8 21

8_20

9_1

(KnotPlot image)	See the full Rolfsen Knot Table. Visit 8 21's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 8 21 at Knotilus!

Knot presentations

Planar diagram presentation	X1425 X3849 X12,6,13,5 X13,16,14,1 X9,14,10,15 X15,10,16,11 X6,12,7,11 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 8, width is 3, Braid index is 3		[{9, 3}, {2, 7}, {6, 8}, {7, 9}, {4, 1}, {3, 6}, {5, 2}, {8, 4}, {1, 5}]

[edit Notes on presentations of 8 21]

Knot 8_21.

A graph: knot 8_21.

A part of a knot and a part of a graph.

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 21"];

In[4]:=

PD[K]

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

Out[4]=

X1425 X3849 X12,6,13,5 X13,16,14,1 X9,14,10,15 X15,10,16,11 X6,12,7,11 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]=

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

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]=

{ 3, 8, 3 }

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

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Three dimensional invariants

Symmetry type Reversible Unknotting number 1 3-genus 2 Bridge index 3 Super bridge index 4 Nakanishi index 1 Maximal Thurston-Bennequin number [-9][1] Hyperbolic Volume 6.78371 A-Polynomial See Data:8 21/A-polynomial

[edit Notes for 8 21'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 8 21's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{2}+4t-5+4t^{-1}-t^{-2}$
Conway polynomial	$1-z^{4}$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 15, -2 }
Jones polynomial	$2q^{-1}-2q^{-2}+3q^{-3}-3q^{-4}+2q^{-5}-2q^{-6}+q^{-7}$
HOMFLY-PT polynomial (db, data sources)	$z^{2}a^{6}+a^{6}-z^{4}a^{4}-3z^{2}a^{4}-3a^{4}+2z^{2}a^{2}+3a^{2}$
Kauffman polynomial (db, data sources)	$z^{4}a^{8}-2z^{2}a^{8}+2z^{5}a^{7}-5z^{3}a^{7}+2za^{7}+z^{6}a^{6}-z^{4}a^{6}-a^{6}+3z^{5}a^{5}-6z^{3}a^{5}+4za^{5}+z^{6}a^{4}-2z^{4}a^{4}+5z^{2}a^{4}-3a^{4}+z^{5}a^{3}-z^{3}a^{3}+2za^{3}+3z^{2}a^{2}-3a^{2}$
The A2 invariant	$q^{22}-2q^{14}-q^{12}-q^{10}+q^{8}+2q^{6}+q^{4}+2q^{2}$
The G2 invariant	$q^{114}-q^{112}+2q^{110}-3q^{108}+q^{104}-4q^{102}+5q^{100}-4q^{98}+3q^{96}+q^{94}-4q^{92}+5q^{90}-2q^{88}+2q^{86}+2q^{84}-4q^{82}+4q^{80}+2q^{78}-2q^{76}+4q^{74}-4q^{72}+3q^{70}-4q^{66}+3q^{64}-7q^{62}+5q^{60}-4q^{58}-3q^{56}+2q^{54}-6q^{52}+4q^{50}-5q^{48}-q^{46}+2q^{44}-3q^{42}+2q^{40}-3q^{36}+7q^{34}-2q^{32}+3q^{28}-3q^{26}+7q^{24}-2q^{22}+2q^{20}+q^{18}-q^{16}+4q^{14}-q^{12}+2q^{10}+q^{8}$

Further Quantum Invariants

Further quantum knot invariants for 8_21.

A1 Invariants.

