10 151

10_150

10_152

(KnotPlot image)

See the full Rolfsen Knot Table.

Visit 10 151's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 10 151 at Knotilus!

[edit 10 151 Quick Notes]

[edit 10 151 Further Notes and Views]

Knot presentations

Planar diagram presentation	X₁₄₂₅ X₃₈₄₉ X_12,6,13,5 X_9,17,10,16 X_17,1,18,20 X_13,19,14,18 X_19,15,20,14 X_15,11,16,10 X_6,12,7,11 X₇₂₈₃
Gauss code	-1, 10, -2, 1, 3, -9, -10, 2, -4, 8, 9, -3, -6, 7, -8, 4, -5, 6, -7, 5
Dowker-Thistlethwaite code	4 8 -12 2 16 -6 18 10 20 14
Conway Notation	[(21,2)(21,2-)]

Minimum Braid Representative

A Morse Link Presentation

An Arc Presentation

Length is 11, width is 4,

Braid index is 4

[{11, 5}, {1, 9}, {8, 10}, {9, 11}, {7, 4}, {5, 8}, {10, 13}, {6, 12}, {13, 7}, {12, 3}, {4, 2}, {3, 1}, {2, 6}]

[edit Notes on presentations of 10 151]

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["10 151"];

In[4]:= PD[K]

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

Out[4]= X₁₄₂₅ X₃₈₄₉ X_12,6,13,5 X_9,17,10,16 X_17,1,18,20 X_13,19,14,18 X_19,15,20,14 X_15,11,16,10 X_6,12,7,11 X₇₂₈₃

In[5]:= GaussCode[K]

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

In[6]:= DTCode[K]

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

(The path below may be different on your system)

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

In[8]:= ConwayNotation[K]

Out[8]= [(21,2)(21,2-)]

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

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

Out[9]= Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \textrm{BR}(4,\{1,1,1,2,-1,-1,3,-2,1,3,-2\})}

In[10]:= {First[br], Crossings[br], BraidIndex[K]}

KnotTheory::credits: The braid index data known to KnotTheory` is taken from Charles Livingston's http://www.indiana.edu/~knotinfo/.

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

Out[10]= { 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[{11, 5}, {1, 9}, {8, 10}, {9, 11}, {7, 4}, {5, 8}, {10, 13}, {6, 12}, {13, 7}, {12, 3}, {4, 2}, {3, 1}, {2, 6}]

In[14]:= Draw[ap]

Out[14]= -Graphics-

Three dimensional invariants

Symmetry type	Chiral
Unknotting number	2
3-genus	3
Bridge index	3
Super bridge index	Missing
Nakanishi index	1
Maximal Thurston-Bennequin number	[-2][-8]
Hyperbolic Volume	11.843
A-Polynomial	See Data:10 151/A-polynomial

[edit Notes for 10 151'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 1}
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}
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 3}
Rasmussen s-Invariant	2

