4q^{67}-21q^{65}-9q^{63}+10q^{61}+17q^{59}-5q^{57}-21q^{55}-7q^{53}+27q^{51}+13q^{49}-30q^{47}-23q^{45}+27q^{43}+27q^{41}-22q^{39}-32q^{37}+12q^{35}+32q^{33}-6q^{31}-23q^{29}-2q^{27}+19q^{25}+7q^{23}-8q^{21}-4q^{19}+4q^{17}+3q^{15}-q^{11}+q^{9}$
4	$q^{180}-2q^{178}-2q^{176}+3q^{174}+3q^{172}+6q^{170}-9q^{168}-13q^{166}+2q^{164}+11q^{162}+30q^{160}-11q^{158}-41q^{156}-22q^{154}+14q^{152}+80q^{150}+22q^{148}-62q^{146}-83q^{144}-27q^{142}+123q^{140}+97q^{138}-31q^{136}-135q^{134}-108q^{132}+102q^{130}+151q^{128}+41q^{126}-124q^{124}-159q^{122}+38q^{120}+138q^{118}+88q^{116}-63q^{114}-137q^{112}-19q^{110}+80q^{108}+92q^{106}-2q^{104}-81q^{102}-57q^{100}+17q^{98}+75q^{96}+49q^{94}-23q^{92}-93q^{90}-46q^{88}+62q^{86}+102q^{84}+34q^{82}-121q^{80}-107q^{78}+32q^{76}+141q^{74}+101q^{72}-112q^{70}-148q^{68}-28q^{66}+124q^{64}+148q^{62}-49q^{60}-133q^{58}-86q^{56}+53q^{54}+136q^{52}+19q^{50}-62q^{48}-83q^{46}-12q^{44}+70q^{42}+38q^{40}-38q^{36}-24q^{34}+16q^{32}+14q^{30}+11q^{28}-5q^{26}-7q^{24}+2q^{22}+q^{20}+3q^{18}-q^{14}+q^{12}$
5	$q^{265}-2q^{263}-2q^{261}+3q^{259}+3q^{257}+3q^{255}-q^{253}-9q^{251}-13q^{249}+2q^{247}+20q^{245}+23q^{243}+6q^{241}-27q^{239}-49q^{237}-33q^{235}+37q^{233}+92q^{231}+71q^{229}-21q^{227}-131q^{225}-155q^{223}-29q^{221}+172q^{219}+255q^{217}+121q^{215}-157q^{213}-369q^{211}-275q^{209}+93q^{207}+447q^{205}+449q^{203}+42q^{201}-464q^{199}-612q^{197}-218q^{195}+403q^{193}+725q^{191}+409q^{189}-288q^{187}-747q^{185}-557q^{183}+129q^{181}+696q^{179}+645q^{177}+23q^{175}-589q^{173}-647q^{171}-147q^{169}+441q^{167}+591q^{165}+229q^{163}-299q^{161}-511q^{159}-260q^{157}+162q^{155}+395q^{153}+289q^{151}-49q^{149}-310q^{147}-287q^{145}-48q^{143}+215q^{141}+319q^{139}+152q^{137}-156q^{135}-340q^{133}-254q^{131}+77q^{129}+394q^{127}+373q^{125}-7q^{123}-426q^{121}-499q^{119}-98q^{117}+448q^{115}+620q^{113}+214q^{111}-413q^{109}-712q^{107}-360q^{105}+339q^{103}+747q^{101}+494q^{99}-202q^{97}-709q^{95}-595q^{93}+27q^{91}+607q^{89}+634q^{87}+132q^{85}-435q^{83}-592q^{81}-272q^{79}+239q^{77}+496q^{75}+313q^{73}-70q^{71}-337q^{69}-309q^{67}-55q^{65}+194q^{63}+240q^{61}+101q^{59}-72q^{57}-154q^{55}-101q^{53}+6q^{51}+77q^{49}+74q^{47}+18q^{45}-27q^{43}-36q^{41}-14q^{39}+4q^{37}+17q^{35}+12q^{33}-q^{31}-4q^{29}-q^{27}-q^{25}+q^{23}+3q^{21}-q^{17}+q^{15}$

A2 Invariants.