Weight	Invariant
1	$q^{15}-q^{13}-q^{9}+q^{5}+2q$
2	$q^{42}-q^{40}-2q^{38}+2q^{36}+q^{34}-2q^{32}+q^{30}+2q^{28}-2q^{26}+q^{22}-q^{20}-q^{18}+2q^{14}-2q^{12}-q^{10}+3q^{8}-q^{6}-q^{4}+3q^{2}+1$
3	$q^{81}-q^{79}-2q^{77}+3q^{73}+3q^{71}-3q^{69}-4q^{67}+q^{65}+5q^{63}+q^{61}-6q^{59}-4q^{57}+5q^{55}+4q^{53}-3q^{51}-4q^{49}+4q^{47}+5q^{45}-2q^{43}-3q^{41}+2q^{37}-q^{35}-2q^{33}-3q^{31}+2q^{29}+4q^{27}-6q^{23}-q^{21}+7q^{19}+3q^{17}-6q^{15}-3q^{13}+4q^{11}+4q^{9}-2q^{7}-4q^{5}+2q^{3}+2q+2q^{-1}$
4	$q^{132}-q^{130}-2q^{128}+q^{124}+5q^{122}+q^{120}-3q^{118}-5q^{116}-6q^{114}+6q^{112}+8q^{110}+4q^{108}-4q^{106}-14q^{104}-4q^{102}+7q^{100}+15q^{98}+8q^{96}-13q^{94}-14q^{92}-3q^{90}+15q^{88}+15q^{86}-6q^{84}-16q^{82}-11q^{80}+9q^{78}+14q^{76}-q^{74}-10q^{72}-8q^{70}+6q^{68}+11q^{66}+2q^{64}-5q^{62}-5q^{60}+2q^{58}+5q^{56}+4q^{54}-2q^{52}-5q^{50}-6q^{48}+11q^{44}+4q^{42}-6q^{40}-15q^{38}-7q^{36}+15q^{34}+14q^{32}+q^{30}-18q^{28}-15q^{26}+11q^{24}+16q^{22}+10q^{20}-9q^{18}-15q^{16}+7q^{12}+10q^{10}-7q^{6}-3q^{4}-q^{2}+3+3q^{-2}+q^{-4}$
5	$q^{195}-q^{193}-2q^{191}+q^{187}+3q^{185}+3q^{183}+q^{181}-5q^{179}-7q^{177}-4q^{175}+q^{173}+8q^{171}+12q^{169}+7q^{167}-8q^{165}-16q^{163}-15q^{161}-3q^{159}+16q^{157}+26q^{155}+18q^{153}-8q^{151}-29q^{149}-32q^{147}-11q^{145}+25q^{143}+42q^{141}+26q^{139}-14q^{137}-45q^{135}-41q^{133}+41q^{129}+49q^{127}+17q^{125}-33q^{123}-49q^{121}-23q^{119}+22q^{117}+46q^{115}+27q^{113}-14q^{111}-40q^{109}-26q^{107}+9q^{105}+30q^{103}+21q^{101}-7q^{99}-24q^{97}-18q^{95}+5q^{93}+18q^{91}+11q^{89}-q^{87}-11q^{85}-12q^{83}+9q^{79}+10q^{77}+9q^{75}-2q^{73}-15q^{71}-15q^{69}-q^{67}+17q^{65}+27q^{63}+12q^{61}-19q^{59}-37q^{57}-25q^{55}+15q^{53}+45q^{51}+36q^{49}-9q^{47}-50q^{45}-48q^{43}-7q^{41}+46q^{39}+54q^{37}+16q^{35}-32q^{33}-52q^{31}-27q^{29}+16q^{27}+43q^{25}+33q^{23}-q^{21}-24q^{19}-25q^{17}-9q^{15}+10q^{13}+21q^{11}+10q^{9}-2q^{7}-6q^{5}-10q^{3}-4q+2q^{-1}+4q^{-3}+2q^{-5}+2q^{-7}$
6	$q^{270}-q^{268}-2q^{266}+q^{262}+3q^{260}+q^{258}+3q^{256}-q^{254}-7q^{252}-6q^{250}-4q^{248}+2q^{246}+5q^{244}+15q^{242}+11q^{240}+q^{238}-10q^{236}-19q^{234}-19q^{232}-16q^{230}+14q^{228}+31q^{226}+34q^{224}+21q^{222}-5q^{220}-35q^{218}-61q^{216}-40q^{214}-q^{212}+46q^{210}+71q^{208}+68q^{206}+18q^{204}-60q^{202}-94q^{200}-86q^{198}-21q^{196}+59q^{194}+123q^{192}+109q^{190}+19q^{188}-76q^{186}-137q^{184}-114q^{182}-24q^{180}+99q^{178}+150q^{176}+105q^{174}-113q^{170}-148q^{168}-94q^{166}+35q^{164}+126q^{162}+130q^{160}+56q^{158}-58q^{156}-121q^{154}-104q^{152}-6q^{150}+78q^{148}+101q^{146}+57q^{144}-24q^{142}-77q^{140}-75q^{138}-11q^{136}+42q^{134}+58q^{132}+33q^{130}-12q^{128}-40q^{126}-40q^{124}-6q^{122}+22q^{120}+32q^{118}+19q^{116}-3q^{114}-20q^{112}-26q^{110}-15q^{108}+3q^{106}+25q^{104}+27q^{102}+19q^{100}-6q^{98}-32q^{96}-47q^{94}-28q^{92}+23q^{90}+55q^{88}+68q^{86}+28q^{84}-36q^{82}-95q^{80}-90q^{78}-11q^{76}+72q^{74}+129q^{72}+96q^{70}-3q^{68}-122q^{66}-156q^{64}-80q^{62}+39q^{60}+148q^{58}+156q^{56}+68q^{54}-81q^{52}-163q^{50}-134q^{48}-39q^{46}+86q^{44}+147q^{42}+118q^{40}+7q^{38}-86q^{36}-112q^{34}-83q^{32}-6q^{30}+64q^{28}+91q^{26}+51q^{24}+3q^{22}-35q^{20}-53q^{18}-35q^{16}-4q^{14}+26q^{12}+27q^{10}+19q^{8}+8q^{6}-8q^{4}-15q^{2}-11-3q^{-2}+q^{-4}+3q^{-6}+3q^{-8}+3q^{-10}+q^{-12}$