[edit Notes for 10 151'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^3-4 t^2+10 t-13+10 t^{-1} -4 t^{-2} + t^{-3} }
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^6+2 z^4+3 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	{ 43, 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 -2 q^6+4 q^5-6 q^4+8 q^3-7 q^2+7 q-5+3 q^{-1} - q^{-2} }
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 z^6 a^{-2} +4 z^4 a^{-2} -z^4 a^{-4} -z^4+6 z^2 a^{-2} -z^2 a^{-4} -2 z^2+3 a^{-2} - a^{-6} -1}
Kauffman polynomial (db, data sources)	$z^{8}a^{-2}+z^{8}a^{-4}+3z^{7}a^{-1}+5z^{7}a^{-3}+2z^{7}a^{-5}+5z^{6}a^{-2}+3z^{6}a^{-4}+z^{6}a^{-6}+3z^{6}+az^{5}-4z^{5}a^{-1}-7z^{5}a^{-3}-2z^{5}a^{-5}-15z^{4}a^{-2}-6z^{4}a^{-4}+2z^{4}a^{-6}-7z^{4}-2az^{3}-3z^{3}a^{-1}+z^{3}a^{-3}+5z^{3}a^{-5}+3z^{3}a^{-7}+10z^{2}a^{-2}+4z^{2}a^{-4}-2z^{2}a^{-6}+4z^{2}+az+2za^{-1}+za^{-3}-3za^{-5}-3za^{-7}-3a^{-2}+a^{-6}-1$
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^6+q^4-q^2+2 q^{-2} - q^{-4} +3 q^{-6} +2 q^{-10} + q^{-12} - q^{-14} + q^{-16} -2 q^{-18} - q^{-20} }
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^{32}-2 q^{30}+5 q^{28}-8 q^{26}+7 q^{24}-4 q^{22}-6 q^{20}+19 q^{18}-30 q^{16}+34 q^{14}-27 q^{12}+q^{10}+27 q^8-52 q^6+62 q^4-46 q^2+15+23 q^{-2} -51 q^{-4} +55 q^{-6} -34 q^{-8} +33 q^{-12} -46 q^{-14} +35 q^{-16} -35 q^{-20} +62 q^{-22} -62 q^{-24} +40 q^{-26} - q^{-28} -43 q^{-30} +75 q^{-32} -80 q^{-34} +64 q^{-36} -24 q^{-38} -18 q^{-40} +56 q^{-42} -68 q^{-44} +57 q^{-46} -25 q^{-48} -11 q^{-50} +42 q^{-52} -45 q^{-54} +24 q^{-56} +13 q^{-58} -42 q^{-60} +55 q^{-62} -42 q^{-64} +5 q^{-66} +29 q^{-68} -56 q^{-70} +60 q^{-72} -45 q^{-74} +15 q^{-76} +13 q^{-78} -34 q^{-80} +34 q^{-82} -28 q^{-84} +15 q^{-86} -2 q^{-88} -7 q^{-90} +7 q^{-92} -8 q^{-94} +6 q^{-96} -2 q^{-98} + q^{-100} + q^{-102} }

Further Quantum Invariants

Further quantum knot invariants for 10_151.

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^5+2 q^3-2 q+2 q^{-1} + q^{-5} +2 q^{-7} -2 q^{-9} +2 q^{-11} -2 q^{-13} }
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^{16}-2 q^{14}-2 q^{12}+7 q^{10}-2 q^8-10 q^6+10 q^4+5 q^2-14+5 q^{-2} +10 q^{-4} -9 q^{-6} - q^{-8} +9 q^{-10} -6 q^{-14} +2 q^{-16} +10 q^{-18} -10 q^{-20} -6 q^{-22} +15 q^{-24} -6 q^{-26} -10 q^{-28} +9 q^{-30} -5 q^{-34} +2 q^{-36} + q^{-38} }
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^{33}+2 q^{31}+2 q^{29}-3 q^{27}-7 q^{25}+2 q^{23}+17 q^{21}+2 q^{19}-25 q^{17}-17 q^{15}+28 q^{13}+38 q^{11}-22 q^9-55 q^7+5 q^5+64 q^3+19 q-67 q^{-1} -37 q^{-3} +56 q^{-5} +52 q^{-7} -40 q^{-9} -55 q^{-11} +26 q^{-13} +57 q^{-15} -10 q^{-17} -47 q^{-19} -5 q^{-21} +40 q^{-23} +21 q^{-25} -31 q^{-27} -39 q^{-29} +17 q^{-31} +56 q^{-33} -3 q^{-35} -65 q^{-37} -20 q^{-39} +71 q^{-41} +35 q^{-43} -61 q^{-45} -50 q^{-47} +41 q^{-49} +52 q^{-51} -20 q^{-53} -45 q^{-55} +5 q^{-57} +28 q^{-59} +6 q^{-61} -14 q^{-63} -6 q^{-65} +7 q^{-67} +2 q^{-69} -2 q^{-73} }