Weight	Invariant
1,0	$q^{32}+q^{30}-2q^{28}-q^{26}-2q^{24}-2q^{22}+q^{20}+3q^{16}+q^{14}+q^{12}+2q^{10}-q^{8}+q^{6}$
1,1	$q^{84}-4q^{82}+10q^{80}-20q^{78}+34q^{76}-54q^{74}+74q^{72}-92q^{70}+105q^{68}-104q^{66}+90q^{64}-60q^{62}+18q^{60}+36q^{58}-90q^{56}+142q^{54}-176q^{52}+200q^{50}-202q^{48}+182q^{46}-153q^{44}+96q^{42}-52q^{40}-10q^{38}+48q^{36}-86q^{34}+104q^{32}-104q^{30}+99q^{28}-72q^{26}+62q^{24}-34q^{22}+25q^{20}-10q^{18}+6q^{16}-2q^{14}+q^{12}$
2,0	$q^{80}+q^{78}-q^{76}-4q^{74}-3q^{72}+2q^{70}+q^{68}+3q^{64}+8q^{62}+4q^{60}-3q^{58}+2q^{54}-4q^{52}-6q^{50}-3q^{48}-3q^{46}-6q^{44}-2q^{42}+q^{40}-3q^{38}+6q^{34}+3q^{32}-3q^{30}+4q^{28}+6q^{26}+q^{24}-2q^{22}+3q^{20}+4q^{18}-q^{16}-q^{14}+q^{12}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{68}-2q^{66}+3q^{62}-6q^{60}+q^{58}+6q^{56}-5q^{54}+4q^{52}+10q^{50}-3q^{48}-q^{46}-q^{44}-7q^{42}-9q^{40}-7q^{38}+q^{36}-q^{34}-2q^{32}+10q^{30}+5q^{28}-4q^{26}+9q^{24}+2q^{22}-3q^{20}+4q^{18}+q^{16}-q^{14}+q^{12}$
1,0,0	$q^{43}+q^{41}+q^{39}-2q^{37}-q^{35}-4q^{33}-2q^{31}-2q^{29}+q^{27}+q^{25}+2q^{23}+3q^{21}+q^{19}+2q^{17}+2q^{13}-q^{11}+q^{9}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{90}+q^{88}-q^{86}-3q^{84}-q^{82}-5q^{78}-4q^{76}+6q^{74}+8q^{72}+3q^{70}+9q^{68}+14q^{66}+5q^{64}-5q^{62}-4q^{60}-9q^{58}-20q^{56}-17q^{54}-7q^{52}-10q^{50}-6q^{48}+9q^{46}+9q^{44}+2q^{42}+7q^{40}+12q^{38}+4q^{36}-q^{34}+4q^{32}+6q^{30}-q^{28}+3q^{24}+q^{22}-q^{20}+q^{18}$
1,0,0,0	$q^{54}+q^{52}+q^{50}+q^{48}-2q^{46}-q^{44}-4q^{42}-4q^{40}-2q^{38}-2q^{36}+q^{34}+q^{32}+3q^{30}+2q^{28}+3q^{26}+q^{24}+2q^{22}+q^{20}+2q^{16}-q^{14}+q^{12}$

B2 Invariants.

Weight	Invariant
0,1	$q^{68}-2q^{66}+4q^{64}-5q^{62}+6q^{60}-7q^{58}+6q^{56}-5q^{54}+2q^{52}-5q^{48}+7q^{46}-11q^{44}+11q^{42}-13q^{40}+11q^{38}-9q^{36}+7q^{34}-2q^{32}+5q^{28}-4q^{26}+7q^{24}-6q^{22}+7q^{20}-4q^{18}+3q^{16}-q^{14}+q^{12}$
1,0	$q^{110}-2q^{106}-2q^{104}+2q^{102}+4q^{100}-q^{98}-6q^{96}-3q^{94}+6q^{92}+6q^{90}-2q^{88}-6q^{86}+q^{84}+8q^{82}+5q^{80}-4q^{78}-3q^{76}+3q^{74}+3q^{72}-5q^{70}-7q^{68}-2q^{66}+2q^{64}-3q^{62}-7q^{60}-3q^{58}+3q^{56}+3q^{54}-4q^{52}-3q^{50}+5q^{48}+9q^{46}-5q^{42}-q^{40}+8q^{38}+6q^{36}-2q^{34}-5q^{32}+q^{30}+4q^{28}+2q^{26}-q^{24}-q^{22}+q^{18}$

D4 Invariants.