A2 Invariants.

Weight	Invariant
1,0	$q^{22}-2q^{14}-q^{12}-q^{10}+q^{8}+2q^{6}+q^{4}+2q^{2}$
1,1	$q^{60}-2q^{58}+4q^{56}-8q^{54}+9q^{52}-10q^{50}+12q^{48}-8q^{46}+4q^{44}-8q^{40}+10q^{38}-15q^{36}+18q^{34}-16q^{32}+20q^{30}-12q^{28}+12q^{26}-6q^{24}-2q^{22}+q^{20}-14q^{18}+8q^{16}-12q^{14}+10q^{12}-4q^{10}+8q^{8}+2q^{6}+4q^{4}+2q^{2}$
2,0	$q^{56}-q^{52}-q^{50}+q^{46}-q^{44}+q^{40}-q^{36}+2q^{32}+2q^{30}+q^{28}+q^{26}-q^{24}-3q^{22}-2q^{20}-2q^{18}-3q^{16}-q^{14}+2q^{12}+2q^{10}+2q^{8}+2q^{6}+4q^{4}+q^{2}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{48}-q^{46}-3q^{40}+q^{38}+q^{36}+3q^{32}+4q^{30}+q^{28}-2q^{24}-4q^{22}-5q^{20}-5q^{18}+2q^{12}+6q^{10}+3q^{8}+q^{6}+3q^{4}$
1,0,0	$q^{29}+q^{25}-2q^{19}-2q^{17}-2q^{15}-q^{13}+q^{11}+2q^{9}+3q^{7}+q^{5}+2q^{3}$
1,0,1	$q^{78}-2q^{76}+3q^{74}-3q^{72}-q^{70}+5q^{68}-8q^{66}+10q^{64}-6q^{62}+2q^{60}+5q^{58}-11q^{56}+12q^{54}-12q^{52}+2q^{50}-q^{48}-6q^{46}+q^{44}+q^{42}+6q^{40}+3q^{38}+18q^{36}-3q^{34}+13q^{32}-5q^{30}-11q^{28}-2q^{26}-25q^{24}-2q^{22}-11q^{20}-4q^{18}+9q^{16}+3q^{14}+12q^{12}+10q^{10}+5q^{8}+5q^{6}+2q^{4}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{62}+q^{56}-q^{54}-4q^{52}-3q^{50}-q^{48}-q^{46}+q^{44}+6q^{42}+9q^{40}+6q^{38}+5q^{36}+3q^{34}-4q^{32}-9q^{30}-8q^{28}-10q^{26}-9q^{24}-4q^{22}+q^{20}+3q^{18}+4q^{16}+8q^{14}+7q^{12}+4q^{10}+2q^{8}+3q^{6}$
1,0,0,0	$q^{36}+q^{32}+q^{30}-2q^{24}-2q^{22}-3q^{20}-2q^{18}-q^{16}+q^{14}+2q^{12}+3q^{10}+3q^{8}+q^{6}+2q^{4}$