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^6+q^4-q^2+2 q^{-2} - q^{-4} +3 q^{-6} +2 q^{-10} + q^{-12} - q^{-14} + q^{-16} -2 q^{-18} - q^{-20} }
1,1	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^{20}-4 q^{18}+12 q^{16}-28 q^{14}+52 q^{12}-86 q^{10}+128 q^8-170 q^6+196 q^4-206 q^2+188-134 q^{-2} +52 q^{-4} +48 q^{-6} -152 q^{-8} +254 q^{-10} -324 q^{-12} +378 q^{-14} -380 q^{-16} +360 q^{-18} -295 q^{-20} +208 q^{-22} -108 q^{-24} +92 q^{-28} -166 q^{-30} +200 q^{-32} -208 q^{-34} +188 q^{-36} -154 q^{-38} +108 q^{-40} -70 q^{-42} +40 q^{-44} -18 q^{-46} +6 q^{-48} -2 q^{-50} +2 q^{-54} }
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^{18}-q^{16}-2 q^{14}+2 q^{12}+2 q^{10}-2 q^8-4 q^6+2 q^4+5 q^2-5-3 q^{-2} +6 q^{-4} + q^{-6} -3 q^{-8} +5 q^{-12} + q^{-16} +6 q^{-18} +3 q^{-20} -2 q^{-22} +4 q^{-24} +4 q^{-26} -7 q^{-28} -3 q^{-30} +3 q^{-32} - q^{-34} -6 q^{-36} -4 q^{-38} +3 q^{-40} - q^{-42} -3 q^{-44} + q^{-46} +2 q^{-48} +2 q^{-50} }

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^{14}-2 q^{12}+q^{10}+3 q^8-7 q^6+3 q^4+3 q^2-11+6 q^{-2} +5 q^{-4} -8 q^{-6} +6 q^{-8} +7 q^{-10} - q^{-12} + q^{-14} +4 q^{-16} +4 q^{-18} - q^{-20} -3 q^{-22} +9 q^{-24} -6 q^{-26} -9 q^{-28} +8 q^{-30} -7 q^{-32} -7 q^{-34} +7 q^{-36} - q^{-38} -2 q^{-40} +3 q^{-42} }

1,0,0

-q^{7}+q^{5}-2q^{3}+q-q^{-1}+2q^{-3}+2q^{-7}+2q^{-9}+q^{-11}+2q^{-13}+2q^{-17}-2q^{-19}+q^{-21}-2q^{-23}-q^{-25}-q^{-27}

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^{16}-q^{14}+3 q^{10}-2 q^8-4 q^6+4 q^4-q^2-9- q^{-2} +6 q^{-4} -3 q^{-6} -7 q^{-8} +9 q^{-10} +10 q^{-12} -4 q^{-14} +2 q^{-16} +16 q^{-18} - q^{-20} -2 q^{-22} +12 q^{-24} +5 q^{-26} -6 q^{-28} +3 q^{-30} +6 q^{-32} -8 q^{-34} -11 q^{-36} -2 q^{-40} -12 q^{-42} -3 q^{-44} +5 q^{-46} - q^{-50} +3 q^{-52} +2 q^{-54} + q^{-56} }

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^8+q^6-2 q^4- q^{-2} +2 q^{-4} +3 q^{-8} + q^{-10} +3 q^{-12} + q^{-14} +2 q^{-16} + q^{-20} + q^{-22} -2 q^{-24} + q^{-26} -2 q^{-28} - q^{-30} - q^{-32} - q^{-34} }

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^{14}+2 q^{12}-5 q^{10}+7 q^8-9 q^6+11 q^4-11 q^2+9-6 q^{-2} +3 q^{-4} +4 q^{-6} -8 q^{-8} +15 q^{-10} -17 q^{-12} +21 q^{-14} -20 q^{-16} +18 q^{-18} -13 q^{-20} +9 q^{-22} -3 q^{-24} -2 q^{-26} +7 q^{-28} -10 q^{-30} +11 q^{-32} -11 q^{-34} +9 q^{-36} -7 q^{-38} +4 q^{-40} -3 q^{-42} }

