10 58

10_57

10_59

(KnotPlot image)

See the full Rolfsen Knot Table.

Visit 10 58's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 10 58 at Knotilus!

[edit 10 58 Quick Notes]

[edit 10 58 Further Notes and Views]

Knot presentations

Planar diagram presentation	X₁₄₂₅ X_7,10,8,11 X₃₉₄₈ X_9,3,10,2 X_5,14,6,15 X_11,19,12,18 X_15,20,16,1 X_19,16,20,17 X_17,13,18,12 X_13,6,14,7
Gauss code	-1, 4, -3, 1, -5, 10, -2, 3, -4, 2, -6, 9, -10, 5, -7, 8, -9, 6, -8, 7
Dowker-Thistlethwaite code	4 8 14 10 2 18 6 20 12 16
Conway Notation	[22,22,2]

Minimum Braid Representative

A Morse Link Presentation

An Arc Presentation

Length is 11, width is 6,

Braid index is 6

[{12, 9}, {10, 8}, {9, 11}, {3, 10}, {7, 2}, {8, 6}, {1, 3}, {4, 7}, {6, 12}, {2, 5}, {11, 4}, {5, 1}]

[edit Notes on presentations of 10 58]

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 58"];

In[4]:= PD[K]

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

Out[4]= X₁₄₂₅ X_7,10,8,11 X₃₉₄₈ X_9,3,10,2 X_5,14,6,15 X_11,19,12,18 X_15,20,16,1 X_19,16,20,17 X_17,13,18,12 X_13,6,14,7

In[5]:= GaussCode[K]

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

In[6]:= DTCode[K]

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

(The path below may be different on your system)

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

In[8]:= ConwayNotation[K]

Out[8]= [22,22,2]

In[9]:= br = BR[K]

KnotTheory::credits: The minimum braids representing the knots with up to 10 crossings were provided by Thomas Gittings. See arXiv:math.GT/0401051.

Out[9]= ${\textrm {BR}}(6,\{1,-2,1,3,-2,-4,-3,-3,5,-4,5\})$

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]= { 6, 11, 6 }

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[{12, 9}, {10, 8}, {9, 11}, {3, 10}, {7, 2}, {8, 6}, {1, 3}, {4, 7}, {6, 12}, {2, 5}, {11, 4}, {5, 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	[-7][-5]
Hyperbolic Volume	12.7213
A-Polynomial	See Data:10 58/A-polynomial

[edit Notes for 10 58's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus	$1$
Topological 4 genus	$1$
Concordance genus	$2$
Rasmussen s-Invariant	0

[edit Notes for 10 58's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$3t^{2}-16t+27-16t^{-1}+3t^{-2}$
Conway polynomial	$3z^{4}-4z^{2}+1$
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	{ 65, 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 q^4-2 q^3+5 q^2-8 q+10-11 q^{-1} +10 q^{-2} -8 q^{-3} +6 q^{-4} -3 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^6-3 z^2 a^4-2 a^4+2 z^4 a^2+3 z^2 a^2+3 a^2+z^4-2 z^2-2-2 z^2 a^{-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 a^3 z^9+a z^9+3 a^4 z^8+6 a^2 z^8+3 z^8+3 a^5 z^7+6 a^3 z^7+7 a z^7+4 z^7 a^{-1} +a^6 z^6-5 a^4 z^6-10 a^2 z^6+3 z^6 a^{-2} -z^6-9 a^5 z^5-23 a^3 z^5-22 a z^5-6 z^5 a^{-1} +2 z^5 a^{-3} -3 a^6 z^4-5 a^4 z^4-4 a^2 z^4-2 z^4 a^{-2} +z^4 a^{-4} -5 z^4+7 a^5 z^3+18 a^3 z^3+21 a z^3+8 z^3 a^{-1} -2 z^3 a^{-3} +3 a^6 z^2+7 a^4 z^2+10 a^2 z^2-2 z^2 a^{-4} +8 z^2-2 a^5 z-4 a^3 z-6 a z-4 z a^{-1} -a^6-2 a^4-3 a^2+ a^{-4} -2}
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^{20}+q^{18}-2 q^{16}+q^{14}-2 q^{10}+3 q^8+q^4-2+ q^{-2} -3 q^{-4} + q^{-6} +2 q^{-8} - q^{-10} + q^{-12} + q^{-14} }
The G2 invariant	Data:10 58/QuantumInvariant/G2/1,0

Further Quantum Invariants

Further quantum knot invariants for 10_58.