Weight	Invariant
1,0,0,0	$q^{94}-2q^{92}+2q^{90}-3q^{88}+4q^{86}-6q^{84}+4q^{82}-5q^{80}+6q^{78}-3q^{76}+4q^{74}+2q^{72}+5q^{70}+6q^{68}-4q^{66}+5q^{64}-10q^{62}+4q^{60}-17q^{58}-16q^{54}+4q^{52}-8q^{50}+5q^{48}-2q^{46}+6q^{44}+7q^{42}+2q^{40}+6q^{38}-2q^{36}+8q^{34}-3q^{32}+6q^{30}-4q^{28}+5q^{26}-q^{24}+2q^{22}-q^{20}+q^{18}$

G2 Invariants.

Weight

Invariant

1,0

$q^{162}-2q^{160}+4q^{158}-6q^{156}+3q^{154}-q^{152}-6q^{150}+14q^{148}-18q^{146}+20q^{144}-12q^{142}-q^{140}+17q^{138}-27q^{136}+34q^{134}-24q^{132}+7q^{130}+10q^{128}-21q^{126}+25q^{124}-16q^{122}+q^{120}+14q^{118}-20q^{116}+12q^{114}-24q^{110}+34q^{108}-36q^{106}+18q^{104}-2q^{102}-24q^{100}+40q^{98}-47q^{96}+34q^{94}-18q^{92}-7q^{90}+25q^{88}-33q^{86}+26q^{84}-9q^{82}-2q^{80}+16q^{78}-17q^{76}+9q^{74}+10q^{72}-21q^{70}+29q^{68}-21q^{66}+6q^{64}+17q^{62}-27q^{60}+33q^{58}-23q^{56}+12q^{54}+2q^{52}-14q^{50}+17q^{48}-14q^{46}+10q^{44}-2q^{42}-q^{40}+3q^{38}-3q^{36}+3q^{34}-q^{32}+q^{30}$

.

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 15"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$3t^{2}-8t+11-8t^{-1}+3t^{-2}$

In[5]:=

Conway[K][z]

Out[5]=

$3z^{4}+4z^{2}+1$

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]=

{ 33, -4 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{4}a^{12}-z^{2}a^{12}+3z^{5}a^{11}-5z^{3}a^{11}+2za^{11}+3z^{6}a^{10}-3z^{4}a^{10}-a^{10}+z^{7}a^{9}+6z^{5}a^{9}-14z^{3}a^{9}+8za^{9}+6z^{6}a^{8}-10z^{4}a^{8}+8z^{2}a^{8}-4a^{8}+z^{7}a^{7}+5z^{5}a^{7}-11z^{3}a^{7}+6za^{7}+3z^{6}a^{6}-5z^{4}a^{6}+5z^{2}a^{6}-3a^{6}+2z^{5}a^{5}-2z^{3}a^{5}+z^{4}a^{4}-2z^{2}a^{4}+a^{4}$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11n65,}

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 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]=

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

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]=

{K11n65,}

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:

(4, -7)

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 $16$ $-56$ $128$ ${\frac {776}{3}}$ ${\frac {112}{3}}$ $-896$ $-{\frac {4016}{3}}$ $-{\frac {704}{3}}$ $-152$ ${\frac {2048}{3}}$ $1568$ ${\frac {12416}{3}}$ ${\frac {1792}{3}}$ ${\frac {107222}{15}}$ ${\frac {6512}{15}}$ ${\frac {107648}{45}}$ ${\frac {250}{9}}$ ${\frac {4502}{15}}$

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 $s=$-4 is the signature of 8 15. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -8 -7 -6 -5 -4 -3 -2 -1 0 χ -3                 1 1 -5               2 1 -1 -7             3     3 -9           2 2     0 -11         4 3       1 -13       2 2         0 -15     2 4           -2 -17   1 2             1 -19   2               -2 -21 1                 1

Integral Khovanov Homology (db, data source)