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^{24}-2 q^{20}-2 q^{18}+3 q^{16}+5 q^{14}-2 q^{12}-8 q^{10}-3 q^8+9 q^6+7 q^4-8 q^2-11+2 q^{-2} +12 q^{-4} +4 q^{-6} -10 q^{-8} -5 q^{-10} +7 q^{-12} +8 q^{-14} -3 q^{-16} -5 q^{-18} +3 q^{-20} +9 q^{-22} + q^{-24} -7 q^{-26} - q^{-28} +9 q^{-30} +5 q^{-32} -6 q^{-34} -7 q^{-36} +6 q^{-38} +9 q^{-40} -4 q^{-42} -13 q^{-44} -2 q^{-46} +10 q^{-48} +4 q^{-50} -9 q^{-52} -10 q^{-54} +2 q^{-56} +8 q^{-58} +2 q^{-60} -4 q^{-62} -3 q^{-64} + q^{-66} +3 q^{-68} }

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^{18}-2 q^{16}+3 q^{14}-4 q^{12}+6 q^{10}-8 q^8+7 q^6-10 q^4+8 q^2-9+4 q^{-2} -4 q^{-4} +3 q^{-6} +3 q^{-8} -4 q^{-10} +12 q^{-12} -6 q^{-14} +16 q^{-16} -13 q^{-18} +17 q^{-20} -13 q^{-22} +17 q^{-24} -12 q^{-26} +10 q^{-28} -7 q^{-30} +7 q^{-32} - q^{-34} -4 q^{-36} + q^{-38} -9 q^{-40} +7 q^{-42} -11 q^{-44} +5 q^{-46} -10 q^{-48} +8 q^{-50} -4 q^{-52} +4 q^{-54} -3 q^{-56} +3 q^{-58} }

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^{32}-2 q^{30}+5 q^{28}-8 q^{26}+7 q^{24}-4 q^{22}-6 q^{20}+19 q^{18}-30 q^{16}+34 q^{14}-27 q^{12}+q^{10}+27 q^8-52 q^6+62 q^4-46 q^2+15+23 q^{-2} -51 q^{-4} +55 q^{-6} -34 q^{-8} +33 q^{-12} -46 q^{-14} +35 q^{-16} -35 q^{-20} +62 q^{-22} -62 q^{-24} +40 q^{-26} - q^{-28} -43 q^{-30} +75 q^{-32} -80 q^{-34} +64 q^{-36} -24 q^{-38} -18 q^{-40} +56 q^{-42} -68 q^{-44} +57 q^{-46} -25 q^{-48} -11 q^{-50} +42 q^{-52} -45 q^{-54} +24 q^{-56} +13 q^{-58} -42 q^{-60} +55 q^{-62} -42 q^{-64} +5 q^{-66} +29 q^{-68} -56 q^{-70} +60 q^{-72} -45 q^{-74} +15 q^{-76} +13 q^{-78} -34 q^{-80} +34 q^{-82} -28 q^{-84} +15 q^{-86} -2 q^{-88} -7 q^{-90} +7 q^{-92} -8 q^{-94} +6 q^{-96} -2 q^{-98} + q^{-100} + q^{-102} }

.

