9 46

9_45

9_47

(KnotPlot image)	See the full Rolfsen Knot Table. Visit 9 46's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 46 at Knotilus!
	9_46 is also known as the pretzel knot P(3,3,-3).

Knot presentations

Planar diagram presentation	X4251 X7,12,8,13 X10,3,11,4 X2,11,3,12 X5,14,6,15 X13,6,14,7 X15,18,16,1 X9,17,10,16 X17,9,18,8
Gauss code	1, -4, 3, -1, -5, 6, -2, 9, -8, -3, 4, 2, -6, 5, -7, 8, -9, 7
Dowker-Thistlethwaite code	4 10 -14 -12 -16 2 -6 -18 -8
Conway Notation	[3,3,21-]

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

[edit Notes on presentations of 9 46]

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["9 46"];

In[4]:=

PD[K]

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

Out[4]=

X4251 X7,12,8,13 X10,3,11,4 X2,11,3,12 X5,14,6,15 X13,6,14,7 X15,18,16,1 X9,17,10,16 X17,9,18,8

In[5]:=

GaussCode[K]

Out[5]=

1, -4, 3, -1, -5, 6, -2, 9, -8, -3, 4, 2, -6, 5, -7, 8, -9, 7

In[6]:=

DTCode[K]

Out[6]=

4 10 -14 -12 -16 2 -6 -18 -8

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[3,3,21-]

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

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 1 Bridge index 3 Super bridge index 4 Nakanishi index 2 Maximal Thurston-Bennequin number [-7][-1] Hyperbolic Volume 4.7517 A-Polynomial See Data:9 46/A-polynomial

[edit Notes for 9 46's three dimensional invariants]

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 9 46'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 -2 t+5-2 t^{-1} }
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 1-2 z^2}
2nd Alexander ideal (db, data sources)	$\{3,t+1\}$
Determinant and Signature	{ 9, 0 }
Jones 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 2- q^{-1} + q^{-2} -2 q^{-3} + q^{-4} - q^{-5} + q^{-6} }
HOMFLY-PT polynomial (db, data sources)	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 a^6-z^2 a^4-a^4-z^2 a^2-a^2+2}
Kauffman polynomial (db, data sources)	$a^{5}z^{7}+a^{3}z^{7}+a^{6}z^{6}+2a^{4}z^{6}+a^{2}z^{6}-5a^{5}z^{5}-5a^{3}z^{5}-5a^{6}z^{4}-9a^{4}z^{4}-4a^{2}z^{4}+7a^{5}z^{3}+8a^{3}z^{3}+az^{3}+6a^{6}z^{2}+9a^{4}z^{2}+3a^{2}z^{2}-4a^{5}z-6a^{3}z-2az-a^{6}-a^{4}+a^{2}+2$
The A2 invariant	$q^{20}+q^{18}-q^{12}-q^{10}-q^{8}-q^{6}+q^{2}+2+2q^{-2}$
The G2 invariant	$q^{94}+q^{90}-q^{88}-q^{82}+2q^{80}+2q^{70}+q^{68}-3q^{66}+q^{62}+2q^{60}+2q^{58}-4q^{56}+3q^{52}+2q^{50}-2q^{48}-5q^{46}+3q^{42}+2q^{40}-4q^{38}-3q^{36}+3q^{32}-5q^{28}-4q^{26}+2q^{24}+2q^{22}-2q^{20}-q^{18}-3q^{16}+3q^{14}+2q^{12}-q^{10}+2q^{4}+3q^{2}+1+q^{-2}+q^{-4}+3q^{-8}+q^{-10}-q^{-12}+q^{-14}$

Further Quantum Invariants

Further quantum knot invariants for 9_46.

A1 Invariants.