B2 Invariants.

Weight	Invariant
0,1	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{48}-q^{46}+2 q^{44}-2 q^{42}+q^{40}-q^{38}+q^{36}-q^{32}+2 q^{30}-3 q^{28}+2 q^{26}-4 q^{24}+2 q^{22}-3 q^{20}+q^{18}+2 q^{12}+3 q^8-q^6+3 q^4}
1,0	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{78}-q^{74}-q^{72}+q^{70}+q^{68}-2 q^{66}-2 q^{64}+2 q^{60}+q^{58}-q^{56}+2 q^{52}+2 q^{50}+q^{48}+q^{44}+q^{42}-3 q^{38}-2 q^{36}-q^{34}-q^{32}-3 q^{30}-3 q^{28}+q^{24}-q^{20}+2 q^{18}+3 q^{16}+3 q^{14}+q^8+3 q^6}

D4 Invariants.

Weight

Invariant

1,0,0,0

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{66}-q^{64}+q^{62}-2 q^{60}+q^{58}-2 q^{56}-q^{52}+q^{50}+q^{48}+q^{46}+4 q^{44}+2 q^{42}+4 q^{40}-q^{38}+2 q^{36}-4 q^{34}-q^{32}-7 q^{30}-4 q^{28}-6 q^{26}-2 q^{24}-q^{22}+3 q^{18}+3 q^{16}+6 q^{14}+3 q^{12}+4 q^{10}+3 q^6}

G2 Invariants.

Weight

Invariant

1,0

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{114}-q^{112}+2 q^{110}-3 q^{108}+q^{104}-4 q^{102}+5 q^{100}-4 q^{98}+3 q^{96}+q^{94}-4 q^{92}+5 q^{90}-2 q^{88}+2 q^{86}+2 q^{84}-4 q^{82}+4 q^{80}+2 q^{78}-2 q^{76}+4 q^{74}-4 q^{72}+3 q^{70}-4 q^{66}+3 q^{64}-7 q^{62}+5 q^{60}-4 q^{58}-3 q^{56}+2 q^{54}-6 q^{52}+4 q^{50}-5 q^{48}-q^{46}+2 q^{44}-3 q^{42}+2 q^{40}-3 q^{36}+7 q^{34}-2 q^{32}+3 q^{28}-3 q^{26}+7 q^{24}-2 q^{22}+2 q^{20}+q^{18}-q^{16}+4 q^{14}-q^{12}+2 q^{10}+q^8}

.

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 21"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-t^{2}+4t-5+4t^{-1}-t^{-2}$

In[5]:=

Conway[K][z]

Out[5]=

$1-z^{4}$

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]=

{ 15, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

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

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_136,}

Same Jones Polynomial (up to mirroring, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 21"];

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]=

{ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -t^2+4 t-5+4 t^{-1} - t^{-2} } , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2 q^{-1} -2 q^{-2} +3 q^{-3} -3 q^{-4} +2 q^{-5} -2 q^{-6} + q^{-7} } }

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_136,}

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:

(0, 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0} $8$ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -16} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 8} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\frac{16}{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\frac{64}{3}} $-24$ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 32} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 136} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\frac{8}{3}} ${\frac {424}{3}}$ ${\frac {40}{3}}$ $8$