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^{13}-2 q^{11}+3 q^9-2 q^7+2 q^5-q^3-q+2 q^{-1} -3 q^{-3} +3 q^{-5} - q^{-7} + q^{-9} }
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^{38}-2 q^{36}-2 q^{34}+8 q^{32}-2 q^{30}-12 q^{28}+13 q^{26}+6 q^{24}-20 q^{22}+7 q^{20}+15 q^{18}-17 q^{16}-2 q^{14}+16 q^{12}-7 q^{10}-9 q^8+8 q^6+8 q^4-11 q^2-5+20 q^{-2} -7 q^{-4} -16 q^{-6} +18 q^{-8} -13 q^{-12} +9 q^{-14} + q^{-16} -5 q^{-18} +3 q^{-20} - q^{-24} + q^{-26} }
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^{75}-2 q^{73}-2 q^{71}+3 q^{69}+8 q^{67}-2 q^{65}-19 q^{63}-4 q^{61}+28 q^{59}+21 q^{57}-31 q^{55}-46 q^{53}+25 q^{51}+67 q^{49}-82 q^{45}-34 q^{43}+82 q^{41}+67 q^{39}-66 q^{37}-93 q^{35}+38 q^{33}+108 q^{31}-10 q^{29}-109 q^{27}-12 q^{25}+100 q^{23}+36 q^{21}-87 q^{19}-50 q^{17}+65 q^{15}+64 q^{13}-42 q^{11}-75 q^9+8 q^7+81 q^5+32 q^3-79 q-66 q^{-1} +64 q^{-3} +96 q^{-5} -38 q^{-7} -109 q^{-9} +9 q^{-11} +104 q^{-13} +12 q^{-15} -79 q^{-17} -28 q^{-19} +55 q^{-21} +27 q^{-23} -31 q^{-25} -18 q^{-27} +14 q^{-29} +11 q^{-31} -7 q^{-33} -4 q^{-35} +3 q^{-37} + q^{-39} -2 q^{-41} + q^{-43} - q^{-49} + q^{-51} }

A2 Invariants.

Weight	Invariant
1,0	$q^{20}+q^{18}-2q^{16}+q^{14}-2q^{10}+3q^{8}+q^{4}-2+q^{-2}-3q^{-4}+q^{-6}+2q^{-8}-q^{-10}+q^{-12}+q^{-14}$

.

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.

In[3]:= K = Knot["10 58"];

In[4]:= Alexander[K][t]

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

Out[4]= $3t^{2}-16t+27-16t^{-1}+3t^{-2}$

In[5]:= Conway[K][z]

Out[5]= $3z^{4}-4z^{2}+1$

In[6]:= Alexander[K, 2][t]

KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.

Out[6]= Failed to parse (SVG (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]= { 65, 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 q^4-2 q^3+5 q^2-8 q+10-11 q^{-1} +10 q^{-2} -8 q^{-3} +6 q^{-4} -3 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^6-3 z^2 a^4-2 a^4+2 z^4 a^2+3 z^2 a^2+3 a^2+z^4-2 z^2-2-2 z^2 a^{-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 a^3 z^9+a z^9+3 a^4 z^8+6 a^2 z^8+3 z^8+3 a^5 z^7+6 a^3 z^7+7 a z^7+4 z^7 a^{-1} +a^6 z^6-5 a^4 z^6-10 a^2 z^6+3 z^6 a^{-2} -z^6-9 a^5 z^5-23 a^3 z^5-22 a z^5-6 z^5 a^{-1} +2 z^5 a^{-3} -3 a^6 z^4-5 a^4 z^4-4 a^2 z^4-2 z^4 a^{-2} +z^4 a^{-4} -5 z^4+7 a^5 z^3+18 a^3 z^3+21 a z^3+8 z^3 a^{-1} -2 z^3 a^{-3} +3 a^6 z^2+7 a^4 z^2+10 a^2 z^2-2 z^2 a^{-4} +8 z^2-2 a^5 z-4 a^3 z-6 a z-4 z a^{-1} -a^6-2 a^4-3 a^2+ a^{-4} -2}