Weight	Invariant
1	$q^{13}-q^{7}-q^{5}+q+2q^{-1}$
2	$q^{38}-q^{34}-q^{28}-q^{26}+q^{22}+q^{20}+q^{18}+2q^{16}-2q^{8}-2q^{6}-q^{4}+q^{2}+1+q^{-2}+q^{-4}+q^{-6}$
3	$q^{75}-q^{71}-q^{69}+q^{65}-q^{61}-q^{59}+q^{55}+2q^{53}+q^{51}+q^{45}+q^{43}-q^{41}-3q^{39}-q^{37}-q^{33}-2q^{31}-q^{29}+q^{27}+q^{25}+q^{23}+2q^{21}+3q^{19}+q^{17}+q^{15}-q^{9}-q^{7}-3q^{5}-2q^{3}+q+3q^{-1}+q^{-3}-3q^{-5}+2q^{-9}+2q^{-11}$
4	$q^{124}-q^{120}-q^{118}-q^{116}+q^{114}+q^{112}+q^{110}-2q^{106}-q^{104}+q^{100}+2q^{98}+2q^{96}+q^{94}-q^{92}-2q^{90}-q^{88}+q^{86}+q^{84}+q^{82}-2q^{80}-4q^{78}-3q^{76}+3q^{72}+q^{70}-2q^{66}-q^{64}+3q^{62}+4q^{60}+3q^{58}-2q^{54}+q^{52}+2q^{50}+2q^{48}-q^{46}-4q^{44}-2q^{42}-q^{40}-q^{38}-2q^{36}-3q^{34}-q^{32}+q^{30}+q^{28}+3q^{22}+3q^{20}+3q^{18}+2q^{16}-2q^{12}-q^{10}+q^{8}+2q^{6}+q^{4}-4q^{2}-3-q^{-2}+4q^{-4}+4q^{-6}-2q^{-8}-3q^{-10}-3q^{-12}+q^{-14}+3q^{-16}+q^{-18}+q^{-20}$
5	$q^{185}-q^{181}-q^{179}-q^{177}+q^{173}+2q^{171}+q^{169}-q^{165}-2q^{163}-2q^{161}+2q^{157}+2q^{155}+2q^{153}+2q^{151}-2q^{147}-3q^{145}-3q^{143}-q^{141}+2q^{139}+3q^{137}+2q^{135}-3q^{131}-5q^{129}-4q^{127}-q^{125}+4q^{123}+6q^{121}+4q^{119}+q^{117}-3q^{115}-4q^{113}-2q^{111}+3q^{109}+6q^{107}+6q^{105}+2q^{103}-3q^{101}-7q^{99}-6q^{97}+4q^{93}+4q^{91}+q^{89}-5q^{87}-8q^{85}-5q^{83}+q^{81}+4q^{79}+4q^{77}-4q^{73}-3q^{71}+q^{69}+6q^{67}+7q^{65}+3q^{63}-2q^{61}-2q^{59}+q^{57}+4q^{55}+3q^{53}-3q^{49}-2q^{47}-q^{45}-3q^{41}-4q^{39}-3q^{37}-q^{35}+q^{33}+q^{31}-q^{27}-q^{25}-q^{23}+q^{21}+3q^{19}+5q^{17}+5q^{15}+3q^{13}-4q^{11}-6q^{9}-5q^{7}+q^{5}+7q^{3}+10q+5q^{-1}-5q^{-3}-10q^{-5}-7q^{-7}+q^{-9}+7q^{-11}+7q^{-13}+q^{-15}-5q^{-17}-5q^{-19}-2q^{-21}+2q^{-25}+2q^{-27}+2q^{-29}$

A2 Invariants.

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

A3 Invariants.

Weight	Invariant
0,1,0	$q^{40}+q^{36}-q^{32}-2q^{30}-2q^{28}+q^{24}+4q^{22}+4q^{20}+4q^{18}-2q^{14}-5q^{12}-6q^{10}-4q^{8}-2q^{6}+2q^{4}+3q^{2}+5+3q^{-2}+2q^{-4}$
1,0,0	$q^{27}+q^{25}+q^{23}-q^{17}-q^{15}-q^{13}-q^{11}-q^{9}-q^{7}+q^{3}+2q+2q^{-1}+2q^{-3}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{54}+q^{52}+q^{50}+q^{48}+q^{46}-q^{44}-2q^{42}-3q^{40}-4q^{38}-4q^{36}-2q^{34}+2q^{32}+4q^{30}+7q^{28}+9q^{26}+8q^{24}+4q^{22}+q^{20}-4q^{18}-8q^{16}-10q^{14}-9q^{12}-7q^{10}-4q^{8}+q^{6}+4q^{4}+6q^{2}+6+6q^{-2}+3q^{-4}+2q^{-6}$
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^{34}+q^{32}+q^{30}+q^{28}-q^{22}-q^{20}-q^{18}-q^{16}-q^{14}-q^{12}-q^{10}-q^8+q^4+2 q^2+2+2 q^{-2} +2 q^{-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^{40}+q^{36}+q^{32}-q^{24}-2 q^{20}-2 q^{16}-q^{12}+2 q^4+q^2+1+ q^{-2} +2 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^{66}+q^{58}-q^{54}-q^{52}-q^{48}-2 q^{46}-q^{44}+q^{40}+q^{38}+2 q^{36}+2 q^{34}+3 q^{32}+2 q^{30}+2 q^{28}-q^{22}-2 q^{20}-4 q^{18}-3 q^{16}-2 q^{14}-2 q^{12}-2 q^{10}-q^8+2 q^6+q^4+2 q^2+2+3 q^{-2} + q^{-4} +2 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^{54}+q^{50}+q^{46}-q^{44}-q^{42}-q^{40}-q^{38}+2 q^{32}+2 q^{30}+4 q^{28}+2 q^{26}+2 q^{24}-q^{22}-q^{20}-4 q^{18}-4 q^{16}-5 q^{14}-4 q^{12}-2 q^{10}-q^8+2 q^6+2 q^4+4 q^2+4+3 q^{-2} +2 q^{-4} +2 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^{94}+q^{90}-q^{88}-q^{82}+2 q^{80}+2 q^{70}+q^{68}-3 q^{66}+q^{62}+2 q^{60}+2 q^{58}-4 q^{56}+3 q^{52}+2 q^{50}-2 q^{48}-5 q^{46}+3 q^{42}+2 q^{40}-4 q^{38}-3 q^{36}+3 q^{32}-5 q^{28}-4 q^{26}+2 q^{24}+2 q^{22}-2 q^{20}-q^{18}-3 q^{16}+3 q^{14}+2 q^{12}-q^{10}+2 q^4+3 q^2+1+ q^{-2} + q^{-4} +3 q^{-8} + q^{-10} - q^{-12} + q^{-14} }

