10 128
From Knot Atlas
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 (KnotPlot image) |
See the full Rolfsen Knot Table. Visit 10 128's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) |
Knot presentations
| Planar diagram presentation | X4251 X8493 X9,17,10,16 X5,15,6,14 X15,7,16,6 X13,1,14,20 X19,11,20,10 X11,19,12,18 X17,13,18,12 X2837 |
| Gauss code | 1, -10, 2, -1, -4, 5, 10, -2, -3, 7, -8, 9, -6, 4, -5, 3, -9, 8, -7, 6 |
| Dowker-Thistlethwaite code | 4 8 -14 2 -16 -18 -20 -6 -12 -10 |
| Conway Notation | [32,3,2-] |
| Minimum Braid Representative | A Morse Link Presentation | An Arc Presentation | ||||
Length is 11, width is 4, Braid index is 4 |
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 [{5, 8}, {4, 6}, {3, 7}, {1, 5}, {9, 4}, {8, 10}, {2, 9}, {10, 3}, {7, 2}, {6, 1}] |
[edit Notes on presentations of 10 128]
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]:=
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AppendTo[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
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Loading KnotTheory` version of May 31, 2006, 14:15:20.091.
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In[3]:=
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K = Knot["10 128"];
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In[4]:=
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PD[K]
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KnotTheory::loading: Loading precomputed data in PD4Knots`.
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Out[4]=
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X4251 X8493 X9,17,10,16 X5,15,6,14 X15,7,16,6 X13,1,14,20 X19,11,20,10 X11,19,12,18 X17,13,18,12 X2837 |
In[5]:=
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GaussCode[K]
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Out[5]=
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1, -10, 2, -1, -4, 5, 10, -2, -3, 7, -8, 9, -6, 4, -5, 3, -9, 8, -7, 6 |
In[6]:=
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DTCode[K]
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Out[6]=
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4 8 -14 2 -16 -18 -20 -6 -12 -10 |
(The path below may be different on your system)
In[7]:=
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AppendTo[$Path, "C:/bin/LinKnot/"];
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In[8]:=
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ConwayNotation[K]
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Out[8]=
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[32,3,2-] |
In[9]:=
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br = BR[K]
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KnotTheory::credits: The minimum braids representing the knots with up to 10 crossings were provided by Thomas Gittings. See arXiv:math.GT/0401051.
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Out[9]=
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Failed to parse (SVG (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,\{1,1,1,2,1,1,2,2,3,-2,3\})} |
In[10]:=
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{First[br], Crossings[br], BraidIndex[K]}
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KnotTheory::credits: The braid index data known to KnotTheory` is taken from Charles Livingston's http://www.indiana.edu/~knotinfo/.
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KnotTheory::loading: Loading precomputed data in IndianaData`.
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Out[10]=
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{ 4, 11, 4 } |
In[11]:=
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Show[BraidPlot[br]]
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Out[11]=
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-Graphics- |
In[12]:=
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Show[DrawMorseLink[K]]
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KnotTheory::credits: "MorseLink was added to KnotTheory` by Siddarth Sankaran at the University of Toronto in the summer of 2005."
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KnotTheory::credits: "DrawMorseLink was written by Siddarth Sankaran at the University of Toronto in the summer of 2005."