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 8 21. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-3$ $i=-1$ $r=-6$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-3$ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{2}\oplus{\mathbb Z}_2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}\oplus{\mathbb Z}_2^{2}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{2}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-1} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}\oplus{\mathbb Z}_2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=0} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}\oplus{\mathbb Z}_2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{2}}

The Coloured Jones Polynomials

The Coloured Jones Polynomials (in the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n+1} -dimensional representation of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle sl(2)} )

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J_n}
2	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{-1} +2 q^{-2} -4 q^{-3} + q^{-4} +6 q^{-5} -8 q^{-6} +10 q^{-8} -10 q^{-9} - q^{-10} +10 q^{-11} -8 q^{-12} -2 q^{-13} +8 q^{-14} -4 q^{-15} -3 q^{-16} +5 q^{-17} - q^{-18} -2 q^{-19} + q^{-20} }
3	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2 q^{-1} -6 q^{-4} +4 q^{-5} +6 q^{-6} -13 q^{-8} + q^{-9} +15 q^{-10} +4 q^{-11} -21 q^{-12} -4 q^{-13} +21 q^{-14} +8 q^{-15} -23 q^{-16} -9 q^{-17} +22 q^{-18} +9 q^{-19} -20 q^{-20} -11 q^{-21} +19 q^{-22} +10 q^{-23} -13 q^{-24} -12 q^{-25} +11 q^{-26} +11 q^{-27} -6 q^{-28} -11 q^{-29} +2 q^{-30} +9 q^{-31} + q^{-32} -7 q^{-33} -2 q^{-34} +4 q^{-35} +2 q^{-36} - q^{-37} -2 q^{-38} + q^{-39} }
4	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1+2 q^{-1} -4 q^{-3} -2 q^{-4} -3 q^{-5} +9 q^{-6} +10 q^{-7} -7 q^{-8} -9 q^{-9} -18 q^{-10} +15 q^{-11} +29 q^{-12} - q^{-13} -14 q^{-14} -44 q^{-15} +12 q^{-16} +48 q^{-17} +12 q^{-18} -13 q^{-19} -66 q^{-20} +4 q^{-21} +57 q^{-22} +22 q^{-23} -6 q^{-24} -77 q^{-25} -2 q^{-26} +58 q^{-27} +25 q^{-28} -76 q^{-30} -5 q^{-31} +51 q^{-32} +25 q^{-33} +7 q^{-34} -67 q^{-35} -10 q^{-36} +37 q^{-37} +23 q^{-38} +16 q^{-39} -52 q^{-40} -15 q^{-41} +17 q^{-42} +18 q^{-43} +26 q^{-44} -31 q^{-45} -15 q^{-46} - q^{-47} +7 q^{-48} +27 q^{-49} -10 q^{-50} -8 q^{-51} -9 q^{-52} -4 q^{-53} +17 q^{-54} -5 q^{-57} -6 q^{-58} +5 q^{-59} + q^{-60} +2 q^{-61} - q^{-62} -2 q^{-63} + q^{-64} }
