10 145

10_144

10_146

(KnotPlot image)

See the full Rolfsen Knot Table.

Visit 10 145's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 10 145 at Knotilus!

[edit 10 145 Quick Notes]

[edit 10 145 Further Notes and Views]

Knot presentations

Planar diagram presentation	X₄₂₅₁ X_5,12,6,13 X₈₃₉₄ X_2,9,3,10 X_11,16,12,17 X_17,10,18,11 X_7,18,8,19 X_13,20,14,1 X_19,14,20,15 X_15,6,16,7
Gauss code	1, -4, 3, -1, -2, 10, -7, -3, 4, 6, -5, 2, -8, 9, -10, 5, -6, 7, -9, 8
Dowker-Thistlethwaite code	4 8 -12 -18 2 -16 -20 -6 -10 -14
Conway Notation	[22,3,3-]

Minimum Braid Representative

A Morse Link Presentation

An Arc Presentation

Length is 11, width is 4,

Braid index is 4

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

[edit Notes on presentations of 10 145]

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["10 145"];

In[4]:= PD[K]

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

Out[4]= X₄₂₅₁ X_5,12,6,13 X₈₃₉₄ X_2,9,3,10 X_11,16,12,17 X_17,10,18,11 X_7,18,8,19 X_13,20,14,1 X_19,14,20,15 X_15,6,16,7

In[5]:= GaussCode[K]

Out[5]= 1, -4, 3, -1, -2, 10, -7, -3, 4, 6, -5, 2, -8, 9, -10, 5, -6, 7, -9, 8

In[6]:= DTCode[K]

Out[6]= 4 8 -12 -18 2 -16 -20 -6 -10 -14

(The path below may be different on your system)

In[7]:= AppendTo[$Path, "C:/bin/LinKnot/"];

In[8]:= ConwayNotation[K]

Out[8]= [22,3,3-]

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

In[14]:= Draw[ap]

Out[14]= -Graphics-

Three dimensional invariants

Symmetry type	Reversible
Unknotting number	2
3-genus	2
Bridge index	3
Super bridge index	Missing
Nakanishi index	1
Maximal Thurston-Bennequin number	[-12][3]
Hyperbolic Volume	5.0449
A-Polynomial	See Data:10 145/A-polynomial

[edit Notes for 10 145'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 2}
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 [1,2]}
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 2}
Rasmussen s-Invariant	4

[edit Notes for 10 145'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+t-3+ 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+5 z^2+1}
2nd Alexander ideal (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 \{1\}}
Determinant and Signature	{ 3, -2 }
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 q^{-2} + q^{-7} - q^{-8} + q^{-9} - q^{-10} }
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^{10}+z^2 a^8+a^8-a^6+z^4 a^4+4 z^2 a^4+2 a^4}
Kauffman 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 z^7 a^{11}-6 z^5 a^{11}+10 z^3 a^{11}-5 z a^{11}+z^8 a^{10}-6 z^6 a^{10}+10 z^4 a^{10}-6 z^2 a^{10}+a^{10}+2 z^7 a^9-12 z^5 a^9+18 z^3 a^9-6 z a^9+z^8 a^8-6 z^6 a^8+9 z^4 a^8-4 z^2 a^8+a^8+z^7 a^7-6 z^5 a^7+8 z^3 a^7-2 z a^7-2 z^2 a^6+a^6-z a^5+z^4 a^4-4 z^2 a^4+2 a^4}
The A2 invariant	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^{32}-q^{30}+q^{24}+q^{14}+q^{10}+q^8+q^6}
The G2 invariant	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^{156}+q^{152}-q^{148}+q^{142}-2 q^{138}-q^{130}-3 q^{128}-q^{126}+q^{124}-q^{122}-q^{120}-q^{118}-2 q^{116}+2 q^{114}-2 q^{112}-q^{110}+q^{108}+2 q^{104}+2 q^{102}+q^{98}+q^{96}+2 q^{92}+q^{88}+2 q^{82}-q^{78}-3 q^{76}+q^{74}-q^{72}-3 q^{66}+2 q^{64}+q^{62}-2 q^{60}-q^{54}+3 q^{52}+2 q^{48}+q^{46}+3 q^{42}+q^{38}+q^{32}+q^{30}}

Further Quantum Invariants

Further quantum knot invariants for 10_145.