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Out[12]=
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-Graphics- |
In[13]:=
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ap = ArcPresentation[K]
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Out[13]=
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ArcPresentation[{5, 8}, {4, 6}, {3, 7}, {1, 5}, {9, 4}, {8, 10}, {2, 9}, {10, 3}, {7, 2}, {6, 1}] |
In[14]:=
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Draw[ap]
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Out[14]=
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-Graphics- |
Three dimensional invariants
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Four dimensional invariants
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Polynomial invariants
| Alexander polynomial | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2 t^3-3 t^2+t+1+ t^{-1} -3 t^{-2} +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 2 z^6+9 z^4+7 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 | { 11, 6 } |
| 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^{10}+q^9-2 q^8+2 q^7-q^6+2 q^5-q^4+q^3} |
| 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^{-6} +z^6 a^{-8} +5 z^4 a^{-6} +5 z^4 a^{-8} -z^4 a^{-10} +6 z^2 a^{-6} +6 z^2 a^{-8} -5 z^2 a^{-10} +2 a^{-6} +2 a^{-8} -4 a^{-10} + a^{-12} } |
| 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^8 a^{-8} +z^8 a^{-10} +z^7 a^{-7} +2 z^7 a^{-9} +z^7 a^{-11} +z^6 a^{-6} -5 z^6 a^{-8} -6 z^6 a^{-10} -4 z^5 a^{-7} -10 z^5 a^{-9} -6 z^5 a^{-11} -5 z^4 a^{-6} +7 z^4 a^{-8} +12 z^4 a^{-10} +2 z^3 a^{-7} +13 z^3 a^{-9} +11 z^3 a^{-11} +6 z^2 a^{-6} -5 z^2 a^{-8} -11 z^2 a^{-10} +z a^{-7} -5 z a^{-9} -6 z a^{-11} -2 a^{-6} +2 a^{-8} +4 a^{-10} + a^{-12} } |
| 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^{-10} + q^{-14} + q^{-16} +2 q^{-18} + q^{-20} + q^{-22} + q^{-24} - q^{-26} - q^{-28} -2 q^{-30} - q^{-32} - q^{-34} + q^{-38} } |
| 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^{-50} + q^{-54} + q^{-58} +3 q^{-64} - q^{-66} +2 q^{-68} +2 q^{-74} + q^{-78} + q^{-80} + q^{-84} +2 q^{-86} - q^{-88} +3 q^{-90} -2 q^{-92} +2 q^{-94} +2 q^{-96} - q^{-98} +3 q^{-100} - q^{-102} +2 q^{-104} - q^{-114} - q^{-116} - q^{-120} -3 q^{-124} - q^{-126} -3 q^{-128} + q^{-130} -3 q^{-132} -2 q^{-134} -3 q^{-138} +2 q^{-140} -2 q^{-142} - q^{-144} - q^{-148} + q^{-152} - q^{-154} +2 q^{-156} + q^{-158} - q^{-160} +2 q^{-162} - q^{-164} + q^{-166} + q^{-168} - q^{-170} + q^{-172} } |
Further Quantum Invariants
Further quantum knot invariants for 10_128.
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} + q^{-9} + q^{-11} + q^{-13} - q^{-17} - q^{-21} } |
| 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^{-10} +2 q^{-16} + q^{-18} + q^{-22} + q^{-24} + q^{-26} - q^{-28} + q^{-32} - q^{-34} - q^{-36} -2 q^{-40} - q^{-42} + q^{-52} - q^{-56} + q^{-60} } |
| 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^{-15} + q^{-21} +2 q^{-23} + q^{-25} - q^{-27} +2 q^{-31} +2 q^{-33} + q^{-35} - q^{-41} + q^{-43} +2 q^{-45} -3 q^{-49} -2 q^{-51} + q^{-53} + q^{-55} -2 q^{-57} -3 q^{-59} + q^{-61} +2 q^{-63} - q^{-65} -3 q^{-67} - q^{-73} - q^{-75} + q^{-77} + q^{-81} + q^{-83} - q^{-85} +3 q^{-89} +2 q^{-91} -3 q^{-93} -3 q^{-95} +3 q^{-97} +3 q^{-99} -2 q^{-101} -3 q^{-103} +3 q^{-107} + q^{-109} - q^{-111} - q^{-113} } |
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^{-10} + q^{-14} + q^{-16} +2 q^{-18} + q^{-20} + q^{-22} + q^{-24} - q^{-26} - q^{-28} -2 q^{-30} - q^{-32} - q^{-34} + q^{-38} } |