"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 58"];

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 3 t^2-16 t+27-16 t^{-1} +3 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^4-2 q^3+5 q^2-8 q+10-11 q^{-1} +10 q^{-2} -8 q^{-3} +6 q^{-4} -3 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]= {}

In[6]:= DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]

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

Out[6]= {}

Vassiliev invariants

V₂ and V₃:

(-4, 1)

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 -16}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 8}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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{376}{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{128}{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{400}{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 -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 -\frac{2048}{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 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{6016}{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{2048}{3}}

-{\frac {22262}{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{1088}{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{45488}{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{1094}{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{3062}{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 10 58. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\		r
	\
j		\

-6

-5

-4

-3

-2

-1

0

1

2

3

4

χ

9

1

7

1

-1

5

4

1

3

4

1

-3

1

6

4

2

-1

6

5

-1

-3

4

5

-1

-5

4

6

2

-7

2

4

-2

-9

1

4

3

-11

2

-2

-13

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

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^{12}-2 q^{11}+q^{10}+4 q^9-10 q^8+7 q^7+12 q^6-32 q^5+20 q^4+30 q^3-66 q^2+29 q+57-91 q^{-1} +23 q^{-2} +76 q^{-3} -91 q^{-4} +6 q^{-5} +78 q^{-6} -68 q^{-7} -12 q^{-8} +63 q^{-9} -36 q^{-10} -20 q^{-11} +36 q^{-12} -10 q^{-13} -13 q^{-14} +11 q^{-15} -3 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 q^{24}-2 q^{23}+q^{22}+2 q^{20}-5 q^{19}+4 q^{18}+2 q^{17}-5 q^{16}-8 q^{15}+22 q^{14}+5 q^{13}-37 q^{12}-21 q^{11}+80 q^{10}+33 q^9-120 q^8-72 q^7+171 q^6+125 q^5-215 q^4-190 q^3+242 q^2+259 q-247-320 q^{-1} +229 q^{-2} +370 q^{-3} -198 q^{-4} -393 q^{-5} +146 q^{-6} +403 q^{-7} -92 q^{-8} -392 q^{-9} +31 q^{-10} +366 q^{-11} +31 q^{-12} -328 q^{-13} -81 q^{-14} +269 q^{-15} +130 q^{-16} -210 q^{-17} -151 q^{-18} +138 q^{-19} +157 q^{-20} -77 q^{-21} -136 q^{-22} +22 q^{-23} +109 q^{-24} +5 q^{-25} -69 q^{-26} -20 q^{-27} +38 q^{-28} +20 q^{-29} -17 q^{-30} -13 q^{-31} +6 q^{-32} +5 q^{-33} -3 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^{40}-2 q^{39}+q^{38}-2 q^{36}+7 q^{35}-8 q^{34}+4 