8 20

8_19

8_21

(KnotPlot image)	See the full Rolfsen Knot Table. Visit 8 20's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 8 20 at Knotilus!
	8_20 is also known as the pretzel knot P(3,-3,2). Its complement contains no complete totally geodesic immersed surfaces.[citation needed] This appears to be the Ashley/oysterman stopper knot of practical knot tying.

The Oysterman's stopper[1]

Knot presentations

Planar diagram presentation	X4251 X8493 X5,12,6,13 X13,16,14,1 X9,14,10,15 X15,10,16,11 X11,6,12,7 X2837
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	[3,21,2-]

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

[edit Notes on presentations of 8 20]

Knot 8_20.

A graph, knot 8_20.

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 20"];

In[4]:=

PD[K]

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

Out[4]=

X4251 X8493 X5,12,6,13 X13,16,14,1 X9,14,10,15 X15,10,16,11 X11,6,12,7 X2837

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]=

[3,21,2-]

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]=

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 \textrm{BR}(3,\{1,1,1,-2,-1,-1,-1,-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[{3, 8}, {2, 4}, {1, 3}, {11, 9}, {8, 10}, {9, 5}, {4, 6}, {5, 7}, {6, 11}, {10, 2}, {7, 1}]

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 [-6][-2] Hyperbolic Volume 4.1249 A-Polynomial See Data:8 20/A-polynomial

[edit Notes for 8 20's three dimensional invariants] 8_20 ribbon diagram from A. Kawauchi's text. Ribbon diagram for 8_20

Four dimensional invariants

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

[edit Notes for 8 20's four dimensional invariants]

Polynomial invariants

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

Further Quantum Invariants

Further quantum knot invariants for 8_20.

A1 Invariants.