| 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} +2 q^{-24} +6 q^{-28} +8 q^{-32} +2 q^{-34} +7 q^{-36} +2 q^{-38} +2 q^{-42} -5 q^{-44} -8 q^{-48} -2 q^{-50} -9 q^{-52} -2 q^{-54} -6 q^{-56} +2 q^{-58} -2 q^{-60} +4 q^{-62} +2 q^{-64} +2 q^{-66} + q^{-68} +2 q^{-70} -2 q^{-78} +2 q^{-80} -4 q^{-82} +3 q^{-84} -2 q^{-86} +2 q^{-88} } |
| 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^{-20} + q^{-26} +2 q^{-28} +2 q^{-30} +2 q^{-32} +2 q^{-34} +3 q^{-36} +2 q^{-38} +2 q^{-40} +2 q^{-42} +2 q^{-44} - q^{-50} -2 q^{-52} -4 q^{-54} -5 q^{-56} -4 q^{-58} -3 q^{-60} -3 q^{-62} + q^{-70} + q^{-72} + q^{-74} +2 q^{-78} +2 q^{-80} +2 q^{-82} + q^{-86} - q^{-88} -2 q^{-90} - q^{-92} + q^{-96} } |
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^{-20} + q^{-24} +2 q^{-26} +2 q^{-28} +2 q^{-30} +3 q^{-32} +3 q^{-34} +3 q^{-36} +2 q^{-38} +3 q^{-40} - q^{-44} -3 q^{-46} -4 q^{-48} -6 q^{-50} -5 q^{-52} -3 q^{-54} -2 q^{-56} + q^{-58} +2 q^{-60} +3 q^{-62} + q^{-64} + q^{-66} } |
| 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^{-15} + q^{-19} + q^{-21} +2 q^{-23} + q^{-25} +2 q^{-27} +2 q^{-29} + q^{-31} + q^{-33} - q^{-35} - q^{-37} -3 q^{-39} -2 q^{-41} -2 q^{-43} - q^{-45} + q^{-49} + q^{-51} } |
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^{-30} + q^{-34} +2 q^{-36} +3 q^{-38} +2 q^{-40} +4 q^{-42} +4 q^{-44} +5 q^{-46} +4 q^{-48} +4 q^{-50} +5 q^{-52} +5 q^{-54} +3 q^{-56} +3 q^{-58} +2 q^{-60} -2 q^{-62} -6 q^{-64} -10 q^{-66} -12 q^{-68} -14 q^{-70} -12 q^{-72} -7 q^{-74} -3 q^{-76} + q^{-78} +7 q^{-80} +8 q^{-82} +6 q^{-84} +5 q^{-86} +4 q^{-88} + q^{-90} - q^{-92} - q^{-94} - q^{-98} - q^{-100} } |
| 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^{-20} + q^{-24} + q^{-26} +2 q^{-28} + q^{-30} +2 q^{-32} +2 q^{-34} +2 q^{-36} +2 q^{-38} + q^{-40} + q^{-42} - q^{-44} - q^{-46} -3 q^{-48} -3 q^{-50} -3 q^{-52} -2 q^{-54} - q^{-56} + q^{-60} + q^{-62} + q^{-64} } |
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^{-20} + q^{-24} +2 q^{-28} + q^{-32} + q^{-34} + q^{-36} +2 q^{-38} - q^{-40} +2 q^{-42} - q^{-44} + q^{-46} -2 q^{-48} - q^{-52} - q^{-54} - q^{-58} - q^{-62} + q^{-64} - q^{-66} } |
| 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^{-30} + q^{-38} + q^{-40} + q^{-42} +2 q^{-46} +2 q^{-48} + q^{-50} +2 q^{-54} +2 q^{-56} +2 q^{-58} + q^{-62} + q^{-64} + q^{-66} - q^{-68} - q^{-70} - q^{-72} - q^{-74} -2 q^{-76} -3 q^{-78} -2 q^{-80} -2 q^{-82} - q^{-84} -2 q^{-86} -2 q^{-88} - q^{-90} + q^{-92} + q^{-98} +2 q^{-100} + q^{-102} + q^{-108} } |
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^{-30} + q^{-34} + q^{-36} +3 q^{-38} + q^{-40} +3 q^{-42} +2 q^{-44} +3 q^{-46} +3 q^{-48} +2 q^{-50} +3 q^{-52} +2 q^{-54} +4 q^{-56} + q^{-58} +2 q^{-60} -2 q^{-62} - q^{-64} -5 q^{-66} -5 q^{-68} -7 q^{-70} -6 q^{-72} -5 q^{-74} -3 q^{-76} - q^{-78} +3 q^{-82} +2 q^{-84} +3 q^{-86} + q^{-88} +2 q^{-90} } |
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^{-50} + q^{-54} + q^{-58} +3 q^{-64} - q^{-66} +2 q^{-68} +2 q^{-74} + q^{-78} + q^{-80} + q^{-84} +2 q^{-86} - q^{-88} +3 q^{-90} -2 q^{-92} +2 q^{-94} +2 q^{-96} - q^{-98} +3 q^{-100} - q^{-102} +2 q^{-104} - q^{-114} - q^{-116} - q^{-120} -3 q^{-124} - q^{-126} -3 q^{-128} + q^{-130} -3 q^{-132} -2 q^{-134} -3 q^{-138} +2 q^{-140} -2 q^{-142} - q^{-144} - q^{-148} + q^{-152} - q^{-154} +2 q^{-156} + q^{-158} - q^{-160} +2 q^{-162} - q^{-164} + q^{-166} + q^{-168} - q^{-170} + q^{-172} } |
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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]:=
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AppendTo[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
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Loading KnotTheory` version of August 31, 2006, 11:25:27.5625.