A1 Invariants.

Weight	Invariant
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^{21}+q^{13}+q^5+q^3}
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^{60}-q^{56}-q^{46}+q^{44}+q^{42}-q^{40}-q^{34}-q^{32}-q^{30}-q^{28}+q^{26}+q^{24}+2 q^{22}-q^{18}+2 q^{16}-q^{12}+q^{10}+q^8+q^6}
3	$-q^{117}+q^{113}+q^{111}-q^{107}+q^{97}-q^{95}-2q^{93}+2q^{89}+2q^{87}-q^{85}-q^{83}+q^{79}+2q^{77}-2q^{73}-q^{71}+q^{69}+q^{67}-2q^{65}-q^{63}+q^{61}-q^{57}-q^{55}+q^{53}-q^{49}-q^{47}-q^{41}-q^{39}+2q^{35}+2q^{33}-q^{29}+2q^{25}+2q^{23}-q^{19}-q^{17}+q^{15}+q^{13}+q^{11}+q^{9}$

A2 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^{32}-q^{30}+q^{24}+q^{14}+q^{10}+q^8+q^6}
1,1	$q^{84}+2q^{80}-2q^{74}-4q^{72}-2q^{68}+2q^{66}+2q^{64}+2q^{62}+6q^{60}-2q^{58}-4q^{54}-3q^{52}-2q^{50}-2q^{48}+2q^{46}-2q^{44}+2q^{42}-4q^{40}+2q^{38}-4q^{36}+2q^{34}+2q^{32}-2q^{30}+4q^{28}+4q^{24}+q^{20}+2q^{18}+2q^{16}+2q^{14}+q^{12}$
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^{82}+q^{80}+q^{78}-q^{76}-q^{74}-q^{72}-q^{70}+q^{64}+q^{60}+q^{58}-q^{56}-q^{54}-q^{52}-2 q^{48}-2 q^{46}-q^{44}-2 q^{42}-q^{40}+2 q^{36}+q^{34}+2 q^{32}+2 q^{30}+q^{28}+q^{26}+q^{24}+q^{22}+2 q^{16}+q^{14}+q^{12}}

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^{66}+q^{62}+q^{60}-2 q^{56}-q^{54}-2 q^{52}-3 q^{50}-q^{46}+2 q^{40}+q^{38}+q^{34}-q^{30}-q^{28}+q^{26}+q^{22}+3 q^{20}+q^{18}+2 q^{16}+q^{14}+q^{12}}
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^{43}-q^{41}-q^{39}+q^{33}+q^{31}-q^{25}+q^{19}+q^{17}+q^{15}+q^{13}+q^{11}+q^9}

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^{88}+q^{86}+q^{84}+q^{82}+q^{80}-q^{74}-2 q^{72}-2 q^{70}-2 q^{68}-3 q^{66}-3 q^{64}-2 q^{62}-2 q^{60}-2 q^{58}-q^{56}+2 q^{54}+2 q^{52}+3 q^{50}+4 q^{48}+2 q^{46}-q^{42}-2 q^{40}-3 q^{38}-q^{36}+q^{34}+2 q^{32}+3 q^{30}+3 q^{28}+4 q^{26}+2 q^{24}+2 q^{22}+q^{20}+q^{18}}

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^{52}-q^{50}-q^{48}+q^{42}+q^{40}+q^{38}-q^{32}-q^{30}+q^{24}+q^{22}+2 q^{20}+q^{18}+q^{16}+q^{14}+q^{12}}