Weight	Invariant
1	$-q^{11}+q^{5}+q^{3}+q+q^{-1}-q^{-3}$
2	$q^{32}-q^{28}-q^{22}-q^{20}+q^{12}+q^{10}+q^{6}+q^{4}+q^{2}+q^{-2}-q^{-6}$
3	$-q^{63}+q^{59}+q^{57}-q^{53}+q^{49}+q^{47}-q^{43}-q^{41}-q^{35}-2q^{33}+q^{29}+q^{27}-q^{25}+q^{21}-q^{17}+q^{15}+q^{13}+q^{5}+2q^{3}+2q+2q^{-7}-q^{-9}-2q^{-11}-q^{-13}+q^{-17}$
4	$q^{104}-q^{100}-q^{98}-q^{96}+q^{94}+q^{92}+q^{90}-2q^{86}-q^{84}+q^{80}+2q^{78}+q^{76}-q^{72}-q^{70}+q^{68}+2q^{66}+q^{64}-q^{62}-3q^{60}-2q^{58}+2q^{54}+q^{52}-2q^{50}-2q^{48}+2q^{44}+2q^{42}-q^{40}-2q^{38}+q^{34}+q^{32}-q^{30}-q^{28}-q^{20}-q^{18}+q^{16}+q^{14}+q^{12}+q^{10}+q^{8}+q^{4}+3q^{2}+4+2q^{-2}-q^{-4}-4q^{-6}-2q^{-8}+3q^{-10}+2q^{-12}-q^{-14}-4q^{-16}-3q^{-18}+q^{-20}+2q^{-22}+2q^{-24}-q^{-28}$
5	$-q^{155}+q^{151}+q^{149}+q^{147}-q^{143}-2q^{141}-q^{139}+q^{135}+2q^{133}+2q^{131}-2q^{127}-2q^{125}-2q^{123}-q^{121}+q^{119}+2q^{117}+2q^{115}+q^{113}-q^{111}-3q^{109}-2q^{107}+3q^{103}+4q^{101}+3q^{99}-3q^{95}-4q^{93}-2q^{91}+2q^{89}+4q^{87}+3q^{85}-q^{83}-4q^{81}-5q^{79}-2q^{77}+3q^{75}+4q^{73}+2q^{71}-q^{69}-4q^{67}-2q^{65}+q^{63}+4q^{61}+3q^{59}-3q^{55}-3q^{53}+2q^{49}+2q^{47}-3q^{43}-2q^{41}+q^{37}-2q^{33}-2q^{31}+q^{27}+q^{25}+q^{15}+2q^{13}+3q^{11}+3q^{9}+2q^{7}-q^{5}-2q^{3}+3q^{-1}+5q^{-3}+5q^{-5}-q^{-7}-7q^{-9}-7q^{-11}-2q^{-13}+3q^{-15}+7q^{-17}+3q^{-19}-3q^{-21}-6q^{-23}-4q^{-25}+q^{-27}+3q^{-29}+3q^{-31}+q^{-33}-q^{-35}-q^{-37}$
6	$q^{216}-q^{212}-q^{210}-q^{208}+2q^{202}+2q^{200}+q^{198}-q^{194}-2q^{192}-3q^{190}-q^{188}+2q^{184}+3q^{182}+3q^{180}+2q^{178}-q^{176}-2q^{174}-3q^{172}-3q^{170}-2q^{168}+3q^{164}+4q^{162}+3q^{160}+q^{158}-2q^{156}-5q^{154}-6q^{152}-3q^{150}+q^{148}+4q^{146}+6q^{144}+6q^{142}+2q^{140}-4q^{138}-6q^{136}-6q^{134}-3q^{132}+3q^{130}+8q^{128}+8q^{126}+4q^{124}-q^{122}-7q^{120}-9q^{118}-5q^{116}+q^{114}+6q^{112}+7q^{110}+5q^{108}-2q^{106}-8q^{104}-8q^{102}-4q^{100}+2q^{98}+7q^{96}+9q^{94}+4q^{92}-3q^{90}-7q^{88}-6q^{86}-q^{84}+4q^{82}+8q^{80}+5q^{78}-2q^{76}-6q^{74}-5q^{72}-q^{70}+3q^{68}+6q^{66}+3q^{64}-3q^{62}-5q^{60}-4q^{58}-q^{56}+2q^{54}+2q^{52}-3q^{48}-2q^{46}+q^{42}+q^{40}-q^{36}-2q^{34}-q^{32}+q^{30}+2q^{28}+2q^{26}+2q^{24}+q^{22}-q^{18}-2q^{16}-q^{14}+q^{12}+4q^{10}+6q^{8}+6q^{6}+2q^{4}-3q^{2}-6-5q^{-2}-2q^{-4}+6q^{-6}+11q^{-8}+9q^{-10}+q^{-12}-8q^{-14}-12q^{-16}-13q^{-18}-2q^{-20}+10q^{-22}+12q^{-24}+7q^{-26}-q^{-28}-7q^{-30}-11q^{-32}-6q^{-34}+q^{-36}+5q^{-38}+5q^{-40}+4q^{-42}+2q^{-44}-2q^{-46}-q^{-48}-q^{-50}-q^{-52}-q^{-54}+q^{-58}$

A2 Invariants.

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

A3 Invariants.

Weight	Invariant
0,1,0	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^{34}+q^{30}-q^{26}-2 q^{24}-4 q^{22}-4 q^{20}-3 q^{18}+2 q^{14}+6 q^{12}+6 q^{10}+6 q^8+3 q^6+q^4-q^2-2-2 q^{-2} - q^{-4} - q^{-6} + q^{-10} }
1,0,0	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^{21}-q^{19}-2 q^{17}-q^{15}+2 q^{11}+3 q^9+3 q^7+2 q^5+q^3- q^{-1} - q^{-5} }
1,0,1	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^{56}+2 q^{52}+q^{48}-q^{44}+q^{42}-2 q^{40}+q^{38}-4 q^{36}-2 q^{34}-6 q^{32}-5 q^{30}-5 q^{28}-6 q^{26}-2 q^{22}+6 q^{20}+5 q^{18}+10 q^{16}+9 q^{14}+9 q^{12}+8 q^{10}+2 q^8+3 q^6-2 q^4-3- q^{-2} - q^{-4} -2 q^{-6} - q^{-8} - q^{-10} + q^{-16} }

A4 Invariants.