.

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["9 46"];

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

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 1-2 z^2}

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]=

$\{3,t+1\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 9, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

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} + q^{-2} -2 q^{-3} + q^{-4} - q^{-5} + q^{-6} }

In[9]:=

HOMFLYPT[K][a, z]

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

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 a^6-z^2 a^4-a^4-z^2 a^2-a^2+2}

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$a^{5}z^{7}+a^{3}z^{7}+a^{6}z^{6}+2a^{4}z^{6}+a^{2}z^{6}-5a^{5}z^{5}-5a^{3}z^{5}-5a^{6}z^{4}-9a^{4}z^{4}-4a^{2}z^{4}+7a^{5}z^{3}+8a^{3}z^{3}+az^{3}+6a^{6}z^{2}+9a^{4}z^{2}+3a^{2}z^{2}-4a^{5}z-6a^{3}z-2az-a^{6}-a^{4}+a^{2}+2$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {6_1, K11n67, K11n97, K11n139,}

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["9 46"];

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

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]=

{6_1, K11n67, K11n97, K11n139,}

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, 3)

V2,1 through V6,9:

V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9 $-8$ 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 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 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{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 \frac{52}{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 -192} 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 -272} 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 -64} 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 -72} 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}} $288$ 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{416}{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{10409}{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{244}{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{8636}{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{775}{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{151}{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 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-2r=s-1} , where 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 s=} 0 is the signature of 9 46. 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 -1             1 1 -3         1 1   0 -5       1       -1 -7       1       -1 -9   1 1         0 -11               0 -13 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=-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 {\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=-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}_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=-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}} 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}\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} ${\mathbb {Z} }_{2}$ 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}} 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^2+ q^{-1} -2 q^{-3} +2 q^{-9} - q^{-10} +2 q^{-12} -2 q^{-13} - q^{-14} +2 q^{-15} - q^{-16} - q^{-17} + q^{-18} }