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^{66}-q^{62}-q^{60}+q^{54}+q^{50}+q^{46}-q^{38}-q^{34}-q^{30}+q^{28}+q^{26}+q^{22}+q^{20}+q^{18}+q^{14}+q^{12}}
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^{108}+q^{100}-q^{90}-2 q^{88}-q^{82}-q^{80}-q^{72}+q^{56}+q^{48}-q^{44}+q^{42}+q^{40}-q^{36}+q^{34}+q^{32}+q^{30}+q^{26}+q^{24}+q^{22}+q^{18}}

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^{90}+q^{86}+q^{84}+q^{82}-q^{78}-q^{76}-3 q^{74}-2 q^{72}-3 q^{70}-2 q^{68}-2 q^{66}+q^{60}+2 q^{58}+2 q^{56}+2 q^{54}+q^{52}+q^{50}-2 q^{44}-q^{42}-2 q^{40}+2 q^{32}+2 q^{30}+3 q^{28}+2 q^{26}+2 q^{24}+q^{22}+q^{20}+q^{18}}

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^{156}+q^{152}-q^{148}+q^{142}-2 q^{138}-q^{130}-3 q^{128}-q^{126}+q^{124}-q^{122}-q^{120}-q^{118}-2 q^{116}+2 q^{114}-2 q^{112}-q^{110}+q^{108}+2 q^{104}+2 q^{102}+q^{98}+q^{96}+2 q^{92}+q^{88}+2 q^{82}-q^{78}-3 q^{76}+q^{74}-q^{72}-3 q^{66}+2 q^{64}+q^{62}-2 q^{60}-q^{54}+3 q^{52}+2 q^{48}+q^{46}+3 q^{42}+q^{38}+q^{32}+q^{30}}

.

Computer Talk

The above data is available with the Mathematica package KnotTheory`, as shown in the (simulated) Mathematica session below. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting. This Mathematica session is also available (albeit only for the knot 5_2) as the notebook PolynomialInvariantsSession.nb.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:= AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of August 31, 2006, 11:25:27.5625.

In[3]:= K = Knot["10 145"];

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+t-3+ 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+5 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]= 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\}}

In[7]:= {KnotDet[K], KnotSignature[K]}

Out[7]= { 3, -2 }

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 q^{-2} + q^{-7} - q^{-8} + q^{-9} - q^{-10} }

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

In[10]:= Kauffman[K][a, z]

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

Out[10]= 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^7 a^{11}-6 z^5 a^{11}+10 z^3 a^{11}-5 z a^{11}+z^8 a^{10}-6 z^6 a^{10}+10 z^4 a^{10}-6 z^2 a^{10}+a^{10}+2 z^7 a^9-12 z^5 a^9+18 z^3 a^9-6 z a^9+z^8 a^8-6 z^6 a^8+9 z^4 a^8-4 z^2 a^8+a^8+z^7 a^7-6 z^5 a^7+8 z^3 a^7-2 z a^7-2 z^2 a^6+a^6-z a^5+z^4 a^4-4 z^2 a^4+2 a^4}

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

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.

In[3]:= K = Knot["10 145"];

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+t-3+ 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^{-7} - q^{-8} + q^{-9} - q^{-10} } }

In[5]:= DeleteCases[Select[AllKnots[], (A === Alexander[#][t]) &], K]

KnotTheory::loading: Loading precomputed data in DTCode4KnotsTo11`.

KnotTheory::credits: The GaussCode to PD conversion was written by Siddarth Sankaran at the University of Toronto in the summer of 2005.