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

The Coloured Jones Polynomials

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

$n$	$J_{n}$
2	$q^{-4}-2q^{-5}+q^{-6}+7q^{-7}-10q^{-8}-2q^{-9}+22q^{-10}-20q^{-11}-10q^{-12}+37q^{-13}-25q^{-14}-19q^{-15}+44q^{-16}-23q^{-17}-22q^{-18}+39q^{-19}-14q^{-20}-19q^{-21}+24q^{-22}-4q^{-23}-11q^{-24}+9q^{-25}-3q^{-27}+q^{-28}$
3	$q^{-6}-2q^{-7}+q^{-8}+3q^{-9}+2q^{-10}-10q^{-11}-3q^{-12}+18q^{-13}+14q^{-14}-31q^{-15}-24q^{-16}+35q^{-17}+52q^{-18}-51q^{-19}-68q^{-20}+45q^{-21}+101q^{-22}-51q^{-23}-118q^{-24}+38q^{-25}+144q^{-26}-37q^{-27}-152q^{-28}+24q^{-29}+160q^{-30}-15q^{-31}-159q^{-32}+5q^{-33}+148q^{-34}+10q^{-35}-136q^{-36}-15q^{-37}+109q^{-38}+30q^{-39}-89q^{-40}-30q^{-41}+60q^{-42}+32q^{-43}-40q^{-44}-25q^{-45}+20q^{-46}+20q^{-47}-11q^{-48}-11q^{-49}+4q^{-50}+5q^{-51}-3q^{-53}+q^{-54}$
4	$q^{-8}-2q^{-9}+q^{-10}+3q^{-11}-2q^{-12}+2q^{-13}-11q^{-14}+3q^{-15}+19q^{-16}+q^{-17}+4q^{-18}-51q^{-19}-11q^{-20}+57q^{-21}+39q^{-22}+36q^{-23}-133q^{-24}-82q^{-25}+78q^{-26}+120q^{-27}+153q^{-28}-216q^{-29}-221q^{-30}+31q^{-31}+204q^{-32}+350q^{-33}-240q^{-34}-373q^{-35}-89q^{-36}+240q^{-37}+563q^{-38}-200q^{-39}-482q^{-40}-228q^{-41}+226q^{-42}+718q^{-43}-132q^{-44}-522q^{-45}-336q^{-46}+179q^{-47}+788q^{-48}-60q^{-49}-496q^{-50}-394q^{-51}+105q^{-52}+764q^{-53}+19q^{-54}-402q^{-55}-406q^{-56}+6q^{-57}+646q^{-58}+93q^{-59}-251q^{-60}-356q^{-61}-94q^{-62}+449q^{-63}+128q^{-64}-86q^{-65}-246q^{-66}-143q^{-67}+239q^{-68}+101q^{-69}+18q^{-70}-118q^{-71}-117q^{-72}+89q^{-73}+45q^{-74}+39q^{-75}-34q^{-76}-59q^{-77}+23q^{-78}+9q^{-79}+20q^{-80}-4q^{-81}-18q^{-82}+4q^{-83}+5q^{-85}-3q^{-87}+q^{-88}$
5	$q^{-10}-2q^{-11}+q^{-12}+3q^{-13}-2q^{-14}-2q^{-15}+q^{-16}-5q^{-17}+4q^{-18}+16q^{-19}+3q^{-20}-15q^{-21}-17q^{-22}-27q^{-23}+13q^{-24}+61q^{-25}+59q^{-26}-12q^{-27}-88q^{-28}-134q^{-29}-40q^{-30}+143q^{-31}+232q^{-32}+127q^{-33}-134q^{-34}-383q^{-35}-294q^{-36}+115q^{-37}+499q^{-38}+510q^{-39}+49q^{-40}-640q^{-41}-805q^{-42}-205q^{-43}+656q^{-44}+1077q^{-45}+551q^{-46}-667q^{-47}-1385q^{-48}-827q^{-49}+542q^{-50}+1584q^{-51}+1247q^{-52}-414q^{-53}-1793q^{-54}-1526q^{-55}+190q^{-56}+1883q^{-57}+1874q^{-58}-8q^{-59}-1965q^{-60}-2072q^{-61}-211q^{-62}+1956q^{-63}+2293q^{-64}+372q^{-65}-1944q^{-66}-2389q^{-67}-542q^{-68}+1870q^{-69}+2477q^{-70}+680q^{-71}-1777q^{-72}-2493q^{-73}-805q^{-74}+1631q^{-75}+2454q^{-76}+941q^{-77}-1439q^{-78}-2387q^{-79}-1038q^{-80}+1209q^{-81}+2203q^{-82}+1153q^{-83}-911q^{-84}-2025q^{-85}-1188q^{-86}+621q^{-87}+1703q^{-88}+1211q^{-89}-299q^{-90}-1403q^{-91}-1137q^