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^4-2 q^2-3 q+6+2 q^{-1} -4 q^{-2} -6 q^{-3} +5 q^{-4} +4 q^{-5} -4 q^{-6} -5 q^{-7} +5 q^{-8} +5 q^{-9} -4 q^{-10} -3 q^{-11} +4 q^{-12} +4 q^{-13} -4 q^{-14} -3 q^{-15} +2 q^{-16} +3 q^{-17} -3 q^{-18} -2 q^{-19} + q^{-20} + q^{-21} - q^{-22} + q^{-24} - q^{-26} + q^{-27} +2 q^{-28} - q^{-29} -2 q^{-30} +2 q^{-32} - q^{-34} - q^{-35} + q^{-36} }
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^8+2 q^6-2 q^5-4 q^4+q^3+q^2+8 q-2-9 q^{-1} - q^{-2} +13 q^{-4} - q^{-5} -10 q^{-6} -3 q^{-7} - q^{-8} +15 q^{-9} + q^{-10} -9 q^{-11} -3 q^{-12} - q^{-13} +12 q^{-14} + q^{-15} -8 q^{-16} -3 q^{-17} -3 q^{-18} +10 q^{-19} +2 q^{-20} -7 q^{-21} -3 q^{-22} -4 q^{-23} +8 q^{-24} +5 q^{-25} -4 q^{-26} -3 q^{-27} -5 q^{-28} +5 q^{-29} +7 q^{-30} - q^{-31} -2 q^{-32} -6 q^{-33} + q^{-34} +6 q^{-35} + q^{-36} - q^{-37} -4 q^{-38} -2 q^{-39} +3 q^{-40} + q^{-42} - q^{-43} -2 q^{-44} +3 q^{-45} -2 q^{-46} +4 q^{-50} -2 q^{-51} - q^{-52} - q^{-53} - q^{-54} +3 q^{-55} - q^{-58} - q^{-59} + q^{-60} }
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^{12}-2 q^9-2 q^8-3 q^7+2 q^6+6 q^5+6 q^4-2 q^3-8 q^2-11 q-1+11 q^{-1} +16 q^{-2} +3 q^{-3} -11 q^{-4} -17 q^{-5} -7 q^{-6} +10 q^{-7} +18 q^{-8} +10 q^{-9} -9 q^{-10} -17 q^{-11} -9 q^{-12} +8 q^{-13} +16 q^{-14} +10 q^{-15} -9 q^{-16} -16 q^{-17} -8 q^{-18} +8 q^{-19} +14 q^{-20} +8 q^{-21} -10 q^{-22} -15 q^{-23} -5 q^{-24} +7 q^{-25} +13 q^{-26} +7 q^{-27} -7 q^{-28} -12 q^{-29} -4 q^{-30} +4 q^{-31} +10 q^{-32} +7 q^{-33} -2 q^{-34} -8 q^{-35} -5 q^{-36} - q^{-37} +6 q^{-38} +6 q^{-39} +2 q^{-40} -4 q^{-41} -5 q^{-42} -4 q^{-43} +3 q^{-45} +5 q^{-46} +2 q^{-47} -2 q^{-48} -4 q^{-49} -4 q^{-50} -3 q^{-51} +4 q^{-52} +6 q^{-53} +3 q^{-54} -4 q^{-56} -6 q^{-57} - q^{-58} +4 q^{-59} +4 q^{-60} +4 q^{-61} - q^{-62} -4 q^{-63} -3 q^{-64} - q^{-65} + q^{-66} +3 q^{-67} + q^{-68} - q^{-69} - q^{-70} - q^{-74} - q^{-75} +2 q^{-77} +2 q^{-78} - q^{-80} - q^{-81} -2 q^{-82} +2 q^{-84} + q^{-85} - q^{-88} - q^{-89} + q^{-90} }
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^{18}+2 q^{16}-2 q^{14}-4 q^{13}-4 q^{12}-q^{11}+3 q^{10}+12 q^9+7 q^8+2 q^7-10 q^6-15 q^5-17 q^4-2 q^3+24 q^2+22 q+19-7 q^{-1} -23 q^{-2} -37 q^{-3} -16 q^{-4} +24 q^{-5} +30 q^{-6} +32 q^{-7} +2 q^{-8} -19 q^{-9} -44 q^{-10} -25 q^{-11} +18 q^{-12} +27 q^{-13} +35 q^{-14} +6 q^{-15} -14 q^{-16} -42 q^{-17} -27 q^{-18} +16 q^{-19} +25 q^{-20} +33 q^{-21} +4 q^{-22} -13 q^{-23} -41 q^{-24} -24 q^{-25} +17 q^{-26} +25 q^{-27} +32 q^{-28} +3 q^{-29} -13 q^{-30} -39 q^{-31} -20 q^{-32} +15 q^{-33} +22 q^{-34} +30 q^{-35} +5 q^{-36} -10 q^{-37} -36 q^{-38} -18 q^{-39} +10 q^{-40} +15 