q^{33}-q^{32}-11 q^{31}+25 q^{30}-15 q^{29}+11 q^{28}-15 q^{27}-45 q^{26}+73 q^{25}+10 q^{24}+36 q^{23}-85 q^{22}-173 q^{21}+155 q^{20}+152 q^{19}+179 q^{18}-229 q^{17}-550 q^{16}+148 q^{15}+464 q^{14}+639 q^{13}-300 q^{12}-1234 q^{11}-182 q^{10}+755 q^9+1456 q^8-29 q^7-1959 q^6-867 q^5+712 q^4+2296 q^3+600 q^2-2318 q-1568+273 q^{-1} +2743 q^{-2} +1280 q^{-3} -2192 q^{-4} -1939 q^{-5} -311 q^{-6} +2691 q^{-7} +1722 q^{-8} -1742 q^{-9} -1928 q^{-10} -840 q^{-11} +2279 q^{-12} +1905 q^{-13} -1119 q^{-14} -1659 q^{-15} -1274 q^{-16} +1629 q^{-17} +1873 q^{-18} -392 q^{-19} -1172 q^{-20} -1561 q^{-21} +812 q^{-22} +1571 q^{-23} +276 q^{-24} -490 q^{-25} -1520 q^{-26} +26 q^{-27} +963 q^{-28} +608 q^{-29} +188 q^{-30} -1071 q^{-31} -412 q^{-32} +277 q^{-33} +482 q^{-34} +529 q^{-35} -456 q^{-36} -383 q^{-37} -135 q^{-38} +148 q^{-39} +446 q^{-40} -56 q^{-41} -143 q^{-42} -175 q^{-43} -53 q^{-44} +198 q^{-45} +41 q^{-46} +5 q^{-47} -71 q^{-48} -62 q^{-49} +47 q^{-50} +15 q^{-51} +20 q^{-52} -10 q^{-53} -20 q^{-54} +6 q^{-55} +5 q^{-57} -3 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 q^{60}-2 q^{59}+q^{58}-2 q^{56}+3 q^{55}+4 q^{54}-8 q^{53}+q^{52}+3 q^{51}-8 q^{50}+10 q^{49}+14 q^{48}-21 q^{47}-9 q^{46}+3 q^{45}-4 q^{44}+36 q^{43}+41 q^{42}-47 q^{41}-82 q^{40}-45 q^{39}+34 q^{38}+184 q^{37}+181 q^{36}-96 q^{35}-365 q^{34}-366 q^{33}+36 q^{32}+668 q^{31}+826 q^{30}+63 q^{29}-1048 q^{28}-1507 q^{27}-542 q^{26}+1488 q^{25}+2637 q^{24}+1327 q^{23}-1766 q^{22}-4006 q^{21}-2804 q^{20}+1707 q^{19}+5681 q^{18}+4795 q^{17}-1109 q^{16}-7200 q^{15}-7358 q^{14}-202 q^{13}+8431 q^{12}+10157 q^{11}+2142 q^{10}-9044 q^9-12831 q^8-4599 q^7+8938 q^6+15092 q^5+7226 q^4-8168 q^3-16662 q^2-9688 q+6860+17481 q^{-1} +11752 q^{-2} -5299 q^{-3} -17577 q^{-4} -13272 q^{-5} +3673 q^{-6} +17126 q^{-7} +14227 q^{-8} -2146 q^{-9} -16230 q^{-10} -14734 q^{-11} +700 q^{-12} +15120 q^{-13} +14863 q^{-14} +628 q^{-15} -13722 q^{-16} -14762 q^{-17} -1986 q^{-18} +12146 q^{-19} +14447 q^{-20} +3341 q^{-21} -10288 q^{-22} -13866 q^{-23} -4733 q^{-24} +8135 q^{-25} +12989 q^{-26} +6033 q^{-27} -5764 q^{-28} -11665 q^{-29} -7050 q^{-30} +3187 q^{-31} +9899 q^{-32} +7679 q^{-33} -758 q^{-34} -7696 q^{-35} -7632 q^{-36} -1427 q^{-37} +5246 q^{-38} +6977 q^{-39} +2970 q^{-40} -2803 q^{-41} -5663 q^{-42} -3832 q^{-43} +685 q^{-44} +4017 q^{-45} +3853 q^{-46} +893 q^{-47} -2264 q^{-48} -3300 q^{-49} -1753 q^{-50} +801 q^{-51} +2302 q^{-52} +1953 q^{-53} +287 q^{-54} -1310 q^{-55} -1672 q^{-56} -786 q^{-57} +446 q^{-58} +1126 q^{-59} +915 q^{-60} +86 q^{-61} -609 q^{-62} -727 q^{-63} -322 q^{-64} +209 q^{-65} +458 q^{-66} +340 q^{-67} +19 q^{-68} -233 q^{-69} -253 q^{-70} -84 q^{-71} +80 q^{-72} +132 q^{-73} +95 q^{-74} -6 q^{-75} -71 q^{-76} -55 q^{-77} -4 q^{-78} +15 q^{-79} +24 q^{-80} +20 q^{-81} -10 q^{-82} -13 q^{-83} - q^{-84} +5 