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In[3]:=
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K = Knot["10 128"];
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In[4]:=
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Alexander[K][t]
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KnotTheory::loading: Loading precomputed data in PD4Knots`.
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Out[4]=
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Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2 t^3-3 t^2+t+1+ t^{-1} -3 t^{-2} +2 t^{-3} } |
In[5]:=
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Conway[K][z]
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Out[5]=
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Failed to parse (SVG (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 z^6+9 z^4+7 z^2+1} |
In[6]:=
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Alexander[K, 2][t]
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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.
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Out[6]=
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Failed to parse (SVG (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]:=
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{KnotDet[K], KnotSignature[K]}
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Out[7]=
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{ 11, 6 } |
In[8]:=
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Jones[K][q]
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KnotTheory::loading: Loading precomputed data in Jones4Knots`.
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Out[8]=
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Failed to parse (SVG (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^{10}+q^9-2 q^8+2 q^7-q^6+2 q^5-q^4+q^3} |
In[9]:=
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HOMFLYPT[K][a, z]
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KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
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Out[9]=
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Failed to parse (SVG (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^{-6} +z^6 a^{-8} +5 z^4 a^{-6} +5 z^4 a^{-8} -z^4 a^{-10} +6 z^2 a^{-6} +6 z^2 a^{-8} -5 z^2 a^{-10} +2 a^{-6} +2 a^{-8} -4 a^{-10} + a^{-12} } |
In[10]:=
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Kauffman[K][a, z]
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KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
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Out[10]=
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Failed to parse (SVG (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^8 a^{-8} +z^8 a^{-10} +z^7 a^{-7} +2 z^7 a^{-9} +z^7 a^{-11} +z^6 a^{-6} -5 z^6 a^{-8} -6 z^6 a^{-10} -4 z^5 a^{-7} -10 z^5 a^{-9} -6 z^5 a^{-11} -5 z^4 a^{-6} +7 z^4 a^{-8} +12 z^4 a^{-10} +2 z^3 a^{-7} +13 z^3 a^{-9} +11 z^3 a^{-11} +6 z^2 a^{-6} -5 z^2 a^{-8} -11 z^2 a^{-10} +z a^{-7} -5 z a^{-9} -6 z a^{-11} -2 a^{-6} +2 a^{-8} +4 a^{-10} + a^{-12} } |
"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`
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Loading KnotTheory` version of May 31, 2006, 14:15:20.091.
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In[3]:=
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K = Knot["10 128"];
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In[4]:=
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{A = Alexander[K][t], J = Jones[K][q]}
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KnotTheory::loading: Loading precomputed data in PD4Knots`.
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KnotTheory::loading: Loading precomputed data in Jones4Knots`.
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Out[4]=
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{ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2 t^3-3 t^2+t+1+ t^{-1} -3 t^{-2} +2 t^{-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^{10}+q^9-2 q^8+2 q^7-q^6+2 q^5-q^4+q^3} } |
In[5]:=
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DeleteCases[Select[AllKnots[], (A === Alexander[#][t]) &], K]
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KnotTheory::loading: Loading precomputed data in DTCode4KnotsTo11`.
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KnotTheory::credits: The GaussCode to PD conversion was written by Siddarth Sankaran at the University of Toronto in the summer of 2005.
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Out[5]=
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{} |
In[6]:=
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DeleteCases[
Select[
AllKnots[],
(J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) &
],
K
]
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KnotTheory::loading: Loading precomputed data in Jones4Knots11`.