Out[5]= {}

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:

(5, -12)

V_2,1 through V_6,9:

V_2,1

V_3,1

V_4,1

V_4,2

V_4,3

V_5,1

V_5,2

V_5,3

V_5,4

V_6,1

V_6,2

V_6,3

V_6,4

V_6,5

V_6,6

V_6,7

V_6,8

V_6,9

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 20}

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 -96}

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 200}

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{1702}{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{266}{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 -1920}

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 -3712}

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 -640}

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 -544}

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{4000}{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 4608}

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{34040}{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{5320}{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{147487}{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 38}

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{93134}{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{4837}{18}}

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{8287}{6}}

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 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} 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=} -2 is the signature of 10 145. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\		r
	\
j		\

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

χ

-3

1

-5

1

-7

1

0

-9

0

-11

1

2

1

0

-13

1

-15

1

0

-17

1

0

-19

0

-21

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=-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 i=-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 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=-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 {\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=-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 {\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=-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 {\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=-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}\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=-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 {\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}_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=-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 {\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}_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 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}}	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^{-4} - q^{-7} + q^{-8} +2 q^{-9} -4 q^{-10} +2 q^{-11} +4 q^{-12} -5 q^{-13} +2 q^{-14} +2 q^{-15} -5 q^{-16} +2 q^{-17} +2 q^{-18} -4 q^{-19} +2 q^{-20} + q^{-21} -2 q^{-22} +2 q^{-23} - q^{-24} - q^{-25} +2 q^{-26} - q^{-27} - q^{-28} + q^{-29} }
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^{-6} - q^{-10} + q^{-12} +2 q^{-13} - q^{-14} -2 q^{-15} +3 q^{-17} + q^{-18} -2 q^{-19} -2 q^{-20} +2 q^{-21} + q^{-22} - q^{-23} -2 q^{-24} + q^{-25} + q^{-26} - q^{-28} - q^{-29} + q^{-30} + q^{-31} -3 q^{-33} +4 q^{-35} -5 q^{-37} - q^{-38} +6 q^{-39} +2 q^{-40} -6 q^{-41} -2 q^{-42} +5 q^{-43} +2 q^{-44} -3 q^{-45} -2 q^{-46} +3 q^{-47} -2 q^{-49} +2 q^{-51} -2 q^{-53} + q^{-55} + q^{-56} - q^{-57} }
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} - q^{-13} + q^{-16} +2 q^{-17} - q^{-18} +2 q^{-19} -4 q^{-20} - q^{-21} + q^{-22} +9 q^{-24} -3 q^{-25} -4 q^{-26} -8 q^{-27} -2 q^{-28} +17 q^{-29} +2 q^{-30} -4 q^{-31} -14 q^{-32} -5 q^{-33} +20 q^{-34} +3 q^{-35} -2 q^{-36} -16 q^{-37} -6 q^{-38} +21 q^{-39} +3 q^{-40} -4 q^{-41} -16 q^{-42} -5 q^{-43} +21 q^{-44} +4 q^{-45} -5 q^{-46} -14 q^{-47} -5 q^{-48} +18 q^{-49} +5 q^{-50} -3 q^{-51} -12 q^{-52} -6 q^{-53} +13 q^{-54} +7 q^{-55} - q^{-56} -10 q^{-57} -6 q^{-58} +8 q^{-59} +6 q^{-60} - q^{-61} -5 q^{-62} -3 q^{-63} +4 q^{-64} +3 q^{-65} -4 q^{-66} -2 q^{-67} + q^{-68} +6 q^{-69} +2 q^{-70} -8 q^{-71} -3 q^{-72} + q^{-73} +7 q^{-74} +3 q^{-75} -5 q^{-76} -3 q^{-77} -2 q^{-78} +5 q^{-79} + q^{-80} -2 q^{-81} - q^{-82} - q^{-83} +4 q^{-84} - q^{-85} - q^{-86} - q^{-87} - q^{-88} +3 q^{-89} - q^{-92} - q^{-93} + q^{-94} }

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).

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Contents

Knot presentations

Three dimensional invariants

Four dimensional invariants

Polynomial invariants

"Similar" Knots (within the Atlas)

Vassiliev invariants

Khovanov Homology

The Coloured Jones Polynomials

Computer Talk

Modifying This Page

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