5	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2 q+2 q^{-1} -2 q^{-2} -6 q^{-3} -6 q^{-4} +6 q^{-5} +4 q^{-6} +14 q^{-7} +9 q^{-8} -17 q^{-9} -25 q^{-10} -10 q^{-11} +5 q^{-12} +37 q^{-13} +43 q^{-14} -7 q^{-15} -52 q^{-16} -53 q^{-17} -20 q^{-18} +57 q^{-19} +91 q^{-20} +31 q^{-21} -60 q^{-22} -106 q^{-23} -61 q^{-24} +55 q^{-25} +132 q^{-26} +76 q^{-27} -51 q^{-28} -136 q^{-29} -101 q^{-30} +43 q^{-31} +150 q^{-32} +107 q^{-33} -36 q^{-34} -146 q^{-35} -119 q^{-36} +29 q^{-37} +150 q^{-38} +120 q^{-39} -25 q^{-40} -145 q^{-41} -120 q^{-42} +20 q^{-43} +138 q^{-44} +121 q^{-45} -15 q^{-46} -133 q^{-47} -113 q^{-48} +7 q^{-49} +115 q^{-50} +115 q^{-51} +2 q^{-52} -105 q^{-53} -104 q^{-54} -14 q^{-55} +80 q^{-56} +101 q^{-57} +28 q^{-58} -64 q^{-59} -85 q^{-60} -38 q^{-61} +35 q^{-62} +75 q^{-63} +44 q^{-64} -14 q^{-65} -53 q^{-66} -46 q^{-67} -6 q^{-68} +34 q^{-69} +40 q^{-70} +17 q^{-71} -13 q^{-72} -30 q^{-73} -23 q^{-74} -2 q^{-75} +19 q^{-76} +20 q^{-77} +8 q^{-78} -4 q^{-79} -15 q^{-80} -12 q^{-81} +8 q^{-83} +7 q^{-84} +4 q^{-85} -7 q^{-87} -4 q^{-88} + q^{-89} +2 q^{-90} + q^{-91} +2 q^{-92} - q^{-93} -2 q^{-94} + q^{-95} }
6	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^3+2 q^2-2 q^{-1} -4 q^{-2} -8 q^{-3} -3 q^{-4} +9 q^{-5} +16 q^{-6} +11 q^{-7} +6 q^{-8} -5 q^{-9} -38 q^{-10} -34 q^{-11} -9 q^{-12} +34 q^{-13} +49 q^{-14} +54 q^{-15} +35 q^{-16} -65 q^{-17} -104 q^{-18} -86 q^{-19} +5 q^{-20} +75 q^{-21} +147 q^{-22} +146 q^{-23} -36 q^{-24} -165 q^{-25} -211 q^{-26} -90 q^{-27} +46 q^{-28} +229 q^{-29} +295 q^{-30} +52 q^{-31} -173 q^{-32} -320 q^{-33} -209 q^{-34} -30 q^{-35} +263 q^{-36} +414 q^{-37} +151 q^{-38} -140 q^{-39} -377 q^{-40} -292 q^{-41} -109 q^{-42} +258 q^{-43} +473 q^{-44} +215 q^{-45} -100 q^{-46} -390 q^{-47} -324 q^{-48} -160 q^{-49} +239 q^{-50} +488 q^{-51} +241 q^{-52} -75 q^{-53} -382 q^{-54} -326 q^{-55} -182 q^{-56} +221 q^{-57} +477 q^{-58} +247 q^{-59} -58 q^{-60} -360 q^{-61} -313 q^{-62} -192 q^{-63} +193 q^{-64} +443 q^{-65} +247 q^{-66} -30 q^{-67} -315 q^{-68} -288 q^{-69} -208 q^{-70} +140 q^{-71} +379 q^{-72} +245 q^{-73} +23 q^{-74} -234 q^{-75} -244 q^{-76} -231 q^{-77} +56 q^{-78} +281 q^{-79} +228 q^{-80} +86 q^{-81} -120 q^{-82} -170 q^{-83} -235 q^{-84} -35 q^{-85} +152 q^{-86} +174 q^{-87} +121 q^{-88} -7 q^{-89} -65 q^{-90} -190 q^{-91} -86 q^{-92} +29 q^{-93} +84 q^{-94} +98 q^{-95} +54 q^{-96} +30 q^{-97} -100 q^{-98} -72 q^{-99} -35 q^{-100} +4 q^{-101} +33 q^{-102} +46 q^{-103} +64 q^{-104} -22 q^{-105} -22 q^{-106} -32 q^{-107} -21 q^{-108} -14 q^{-109} +7 q^{-110} +43 q^{-111} +4 q^{-112} +8 q^{-113} -6 q^{-114} -8 q^{-115} -17 q^{-116} -10 q^{-117} +13 q^{-118} + q^{-119} +8 q^{-120} +3 q^{-121} +3 q^{-122} -7 q^{-123} -6 q^{-124} +3 q^{-125} -2 q^{-126} +2 q^{-127} + q^{-128} +2 q^{-129} - q^{-130} -2 q^{-131} + q^{-132} }