Weight	Invariant
0,1,0,0	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^{44}+q^{42}+2 q^{40}+2 q^{38}+q^{36}-q^{34}-3 q^{32}-6 q^{30}-9 q^{28}-9 q^{26}-7 q^{24}-3 q^{22}+q^{20}+8 q^{18}+12 q^{16}+13 q^{14}+12 q^{12}+9 q^{10}+3 q^8-q^6-4 q^4-5 q^2-5-3 q^{-2} - q^{-4} + q^{-10} + q^{-12} }
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^{26}-q^{24}-2 q^{22}-2 q^{20}-q^{18}+2 q^{14}+3 q^{12}+4 q^{10}+3 q^8+2 q^6+q^4-1- q^{-2} - q^{-6} }

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^{34}-q^{30}-q^{26}+q^{18}+2 q^{14}+2 q^{10}+q^6+q^4+q^2+ q^{-4} - q^{-6} - q^{-10} }
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^{56}+q^{48}-q^{44}-q^{42}-q^{38}-2 q^{36}-2 q^{34}-2 q^{32}-q^{30}-q^{28}+2 q^{22}+2 q^{20}+3 q^{18}+2 q^{16}+3 q^{14}+3 q^{12}+2 q^{10}+q^6-1- q^{-2} - q^{-4} - q^{-8} - q^{-10} + q^{-16} }

D4 Invariants.

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

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^{80}+q^{76}-q^{74}-2 q^{68}+q^{66}-q^{64}-q^{62}-q^{60}-2 q^{58}-3 q^{52}-q^{50}-q^{48}+q^{44}-2 q^{42}+q^{40}+q^{38}+2 q^{36}+q^{34}+2 q^{30}+2 q^{28}+3 q^{26}+2 q^{22}+3 q^{20}+q^{18}+q^{16}+q^{14}+3 q^{10}-2 q^6+q^4+1- q^{-2} -2 q^{-4} - q^{-12} - q^{-14} - q^{-20} + q^{-24} }

.

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 20"];

In[4]:=

Alexander[K][t]

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

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^2-2 t+3-2 t^{-1} + t^{-2} }

In[5]:=

Conway[K][z]

Out[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 z^4+2 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]=

{ 9, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

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

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_140, K11n73, K11n74,}

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 20"];

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

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_140, K11n73, K11n74,}

In[6]:=

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

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

Out[6]=

{}

Vassiliev invariants

V2 and V3:

(2, -2)

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 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 -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 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 \frac{172}{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{20}{3}} $-128$ 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{640}{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}} 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 -48} 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{256}{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 128} 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{1376}{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{160}{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{11911}{15}} $-{\frac {524}{15}}$ 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{15484}{45}} 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{233}{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 \frac{631}{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 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^rq^j} are shown, along with their alternating sums 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 \chi} (fixed 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} , alternation over 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} ). 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 20. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

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 \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\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 i=-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 i=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 r=-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 {\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=-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 {\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=-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}} 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} 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=-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}_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}^{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=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}_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}}