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["10 151"];

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^3-4 t^2+10 t-13+10 t^{-1} -4 t^{-2} + t^{-3} }

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^6+2 z^4+3 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]= { 43, 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 -2 q^6+4 q^5-6 q^4+8 q^3-7 q^2+7 q-5+3 q^{-1} - q^{-2} }

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 z^6 a^{-2} +4 z^4 a^{-2} -z^4 a^{-4} -z^4+6 z^2 a^{-2} -z^2 a^{-4} -2 z^2+3 a^{-2} - a^{-6} -1}

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

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

Out[10]= $z^{8}a^{-2}+z^{8}a^{-4}+3z^{7}a^{-1}+5z^{7}a^{-3}+2z^{7}a^{-5}+5z^{6}a^{-2}+3z^{6}a^{-4}+z^{6}a^{-6}+3z^{6}+az^{5}-4z^{5}a^{-1}-7z^{5}a^{-3}-2z^{5}a^{-5}-15z^{4}a^{-2}-6z^{4}a^{-4}+2z^{4}a^{-6}-7z^{4}-2az^{3}-3z^{3}a^{-1}+z^{3}a^{-3}+5z^{3}a^{-5}+3z^{3}a^{-7}+10z^{2}a^{-2}+4z^{2}a^{-4}-2z^{2}a^{-6}+4z^{2}+az+2za^{-1}+za^{-3}-3za^{-5}-3za^{-7}-3a^{-2}+a^{-6}-1$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11n54, K11n129,}

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["10 151"];

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^3-4 t^2+10 t-13+10 t^{-1} -4 t^{-2} + t^{-3} } , $-2q^{6}+4q^{5}-6q^{4}+8q^{3}-7q^{2}+7q-5+3q^{-1}-q^{-2}$ }

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]= {K11n54, K11n129,}

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:

(3, 4)

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 12}

Failed to parse (SVG (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 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 142}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 384}

Failed to parse (SVG (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{1952}{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{320}{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 96}

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 512}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1704}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 216}

Failed to parse (SVG (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{30591}{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 \frac{614}{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{18142}{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{193}{6}}