{-92}+54q^{-93}+1017q^{-94}+1021q^{-95}+160q^{-96}-706q^{-97}-821q^{-98}-268q^{-99}+396q^{-100}+627q^{-101}+308q^{-102}-200q^{-103}-414q^{-104}-270q^{-105}+42q^{-106}+259q^{-107}+214q^{-108}+12q^{-109}-136q^{-110}-136q^{-111}-41q^{-112}+58q^{-113}+88q^{-114}+36q^{-115}-26q^{-116}-44q^{-117}-20q^{-118}+3q^{-119}+18q^{-120}+20q^{-121}-4q^{-122}-11q^{-123}-3q^{-124}+5q^{-127}-3q^{-129}+q^{-130}$
6	$q^{-12}-2q^{-13}+q^{-14}+3q^{-15}-2q^{-16}-2q^{-17}-3q^{-18}+7q^{-19}-4q^{-20}+q^{-21}+18q^{-22}-6q^{-23}-14q^{-24}-25q^{-25}+9q^{-26}-4q^{-27}+20q^{-28}+82q^{-29}+17q^{-30}-42q^{-31}-124q^{-32}-60q^{-33}-73q^{-34}+60q^{-35}+299q^{-36}+216q^{-37}+49q^{-38}-301q^{-39}-346q^{-40}-472q^{-41}-113q^{-42}+618q^{-43}+804q^{-44}+644q^{-45}-179q^{-46}-714q^{-47}-1468q^{-48}-1020q^{-49}+496q^{-50}+1564q^{-51}+2011q^{-52}+879q^{-53}-459q^{-54}-2719q^{-55}-2904q^{-56}-790q^{-57}+1677q^{-58}+3681q^{-59}+3049q^{-60}+1117q^{-61}-3312q^{-62}-5153q^{-63}-3295q^{-64}+491q^{-65}+4699q^{-66}+5613q^{-67}+3891q^{-68}-2673q^{-69}-6793q^{-70}-6189q^{-71}-1759q^{-72}+4584q^{-73}+7623q^{-74}+6936q^{-75}-1104q^{-76}-7386q^{-77}-8535q^{-78}-4188q^{-79}+3648q^{-80}+8682q^{-81}+9364q^{-82}+622q^{-83}-7176q^{-84}-9954q^{-85}-6082q^{-86}+2500q^{-87}+8961q^{-88}+10872q^{-89}+1999q^{-90}-6597q^{-91}-10565q^{-92}-7282q^{-93}+1444q^{-94}+8747q^{-95}+11598q^{-96}+3029q^{-97}-5800q^{-98}-10582q^{-99}-7983q^{-100}+381q^{-101}+8084q^{-102}+11714q^{-103}+3946q^{-104}-4607q^{-105}-9983q^{-106}-8338q^{-107}-922q^{-108}+6758q^{-109}+11131q^{-110}+4841q^{-111}-2802q^{-112}-8521q^{-113}-8179q^{-114}-2448q^{-115}+4601q^{-116}+9567q^{-117}+5393q^{-118}-556q^{-119}-6075q^{-120}-7120q^{-121}-3682q^{-122}+1937q^{-123}+6946q^{-124}+5054q^{-125}+1386q^{-126}-3121q^{-127}-5044q^{-128}-3915q^{-129}-317q^{-130}+3866q^{-131}+3649q^{-132}+2197q^{-133}-685q^{-134}-2574q^{-135}-2977q^{-136}-1340q^{-137}+1399q^{-138}+1826q^{-139}+1782q^{-140}+478q^{-141}-716q^{-142}-1572q^{-143}-1173q^{-144}+170q^{-145}+506q^{-146}+898q^{-147}+546q^{-148}+87q^{-149}-552q^{-150}-592q^{-151}-98q^{-152}-17q^{-153}+280q^{-154}+257q^{-155}+183q^{-156}-121q^{-157}-198q^{-158}-47q^{-159}-74q^{-160}+47q^{-161}+69q^{-162}+91q^{-163}-18q^{-164}-47q^{-165}-5q^{-166}-31q^{-167}+3q^{-168}+9q^{-169}+29q^{-170}-4q^{-171}-11q^{-172}+4q^{-173}-7q^{-174}+5q^{-177}-3q^{-179}+q^{-180}$