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^2+q+1-2 q^{-1} +2 q^{-2} + q^{-3} -2 q^{-4} + q^{-5} +2 q^{-6} -2 q^{-7} +2 q^{-9} -2 q^{-10} - q^{-11} +2 q^{-12} - q^{-13} - q^{-14} + q^{-15} }
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 q^7-q^6-q^5-q^4+2 q^3+2 q^2-3 q-1+2 q^{-1} +4 q^{-2} -3 q^{-3} -2 q^{-4} + q^{-5} +4 q^{-6} -3 q^{-7} - q^{-8} + q^{-9} +2 q^{-10} -2 q^{-11} + q^{-14} - q^{-17} - q^{-18} + q^{-19} + q^{-20} - q^{-21} -2 q^{-22} + q^{-23} +2 q^{-24} -2 q^{-26} + q^{-28} + q^{-29} - q^{-30} }
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 -q^{12}+q^{11}+2 q^{10}-q^8-5 q^7+5 q^5+3 q^4-10 q^2-2 q+8+6 q^{-1} +2 q^{-2} -11 q^{-3} -4 q^{-4} +7 q^{-5} +7 q^{-6} +2 q^{-7} -11 q^{-8} -4 q^{-9} +7 q^{-10} +5 q^{-11} +2 q^{-12} -10 q^{-13} -4 q^{-14} +7 q^{-15} +4 q^{-16} +2 q^{-17} -8 q^{-18} -4 q^{-19} +6 q^{-20} +2 q^{-21} +3 q^{-22} -5 q^{-23} -4 q^{-24} +4 q^{-25} +3 q^{-27} -2 q^{-28} -3 q^{-29} +2 q^{-30} -2 q^{-31} +2 q^{-32} - q^{-34} +3 q^{-35} -3 q^{-36} +4 q^{-40} -2 q^{-41} - q^{-42} - q^{-43} - q^{-44} +3 q^{-45} - q^{-48} - q^{-49} + q^{-50} }
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 -q^{16}+2 q^{14}+2 q^{13}-2 q^{11}-6 q^{10}-2 q^9+5 q^8+8 q^7+4 q^6-6 q^5-11 q^4-7 q^3+5 q^2+14 q+10-6 q^{-1} -13 q^{-2} -10 q^{-3} +3 q^{-4} +15 q^{-5} +13 q^{-6} -5 q^{-7} -13 q^{-8} -11 q^{-9} +2 q^{-10} +14 q^{-11} +13 q^{-12} -5 q^{-13} -13 q^{-14} -10 q^{-15} +2 q^{-16} +13 q^{-17} +11 q^{-18} -5 q^{-19} -11 q^{-20} -9 q^{-21} + q^{-22} +11 q^{-23} +10 q^{-24} -2 q^{-25} -9 q^{-26} -9 q^{-27} - q^{-28} +8 q^{-29} +10 q^{-30} + q^{-31} -6 q^{-32} -8 q^{-33} -4 q^{-34} +5 q^{-35} +8 q^{-36} +4 q^{-37} -3 q^{-38} -6 q^{-39} -5 q^{-40} +5 q^{-42} +5 q^{-43} -2 q^{-45} -4 q^{-46} -2 q^{-47} + q^{-48} +3 q^{-49} + q^{-50} + q^{-51} - q^{-52} - q^{-53} - q^{-56} + q^{-58} + q^{-59} + q^{-60} -2 q^{-62} -2 q^{-63} + q^{-65} + q^{-66} +2 q^{-67} -2 q^{-69} - q^{-70} + q^{-73} + q^{-74} - q^{-75} }
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^{26}-q^{25}-q^{24}-q^{20}+5 q^{19}+q^{18}-4 q^{15}-7 q^{14}-6 q^{13}+9 q^{12}+7 q^{11}+8 q^{10}+5 q^9-6 q^8-19 q^7-17 q^6+10 q^5+11 q^4+17 q^3+13 q^2-4 q-24-25 q^{-1} +7 q^{-2} +10 q^{-3} +20 q^{-4} +18 q^{-5} -24 q^{-7} -27 q^{-8} +4 q^{-9} +8 q^{-10} +19 q^{-11} +19 q^{-12} + q^{-13} -23 q^{-14} -26 q^{-15} +4 q^{-16} +8 q^{-17} +18 q^{-18} +17 q^{-19} -22 q^{-21} -25 q^{-22} +5 q^{-23} +8 q^{-24} +17 q^{-25} +15 q^{-26} - q^{-27} -19 q^{-28} -23 q^{-29} +5 q^{-30} +5 q^{-31} +14 q^{-32} +14 