7	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2 q^5+2 q^3-2 q-6-4 q^{-1} -6 q^{-2} +4 q^{-4} +16 q^{-5} +24 q^{-6} +15 q^{-7} -9 q^{-8} -23 q^{-9} -37 q^{-10} -45 q^{-11} -25 q^{-12} +17 q^{-13} +81 q^{-14} +98 q^{-15} +55 q^{-16} +8 q^{-17} -74 q^{-18} -151 q^{-19} -161 q^{-20} -87 q^{-21} +83 q^{-22} +223 q^{-23} +236 q^{-24} +188 q^{-25} +3 q^{-26} -240 q^{-27} -376 q^{-28} -357 q^{-29} -83 q^{-30} +249 q^{-31} +454 q^{-32} +510 q^{-33} +257 q^{-34} -187 q^{-35} -546 q^{-36} -691 q^{-37} -415 q^{-38} +115 q^{-39} +563 q^{-40} +832 q^{-41} +601 q^{-42} +6 q^{-43} -578 q^{-44} -954 q^{-45} -745 q^{-46} -115 q^{-47} +541 q^{-48} +1023 q^{-49} +886 q^{-50} +224 q^{-51} -512 q^{-52} -1072 q^{-53} -964 q^{-54} -303 q^{-55} +451 q^{-56} +1086 q^{-57} +1036 q^{-58} +372 q^{-59} -424 q^{-60} -1093 q^{-61} -1056 q^{-62} -406 q^{-63} +378 q^{-64} +1078 q^{-65} +1082 q^{-66} +436 q^{-67} -359 q^{-68} -1072 q^{-69} -1078 q^{-70} -446 q^{-71} +336 q^{-72} +1051 q^{-73} +1075 q^{-74} +455 q^{-75} -314 q^{-76} -1032 q^{-77} -1068 q^{-78} -461 q^{-79} +299 q^{-80} +1000 q^{-81} +1043 q^{-82} +468 q^{-83} -255 q^{-84} -956 q^{-85} -1032 q^{-86} -482 q^{-87} +220 q^{-88} +897 q^{-89} +985 q^{-90} +499 q^{-91} -139 q^{-92} -812 q^{-93} -954 q^{-94} -527 q^{-95} +70 q^{-96} +711 q^{-97} +879 q^{-98} +546 q^{-99} +40 q^{-100} -572 q^{-101} -813 q^{-102} -564 q^{-103} -136 q^{-104} +432 q^{-105} +694 q^{-106} +552 q^{-107} +245 q^{-108} -268 q^{-109} -572 q^{-110} -524 q^{-111} -311 q^{-112} +113 q^{-113} +415 q^{-114} +454 q^{-115} +361 q^{-116} +24 q^{-117} -269 q^{-118} -352 q^{-119} -353 q^{-120} -123 q^{-121} +117 q^{-122} +236 q^{-123} +315 q^{-124} +173 q^{-125} -7 q^{-126} -119 q^{-127} -232 q^{-128} -178 q^{-129} -65 q^{-130} +15 q^{-131} +146 q^{-132} +147 q^{-133} +92 q^{-134} +45 q^{-135} -69 q^{-136} -87 q^{-137} -78 q^{-138} -78 q^{-139} +3 q^{-140} +44 q^{-141} +56 q^{-142} +69 q^{-143} +14 q^{-144} - q^{-145} -12 q^{-146} -54 q^{-147} -29 q^{-148} -15 q^{-149} +2 q^{-150} +29 q^{-151} +12 q^{-152} +15 q^{-153} +17 q^{-154} -8 q^{-155} -11 q^{-156} -14 q^{-157} -12 q^{-158} +6 q^{-159} - q^{-160} +3 q^{-161} +10 q^{-162} +3 q^{-163} +2 q^{-164} -4 q^{-165} -6 q^{-166} + q^{-167} -2 q^{-169} +2 q^{-170} + q^{-171} +2 q^{-172} - q^{-173} -2 q^{-174} + q^{-175} }

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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