7	$q^{-14}-2q^{-15}+q^{-16}+3q^{-17}-2q^{-18}-2q^{-19}-3q^{-20}+3q^{-21}+8q^{-22}-7q^{-23}+3q^{-24}+9q^{-25}-5q^{-26}-12q^{-27}-22q^{-28}+35q^{-30}+4q^{-31}+29q^{-32}+35q^{-33}-13q^{-34}-53q^{-35}-126q^{-36}-76q^{-37}+47q^{-38}+77q^{-39}+199q^{-40}+233q^{-41}+88q^{-42}-100q^{-43}-440q^{-44}-525q^{-45}-273q^{-46}-5q^{-47}+593q^{-48}+972q^{-49}+869q^{-50}+389q^{-51}-720q^{-52}-1582q^{-53}-1698q^{-54}-1248q^{-55}+377q^{-56}+2113q^{-57}+2971q^{-58}+2752q^{-59}+538q^{-60}-2251q^{-61}-4283q^{-62}-4961q^{-63}-2506q^{-64}+1649q^{-65}+5479q^{-66}+7594q^{-67}+5418q^{-68}+195q^{-69}-5775q^{-70}-10470q^{-71}-9460q^{-72}-3321q^{-73}+5123q^{-74}+12813q^{-75}+13835q^{-76}+7927q^{-77}-2683q^{-78}-14330q^{-79}-18648q^{-80}-13455q^{-81}-932q^{-82}+14463q^{-83}+22554q^{-84}+19601q^{-85}+6291q^{-86}-13197q^{-87}-25969q^{-88}-25701q^{-89}-12026q^{-90}+10580q^{-91}+27728q^{-92}+31256q^{-93}+18529q^{-94}-6954q^{-95}-28747q^{-96}-35892q^{-97}-24396q^{-98}+2813q^{-99}+28289q^{-100}+39474q^{-101}+30019q^{-102}+1428q^{-103}-27450q^{-104}-41993q^{-105}-34422q^{-106}-5428q^{-107}+25855q^{-108}+43610q^{-109}+38187q^{-110}+8921q^{-111}-24374q^{-112}-44475q^{-113}-40815q^{-114}-11853q^{-115}+22629q^{-116}+44836q^{-117}+42923q^{-118}+14253q^{-119}-21172q^{-120}-44808q^{-121}-44255q^{-122}-16241q^{-123}+19570q^{-124}+44457q^{-125}+45310q^{-126}+18010q^{-127}-18022q^{-128}-43820q^{-129}-45938q^{-130}-19653q^{-131}+16181q^{-132}+42728q^{-133}+46276q^{-134}+21395q^{-135}-13920q^{-136}-41152q^{-137}-46293q^{-138}-23147q^{-139}+11205q^{-140}+38759q^{-141}+45645q^{-142}+25027q^{-143}-7705q^{-144}-35542q^{-145}-44468q^{-146}-26695q^{-147}+3879q^{-148}+31235q^{-149}+42093q^{-150}+28009q^{-151}+598q^{-152}-26035q^{-153}-38856q^{-154}-28555q^{-155}-4784q^{-156}+20059q^{-157}+34166q^{-158}+28072q^{-159}+8765q^{-160}-13734q^{-161}-28722q^{-162}-26344q^{-163}-11572q^{-164}+7629q^{-165}+22422q^{-166}+23332q^{-167}+13290q^{-168}-2311q^{-169}-16171q^{-170}-19357q^{-171}-13373q^{-172}-1746q^{-173}+10239q^{-174}+14864q^{-175}+12281q^{-176}+4278q^{-177}-5489q^{-178}-10336q^{-179}-10095q^{-180}-5371q^{-181}+1885q^{-182}+6422q^{-183}+7634q^{-184}+5202q^{-185}+204q^{-186}-3350q^{-187}-5056q^{-188}-4347q^{-189}-1288q^{-190}+1312q^{-191}+3053q^{-192}+3187q^{-193}+1417q^{-194}-167q^{-195}-1526q^{-196}-2043q^{-197}-1225q^{-198}-356q^{-199}+639q^{-200}+1216q^{-201}+838q^{-202}+408q^{-203}-166q^{-204}-603q^{-205}-476q^{-206}-375q^{-207}-46q^{-208}+307q^{-209}+268q^{-210}+227q^{-211}+52q^{-212}-117q^{-213}-89q^{-214}-137q^{-215}-85q^{-216}+50q^{-217}+59q^{-218}+72q^{-219}+26q^{-220}-28q^{-221}+3q^{-222}-27q^{-223}-31q^{-224}+3q^{-225}+9q^{-226}+20q^{-227}+5q^{-228}-11q^{-229}+4q^{-230}-7q^{-232}+5q^{-235}-3q^{-237}+q^{-238}$

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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