q^{-87} -3 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^{84}-2 q^{83}+q^{82}-2 q^{80}+3 q^{79}+4 q^{77}-11 q^{76}+5 q^{75}+6 q^{74}-13 q^{73}+9 q^{72}+4 q^{71}+8 q^{70}-35 q^{69}+14 q^{68}+33 q^{67}-31 q^{66}+14 q^{65}+8 q^{64}-7 q^{63}-104 q^{62}+35 q^{61}+134 q^{60}+q^{59}+56 q^{58}-24 q^{57}-166 q^{56}-376 q^{55}+11 q^{54}+473 q^{53}+386 q^{52}+433 q^{51}-56 q^{50}-866 q^{49}-1518 q^{48}-590 q^{47}+1143 q^{46}+1977 q^{45}+2404 q^{44}+770 q^{43}-2440 q^{42}-5191 q^{41}-3931 q^{40}+896 q^{39}+5519 q^{38}+8774 q^{37}+5840 q^{36}-3087 q^{35}-12736 q^{34}-14275 q^{33}-5126 q^{32}+8535 q^{31}+21464 q^{30}+20913 q^{29}+3711 q^{28}-20763 q^{27}-33790 q^{26}-23968 q^{25}+2797 q^{24}+35737 q^{23}+47742 q^{22}+25842 q^{21}-19296 q^{20}-55795 q^{19}-56250 q^{18}-19856 q^{17}+39828 q^{16}+77149 q^{15}+62162 q^{14}-176 q^{13}-66747 q^{12}-90348 q^{11}-56332 q^{10}+25849 q^9+94225 q^8+98984 q^7+31972 q^6-59204 q^5-110932 q^4-91890 q^3-565 q^2+92403 q+121569+62685 q^{-1} -39334 q^{-2} -112917 q^{-3} -113494 q^{-4} -26119 q^{-5} +78166 q^{-6} +126576 q^{-7} +81477 q^{-8} -18714 q^{-9} -102975 q^{-10} -119790 q^{-11} -43052 q^{-12} +61269 q^{-13} +120653 q^{-14} +88906 q^{-15} -2616 q^{-16} -89116 q^{-17} -117304 q^{-18} -53331 q^{-19} +44956 q^{-20} +110170 q^{-21} +91149 q^{-22} +11750 q^{-23} -72997 q^{-24} -110909 q^{-25} -62305 q^{-26} +26139 q^{-27} +95418 q^{-28} +91347 q^{-29} +28507 q^{-30} -51354 q^{-31} -99432 q^{-32} -71019 q^{-33} +2037 q^{-34} +72749 q^{-35} +86347 q^{-36} +46392 q^{-37} -22402 q^{-38} -78314 q^{-39} -73965 q^{-40} -23928 q^{-41} +40923 q^{-42} +70135 q^{-43} +57488 q^{-44} +8709 q^{-45} -46453 q^{-46} -63587 q^{-47} -41731 q^{-48} +6538 q^{-49} +41592 q^{-50} +52988 q^{-51} +30341 q^{-52} -11733 q^{-53} -39075 q^{-54} -41933 q^{-55} -17472 q^{-56} +9876 q^{-57} +32877 q^{-58} +33080 q^{-59} +12006 q^{-60} -10965 q^{-61} -25847 q^{-62} -22234 q^{-63} -10742 q^{-64} +9074 q^{-65} +19913 q^{-66} +16939 q^{-67} +6488 q^{-68} -6498 q^{-69} -12371 q^{-70} -14007 q^{-71} -4760 q^{-72} +4437 q^{-73} +9148 q^{-74} +8804 q^{-75} +3734 q^{-76} -1233 q^{-77} -7148 q^{-78} -5993 q^{-79} -2838 q^{-80} +1026 q^{-81} +3675 q^{-82} +4012 q^{-83} +3001 q^{-84} -1002 q^{-85} -2170 q^{-86} -2599 q^{-87} -1575 q^{-88} -153 q^{-89} +1214 q^{-90} +2087 q^{-91} +732 q^{-92} +179 q^{-93} -683 q^{-94} -878 q^{-95} -809 q^{-96} -167 q^{-97} +595 q^{-98} +350 q^{-99} +416 q^{-100} +81 q^{-101} -106 q^{-102} -346 q^{-103} -228 q^{-104} +62 q^{-105} +12 q^{-106} +132 q^{-107} +86 q^{-108} +59 q^{-109} -71 q^{-110} -66 q^{-111} +3 q^{-112} -27 q^{-113} +15 q^{-114} +15 q^{-115} +29 q^{-116} -10 q^{-117} -13 q^{-118} +6 q^{-119} -7 q^{-120} +5 q^{-123} -3 q^{-125} + q^{-126} }

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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