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Out[6]=
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{} |
Vassiliev invariants
| V2 and V3: | (7, 17) |
| V2,1 through V6,9: |
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V2,1 through V6,9 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=} 6 is the signature of 10 128. Nonzero entries off the critical diagonals (if any exist) are highlighted in red. |
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| Integral Khovanov Homology
(db, data source) |
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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^{29}-q^{28}-q^{27}+2 q^{26}-2 q^{24}+2 q^{23}-2 q^{21}+q^{20}-q^{19}-q^{16}+2 q^{15}-q^{14}-2 q^{13}+4 q^{12}-q^{11}-2 q^{10}+3 q^9-q^7+q^6} |
| 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^{55}+2 q^{53}+2 q^{52}-4 q^{51}-3 q^{50}+3 q^{49}+7 q^{48}-4 q^{47}-9 q^{46}+3 q^{45}+12 q^{44}-3 q^{43}-12 q^{42}+2 q^{41}+14 q^{40}-3 q^{39}-13 q^{38}+3 q^{37}+12 q^{36}-3 q^{35}-12 q^{34}+3 q^{33}+9 q^{32}-q^{31}-9 q^{30}+2 q^{29}+5 q^{28}-6 q^{26}+2 q^{25}+2 q^{24}-q^{23}-3 q^{22}+4 q^{21}+q^{20}-3 q^{19}-2 q^{18}+4 q^{17}+2 q^{16}-2 q^{15}-2 q^{14}+2 q^{13}+q^{12}-q^{10}+q^9} |
| 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^{90}-q^{89}-q^{87}-q^{86}+3 q^{85}-q^{84}+4 q^{83}-2 q^{82}-6 q^{81}+q^{80}-3 q^{79}+12 q^{78}+3 q^{77}-8 q^{76}-6 q^{75}-11 q^{74}+19 q^{73}+10 q^{72}-6 q^{71}-9 q^{70}-19 q^{69}+20 q^{68}+13 q^{67}-3 q^{66}-8 q^{65}-23 q^{64}+20 q^{63}+14 q^{62}-2 q^{61}-7 q^{60}-22 q^{59}+16 q^{58}+15 q^{57}-q^{56}-6 q^{55}-21 q^{54}+9 q^{53}+15 q^{52}+3 q^{51}-3 q^{50}-20 q^{49}+q^{48}+13 q^{47}+8 q^{46}+2 q^{45}-17 q^{44}-7 q^{43}+8 q^{42}+8 q^{41}+7 q^{40}-10 q^{39}-10 q^{38}+4 q^{37}+4 q^{36}+6 q^{35}-4 q^{34}-7 q^{33}+5 q^{32}+2 q^{30}-3 q^{29}-5 q^{28}+7 q^{27}+q^{26}+q^{25}-3 q^{24}-5 q^{23}+5 q^{22}+q^{21}+2 q^{20}-q^{19}-3 q^{18}+2 q^{17}+q^{15}-q^{13}+q^{12}} |
| 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^{132}+q^{130}+2 q^{129}+q^{128}-4 q^{126}-5 q^{125}-q^{124}+4 q^{123}+7 q^{122}+7 q^{121}-10 q^{119}-13 q^{118}-6 q^{117}+8 q^{116}+18 q^{115}+16 q^{114}-2 q^{113}-22 q^{112}-25 q^{111}-7 q^{110}+22 q^{109}+33 q^{108}+15 q^{107}-19 q^{106}-38 q^{105}-22 q^{104}+17 q^{103}+41 q^{102}+24 q^{101}-15 q^{100}-40 q^{99}-27 q^{98}+14 q^{97}+41 q^{96}+25 q^{95}-14 q^{94}-40 q^{93}-25 q^{92}+14 q^{91}+40 q^{90}+24 q^{89}-14 q^{88}-40 q^{87}-22 q^{86}+14 q^{85}+36 q^{84}+23 q^{83}-11 q^{82}-35 q^{81}-19 