q^{-33} + q^{-34} -13 q^{-35} -20 q^{-36} +2 q^{-37} + q^{-38} +10 q^{-39} +13 q^{-40} +5 q^{-41} -6 q^{-42} -17 q^{-43} -2 q^{-44} -4 q^{-45} +5 q^{-46} +12 q^{-47} +9 q^{-48} + q^{-49} -12 q^{-50} -4 q^{-51} -9 q^{-52} - q^{-53} +8 q^{-54} +9 q^{-55} +7 q^{-56} -5 q^{-57} -2 q^{-58} -10 q^{-59} -6 q^{-60} +2 q^{-61} +5 q^{-62} +9 q^{-63} + q^{-64} +3 q^{-65} -6 q^{-66} -6 q^{-67} -3 q^{-68} - q^{-69} +6 q^{-70} + q^{-71} +5 q^{-72} -2 q^{-74} -3 q^{-75} -3 q^{-76} +3 q^{-77} -3 q^{-78} +2 q^{-79} + q^{-80} + q^{-81} - q^{-83} +4 q^{-84} -4 q^{-85} - q^{-86} - q^{-87} +5 q^{-91} - q^{-92} - q^{-94} - q^{-95} -2 q^{-96} - q^{-97} +3 q^{-98} + q^{-100} - q^{-103} - q^{-104} + q^{-105} }
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 -q^{35}+q^{34}+q^{33}+q^{32}-2 q^{30}-q^{29}-2 q^{28}-3 q^{27}+4 q^{25}+6 q^{24}+7 q^{23}-q^{21}-7 q^{20}-14 q^{19}-10 q^{18}+11 q^{16}+21 q^{15}+16 q^{14}+6 q^{13}-9 q^{12}-28 q^{11}-26 q^{10}-16 q^9+6 q^8+34 q^7+35 q^6+21 q^5-3 q^4-33 q^3-38 q^2-30 q-3+34 q^{-1} +44 q^{-2} +31 q^{-3} +6 q^{-4} -31 q^{-5} -39 q^{-6} -35 q^{-7} -10 q^{-8} +30 q^{-9} +42 q^{-10} +32 q^{-11} +10 q^{-12} -28 q^{-13} -38 q^{-14} -34 q^{-15} -11 q^{-16} +29 q^{-17} +40 q^{-18} +31 q^{-19} +9 q^{-20} -28 q^{-21} -38 q^{-22} -33 q^{-23} -9 q^{-24} +30 q^{-25} +40 q^{-26} +30 q^{-27} +6 q^{-28} -28 q^{-29} -37 q^{-30} -31 q^{-31} -7 q^{-32} +28 q^{-33} +38 q^{-34} +27 q^{-35} +5 q^{-36} -24 q^{-37} -34 q^{-38} -28 q^{-39} -7 q^{-40} +23 q^{-41} +33 q^{-42} +24 q^{-43} +6 q^{-44} -17 q^{-45} -28 q^{-46} -25 q^{-47} -9 q^{-48} +14 q^{-49} +26 q^{-50} +22 q^{-51} +10 q^{-52} -8 q^{-53} -20 q^{-54} -21 q^{-55} -13 q^{-56} +3 q^{-57} +16 q^{-58} +19 q^{-59} +13 q^{-60} +3 q^{-61} -10 q^{-62} -15 q^{-63} -14 q^{-64} -8 q^{-65} +4 q^{-66} +11 q^{-67} +14 q^{-68} +11 q^{-69} -5 q^{-71} -10 q^{-72} -13 q^{-73} -7 q^{-74} +7 q^{-76} +13 q^{-77} +6 q^{-78} +6 q^{-79} -10 q^{-81} -9 q^{-82} -8 q^{-83} -3 q^{-84} +5 q^{-85} +5 q^{-86} +10 q^{-87} +9 q^{-88} -2 q^{-89} -3 q^{-90} -6 q^{-91} -9 q^{-92} -3 q^{-93} -3 q^{-94} +5 q^{-95} +9 q^{-96} +2 q^{-97} +4 q^{-98} + q^{-99} -5 q^{-100} -3 q^{-101} -6 q^{-102} -2 q^{-103} +4 q^{-104} - q^{-105} +3 q^{-106} +3 q^{-107} +2 q^{-109} -2 q^{-110} -2 q^{-111} +2 q^{-112} -3 q^{-113} - q^{-114} -2 q^{-116} +3 q^{-117} + q^{-118} +3 q^{-120} - q^{-123} -4 q^{-124} - q^{-127} +2 q^{-128} + q^{-129} +2 q^{-130} + q^{-131} -2 q^{-132} - q^{-133} - q^{-135} + q^{-138} + q^{-139} - q^{-140} }

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_19 8_21