q^{-41} +27 q^{-42} +8 q^{-43} -4 q^{-44} -32 q^{-45} -17 q^{-46} +2 q^{-47} +8 q^{-48} +23 q^{-49} +11 q^{-50} +5 q^{-51} -24 q^{-52} -15 q^{-53} -6 q^{-54} - q^{-55} +17 q^{-56} +13 q^{-57} +14 q^{-58} -13 q^{-59} -9 q^{-60} -10 q^{-61} -10 q^{-62} +8 q^{-63} +9 q^{-64} +16 q^{-65} -3 q^{-66} + q^{-67} -7 q^{-68} -12 q^{-69} -2 q^{-70} +9 q^{-72} +9 q^{-74} + q^{-75} -6 q^{-76} -2 q^{-77} -6 q^{-78} -2 q^{-79} -5 q^{-80} +7 q^{-81} +5 q^{-82} +2 q^{-83} +5 q^{-84} -2 q^{-85} -4 q^{-86} -8 q^{-87} - q^{-88} +2 q^{-90} +8 q^{-91} +3 q^{-92} + q^{-93} -3 q^{-94} -3 q^{-95} -4 q^{-96} -3 q^{-97} +5 q^{-98} +2 q^{-100} + q^{-101} - q^{-103} -2 q^{-104} +4 q^{-105} -3 q^{-106} - q^{-107} - q^{-108} +5 q^{-112} - q^{-113} - q^{-115} - q^{-116} -2 q^{-117} - q^{-118} +3 q^{-119} + q^{-121} - q^{-124} - q^{-125} + q^{-126} }
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^{24}-4 q^{19}-4 q^{18}-5 q^{17}+2 q^{16}+6 q^{15}+8 q^{14}+12 q^{13}+8 q^{12}-4 q^{11}-20 q^{10}-29 q^9-15 q^8+6 q^7+19 q^6+37 q^5+40 q^4+11 q^3-27 q^2-58 q-47-17 q^{-1} +14 q^{-2} +56 q^{-3} +69 q^{-4} +38 q^{-5} -12 q^{-6} -63 q^{-7} -71 q^{-8} -40 q^{-9} -4 q^{-10} +53 q^{-11} +76 q^{-12} +52 q^{-13} +5 q^{-14} -53 q^{-15} -69 q^{-16} -47 q^{-17} -15 q^{-18} +46 q^{-19} +70 q^{-20} +49 q^{-21} +10 q^{-22} -47 q^{-23} -65 q^{-24} -44 q^{-25} -12 q^{-26} +46 q^{-27} +67 q^{-28} +44 q^{-29} +9 q^{-30} -48 q^{-31} -65 q^{-32} -43 q^{-33} -6 q^{-34} +48 q^{-35} +66 q^{-36} +40 q^{-37} +6 q^{-38} -46 q^{-39} -63 q^{-40} -41 q^{-41} -4 q^{-42} +44 q^{-43} +60 q^{-44} +35 q^{-45} +7 q^{-46} -38 q^{-47} -57 q^{-48} -39 q^{-49} -6 q^{-50} +35 q^{-51} +49 q^{-52} +33 q^{-53} +13 q^{-54} -25 q^{-55} -48 q^{-56} -37 q^{-57} -13 q^{-58} +21 q^{-59} +39 q^{-60} +32 q^{-61} +22 q^{-62} -8 q^{-63} -35 q^{-64} -34 q^{-65} -24 q^{-66} +4 q^{-67} +25 q^{-68} +29 q^{-69} +28 q^{-70} +10 q^{-71} -17 q^{-72} -26 q^{-73} -29 q^{-74} -14 q^{-75} +5 q^{-76} +16 q^{-77} +28 q^{-78} +22 q^{-79} +3 q^{-80} -8 q^{-81} -22 q^{-82} -24 q^{-83} -14 q^{-84} -2 q^{-85} +14 q^{-86} +22 q^{-87} +17 q^{-88} +12 q^{-89} -4 q^{-90} -17 q^{-91} -18 q^{-92} -16 q^{-93} -5 q^{-94} +7 q^{-95} +13 q^{-96} +18 q^{-97} +13 q^{-98} -5 q^{-100} -12 q^{-101} -14 q^{-102} -7 q^{-103} -2 q^{-104} +6 q^{-105} +10 q^{-106} +7 q^{-107} +8 q^{-108} +3 q^{-109} -5 q^{-110} -5 q^{-111} -6 q^{-112} -7 q^{-113} -3 q^{-114} -2 q^{-115} +4 q^{-116} +7 q^{-117} +4 q^{-118} +4 q^{-119} +4 q^{-120} -2 q^{-121} -4 q^{-122} -8 q^{-123} -6 q^{-124} - q^{-126} +3 q^{-127} +8 q^{-128} +5 q^{-129} +3 q^{-130} - q^{-131} -5 q^{-132} -2 q^{-133} -4 q^{-134} -3 q^{-135} +2 q^{-136} + q^{-137} +3 q^{-138} +2 q^{-139} -2 q^{-140} +2 q^{-141} - q^{-143} +2 q^{-144} -2 q^{-145} - q^{-146} -3 q^{-148} + q^{-151} +4 q^{-152} + q^{-155} -2 q^{-156} - q^{-157} -2 q^{-158} - q^{-159} +2 q^{-160} + q^{-161} + q^{-163} - q^{-166} - q^{-167} + q^{-168} }

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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