{\frac {1471}{10}}

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

\		r
	\
j		\

-3

-2

-1

0

1

2

3

4

5

χ

13

2

-2

11

2

9

4

2

-2

7

4

2

5

3

4

1

3

4

0

1

2

4

2

-1

1

3

-2

-3

2

-5

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=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 r=-3}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-2}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{2}\oplus{\mathbb Z}_2}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-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}^{3}\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=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}^{4}\oplus{\mathbb Z}_2^{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}}
Failed to parse (SVG (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^{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}^{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 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}^{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=3}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{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=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}\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=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^{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^{18}+q^{17}-7 q^{16}+6 q^{15}+10 q^{14}-26 q^{13}+10 q^{12}+31 q^{11}-47 q^{10}+6 q^9+51 q^8-55 q^7-2 q^6+57 q^5-46 q^4-12 q^3+49 q^2-27 q-17+30 q^{-1} -8 q^{-2} -12 q^{-3} +10 q^{-4} -3 q^{-6} + q^{-7} }
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^{35}+2 q^{34}+2 q^{33}+5 q^{32}-15 q^{31}-6 q^{30}+22 q^{29}+27 q^{28}-38 q^{27}-56 q^{26}+47 q^{25}+99 q^{24}-49 q^{23}-147 q^{22}+36 q^{21}+195 q^{20}-13 q^{19}-238 q^{18}-9 q^{17}+257 q^{16}+46 q^{15}-277 q^{14}-65 q^{13}+265 q^{12}+98 q^{11}-258 q^{10}-110 q^9+223 q^8+135 q^7-191 q^6-141 q^5+142 q^4+150 q^3-99 q^2-137 q+49+120 q^{-1} -13 q^{-2} -92 q^{-3} -10 q^{-4} +60 q^{-5} +20 q^{-6} -32 q^{-7} -20 q^{-8} +15 q^{-9} +12 q^{-10} -5 q^{-11} -5 q^{-12} +3 q^{-14} - q^{-15} }
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^{58}+q^{57}-3 q^{56}-6 q^{55}+4 q^{54}+7 q^{53}+17 q^{52}-7 q^{51}-51 q^{50}-9 q^{49}+27 q^{48}+107 q^{47}+37 q^{46}-168 q^{45}-131 q^{44}-18 q^{43}+315 q^{42}+271 q^{41}-260 q^{40}-414 q^{39}-290 q^{38}+515 q^{37}+725 q^{36}-142 q^{35}-700 q^{34}-794 q^{33}+519 q^{32}+1196 q^{31}+179 q^{30}-795 q^{29}-1295 q^{28}+328 q^{27}+1457 q^{26}+514 q^{25}-685 q^{24}-1592 q^{23}+79 q^{22}+1475 q^{21}+735 q^{20}-473 q^{19}-1657 q^{18}-148 q^{17}+1305 q^{16}+848 q^{15}-195 q^{14}-1540 q^{13}-368 q^{12}+978 q^{11}+873 q^{10}+140 q^9-1245 q^8-553 q^7+519 q^6+756 q^5+449 q^4-784 q^3-583 q^2+62 q+461+558 q^{-1} -294 q^{-2} -394 q^{-3} -191 q^{-4} +121 q^{-5} +409 q^{-6} -130 q^{-8} -175 q^{-9} -60 q^{-10} +171 q^{-11} +52 q^{-12} +11 q^{-13} -64 q^{-14} -61 q^{-15} +38 q^{-16} +15 q^{-17} +20 q^{-18} -8 q^{-19} -19 q^{-20} +5 q^{-21} +5 q^{-23} -3 q^{-25} + q^{-26} }