q^{80}+9 q^{79}+27 q^{78}+21 q^{77}-4 q^{76}-24 q^{75}-17 q^{74}+14 q^{72}+16 q^{71}+5 q^{70}-8 q^{69}-10 q^{68}-8 q^{67}-q^{66}+6 q^{65}+6 q^{64}+8 q^{63}+2 q^{62}-6 q^{61}-10 q^{60}-8 q^{59}-q^{58}+11 q^{57}+13 q^{56}+5 q^{55}-7 q^{54}-14 q^{53}-11 q^{52}+4 q^{51}+12 q^{50}+11 q^{49}-8 q^{47}-11 q^{46}-q^{45}+5 q^{44}+6 q^{43}+q^{42}-3 q^{41}-5 q^{40}+q^{39}+4 q^{38}+3 q^{37}-2 q^{36}-4 q^{35}-4 q^{34}+2 q^{33}+4 q^{32}+4 q^{31}-3 q^{29}-4 q^{28}+q^{27}+q^{26}+2 q^{25}+2 q^{24}-q^{23}-2 q^{22}+q^{21}+q^{18}-q^{16}+q^{15}} |
| 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^{183}-q^{182}-q^{179}-q^{178}-q^{177}+3 q^{176}+q^{175}+4 q^{174}+2 q^{173}-q^{172}-7 q^{171}-8 q^{170}-4 q^{169}-3 q^{168}+13 q^{167}+15 q^{166}+15 q^{165}-2 q^{164}-13 q^{163}-26 q^{162}-30 q^{161}+q^{160}+21 q^{159}+46 q^{158}+31 q^{157}+13 q^{156}-38 q^{155}-68 q^{154}-40 q^{153}-5 q^{152}+60 q^{151}+69 q^{150}+64 q^{149}-22 q^{148}-85 q^{147}-79 q^{146}-44 q^{145}+50 q^{144}+84 q^{143}+103 q^{142}-q^{141}-81 q^{140}-94 q^{139}-66 q^{138}+37 q^{137}+83 q^{136}+118 q^{135}+6 q^{134}-75 q^{133}-95 q^{132}-72 q^{131}+33 q^{130}+81 q^{129}+123 q^{128}+4 q^{127}-74 q^{126}-95 q^{125}-74 q^{124}+33 q^{123}+82 q^{122}+122 q^{121}+3 q^{120}-69 q^{119}-94 q^{118}-74 q^{117}+30 q^{116}+78 q^{115}+117 q^{114}+6 q^{113}-58 q^{112}-88 q^{111}-74 q^{110}+20 q^{109}+64 q^{108}+109 q^{107}+16 q^{106}-34 q^{105}-75 q^{104}-76 q^{103}-q^{102}+41 q^{101}+94 q^{100}+28 q^{99}-q^{98}-50 q^{97}-71 q^{96}-25 q^{95}+11 q^{94}+67 q^{93}+29 q^{92}+30 q^{91}-17 q^{90}-50 q^{89}-33 q^{88}-15 q^{87}+30 q^{86}+11 q^{85}+38 q^{84}+8 q^{83}-18 q^{82}-16 q^{81}-16 q^{80}+4 q^{79}-14 q^{78}+20 q^{77}+6 q^{76}-2 q^{75}+7 q^{74}+4 q^{73}+7 q^{72}-19 q^{71}-12 q^{69}-12 q^{68}+11 q^{67}+15 q^{66}+18 q^{65}-3 q^{64}+q^{63}-16 q^{62}-22 q^{61}+2 q^{60}+6 q^{59}+14 q^{58}+5 q^{57}+8 q^{56}-7 q^{55}-16 q^{54}+2 q^{53}-q^{52}+6 q^{51}+q^{50}+5 q^{49}-5 q^{48}-10 q^{47}+6 q^{46}+6 q^{44}+q^{43}+3 q^{42}-6 q^{41}-9 q^{40}+4 q^{39}-q^{38}+4 q^{37}+3 q^{36}+4 q^{35}-3 q^{34}-5 q^{33}+2 q^{32}-2 q^{31}+q^{30}+q^{29}+3 q^{28}-q^{27}-2 q^{26}+2 q^{25}-q^{24}+q^{21}-q^{19}+q^{18}} |
Computer Talk
Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session, or any of the Computer Talk sections above.
Modifying This Page
| Read me first: Modifying Knot Pages
See/edit the Rolfsen Knot Page master template (intermediate). See/edit the Rolfsen_Splice_Base (expert). Back to the top. |
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