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^{86}+2 q^{85}+4 q^{83}+5 q^{82}-7 q^{81}-22 q^{80}-q^{79}+10 q^{78}+34 q^{77}+60 q^{76}-18 q^{75}-115 q^{74}-113 q^{73}-24 q^{72}+158 q^{71}+325 q^{70}+168 q^{69}-262 q^{68}-572 q^{67}-473 q^{66}+175 q^{65}+958 q^{64}+1030 q^{63}+70 q^{62}-1290 q^{61}-1796 q^{60}-678 q^{59}+1477 q^{58}+2730 q^{57}+1616 q^{56}-1360 q^{55}-3665 q^{54}-2850 q^{53}+886 q^{52}+4421 q^{51}+4240 q^{50}-67 q^{49}-4887 q^{48}-5597 q^{47}-990 q^{46}+5020 q^{45}+6757 q^{44}+2119 q^{43}-4843 q^{42}-7587 q^{41}-3238 q^{40}+4461 q^{39}+8157 q^{38}+4098 q^{37}-3939 q^{36}-8344 q^{35}-4864 q^{34}+3383 q^{33}+8418 q^{32}+5307 q^{31}-2827 q^{30}-8185 q^{29}-5723 q^{28}+2252 q^{27}+7951 q^{26}+5910 q^{25}-1656 q^{24}-7456 q^{23}-6134 q^{22}+971 q^{21}+6930 q^{20}+6187 q^{19}-209 q^{18}-6116 q^{17}-6228 q^{16}-640 q^{15}+5202 q^{14}+6016 q^{13}+1502 q^{12}-4009 q^{11}-5657 q^{10}-2276 q^9+2758 q^8+4966 q^7+2820 q^6-1410 q^5-4063 q^4-3078 q^3+267 q^2+2944 q+2942+649 q^{-1} -1809 q^{-2} -2498 q^{-3} -1166 q^{-4} +806 q^{-5} +1834 q^{-6} +1306 q^{-7} -79 q^{-8} -1114 q^{-9} -1148 q^{-10} -340 q^{-11} +528 q^{-12} +819 q^{-13} +458 q^{-14} -126 q^{-15} -464 q^{-16} -406 q^{-17} -76 q^{-18} +212 q^{-19} +268 q^{-20} +116 q^{-21} -54 q^{-22} -132 q^{-23} -106 q^{-24} -6 q^{-25} +62 q^{-26} +56 q^{-27} +12 q^{-28} -12 q^{-29} -24 q^{-30} -20 q^{-31} +8 q^{-32} +12 q^{-33} +2 q^{-34} -5 q^{-37} +3 q^{-39} - q^{-40} }
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^{120}+q^{119}-3 q^{118}-2 q^{117}-2 q^{116}+q^{115}+2 q^{114}+13 q^{113}+21 q^{112}-8 q^{111}-41 q^{110}-48 q^{109}-17 q^{108}+6 q^{107}+113 q^{106}+183 q^{105}+92 q^{104}-146 q^{103}-357 q^{102}-341 q^{101}-243 q^{100}+327 q^{99}+915 q^{98}+970 q^{97}+231 q^{96}-928 q^{95}-1710 q^{94}-2000 q^{93}-429 q^{92}+2055 q^{91}+3803 q^{90}+3118 q^{89}+94 q^{88}-3635 q^{87}-6799 q^{86}-5086 q^{85}+735 q^{84}+7577 q^{83}+10229 q^{82}+6481 q^{81}-2214 q^{80}-12818 q^{79}-15176 q^{78}-7357 q^{77}+7382 q^{76}+18698 q^{75}+19187 q^{74}+6932 q^{73}-14315 q^{72}-26637 q^{71}-22143 q^{70}-864 q^{69}+22214 q^{68}+32959 q^{67}+22463 q^{66}-7769 q^{65}-32841 q^{64}-37349 q^{63}-14781 q^{62}+17946 q^{61}+41183 q^{60}+37568 q^{59}+3743 q^{58}-31588 q^{57}-46734 q^{56}-27737 q^{55}+9250 q^{54}+42256 q^{53}+46837 q^{52}+14301 q^{51}-26107 q^{50}-49357 q^{49}-35575 q^{48}+1077 q^{47}+39177 q^{46}+50048 q^{45}+20995 q^{44}-20269 q^{43}-47977 q^{42}-38806 q^{41}-4769 q^{40}+34917 q^{39}+49793 q^{38}+24848 q^{37}-14993 q^{36}-44855 q^{35}-39861 q^{34}-9656 q^{33}+29765 q^{32}+47813 q^{31}+28065 q^{30}-8734 q^{29}-39836 q^{28}-39980 q^{27}-15505 q^{26}+22101 q^{25}+43550 q^{24}+31297 q^{23}+47 q^{22}-31223 q^{21}-37992 q^{20}-22209 q^{19}+10784 q^{18}+35021 q^{17}+32471 q^{16}+10341 q^{15}-18165 q^{14}-31284 q^{13}-26575 q^{12}-2302 q^{11}+21452 q^{10}+28161 q^9+17824 q^8-3343 q^7-19019 q^6-24437 q^5-11863 q^4+6209 q^3+17586 q^2+17892 q+7230-5189 q^{-1} -15429 q^{-2} -13277 q^{-3} -4234 q^{-4} +5443 q^{-5} +10909 q^{-6} +9321 q^{-7} +3615 q^{-8} -5026 q^{-9} -7871 q^{-10} -6366 q^{-11} -1807 q^{-12} +2900 q^{-13} +5254 q^{-14} +4911 q^{-15} +740 q^{-16} -1901 q^{-17} -3355 q^{-18} -2721 q^{-19} -952 q^{-20} +1066 q^{-21} +2402 q^{-22} +1415 q^{-23} +571 q^{-24} -589 q^{-25} -1072 q^{-26} -1070 q^{-27} -375 q^{-28} +490 q^{-29} +457 q^{-30} +526 q^{-31} +183 q^{-32} -73 q^{-33} -351 q^{-34} -274 q^{-35} + q^{-36} +3 q^{-37} +131 q^{-38} +102 q^{-39} +72 q^{-40} -56 q^{-41} -65 q^{-42} -7 q^{-43} -29 q^{-44} +12 q^{-45} +15 q^{-46} +29 q^{-47} -8 q^{-48} -12 q^{-49} +5 q^{-50} -7 q^{-51} +5 q^{-54} -3 q^{-56} + q^{-57} }

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

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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

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