10 101
From Knot Atlas
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 (KnotPlot image) |
See the full Rolfsen Knot Table. Visit 10 101's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) |
Knot presentations
| Planar diagram presentation | X4251 X10,4,11,3 X14,6,15,5 X20,16,1,15 X16,12,17,11 X12,20,13,19 X18,8,19,7 X6,14,7,13 X8,18,9,17 X2,10,3,9 |
| Gauss code | 1, -10, 2, -1, 3, -8, 7, -9, 10, -2, 5, -6, 8, -3, 4, -5, 9, -7, 6, -4 |
| Dowker-Thistlethwaite code | 4 10 14 18 2 16 6 20 8 12 |
| Conway Notation | [21:2:2] |
| Minimum Braid Representative | A Morse Link Presentation | An Arc Presentation | |||||
Length is 14, width is 5, Braid index is 5 |
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 [{3, 9}, {2, 5}, {1, 3}, {10, 7}, {8, 6}, {7, 4}, {9, 11}, {5, 10}, {12, 8}, {11, 2}, {4, 12}, {6, 1}] |
[edit Notes on presentations of 10 101]
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[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
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Loading KnotTheory` version of May 31, 2006, 14:15:20.091.
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In[3]:=
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K = Knot["10 101"];
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In[4]:=
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PD[K]
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KnotTheory::loading: Loading precomputed data in PD4Knots`.
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Out[4]=
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X4251 X10,4,11,3 X14,6,15,5 X20,16,1,15 X16,12,17,11 X12,20,13,19 X18,8,19,7 X6,14,7,13 X8,18,9,17 X2,10,3,9 |
In[5]:=
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GaussCode[K]
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Out[5]=
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1, -10, 2, -1, 3, -8, 7, -9, 10, -2, 5, -6, 8, -3, 4, -5, 9, -7, 6, -4 |
In[6]:=
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DTCode[K]
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Out[6]=
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4 10 14 18 2 16 6 20 8 12 |
(The path below may be different on your system)
In[7]:=
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AppendTo[$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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[21:2:2] |
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}(5,\{1,1,1,2,-1,3,-2,1,3,2,2,4,-3,4\})} |
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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{ 5, 14, 5 } |
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[{3, 9}, {2, 5}, {1, 3}, {10, 7}, {8, 6}, {7, 4}, {9, 11}, {5, 10}, {12, 8}, {11, 2}, {4, 12}, {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 7 t^2-21 t+29-21 t^{-1} +7 t^{-2} } |
| Conway polynomial | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 7 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 | { 85, 4 } |
| 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^{12}-4 q^{11}+7 q^{10}-11 q^9+13 q^8-14 q^7+14 q^6-10 q^5+7 q^4-3 q^3+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^4 a^{-4} +3 z^4 a^{-6} +3 z^4 a^{-8} +z^2 a^{-4} +5 z^2 a^{-6} +5 z^2 a^{-8} -4 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 2 z^9 a^{-9} +2 z^9 a^{-11} +6 z^8 a^{-8} +11 z^8 a^{-10} +5 z^8 a^{-12} +7 z^7 a^{-7} +10 z^7 a^{-9} +7 z^7 a^{-11} +4 z^7 a^{-13} +6 z^6 a^{-6} -6 z^6 a^{-8} -24 z^6 a^{-10} -11 z^6 a^{-12} +z^6 a^{-14} +3 z^5 a^{-5} -8 z^5 a^{-7} -31 z^5 a^{-9} -31 z^5 a^{-11} -11 z^5 a^{-13} +z^4 a^{-4} -8 z^4 a^{-6} +z^4 a^{-8} +15 z^4 a^{-10} +3 z^4 a^{-12} -2 z^4 a^{-14} -2 z^3 a^{-5} +4 z^3 a^{-7} +26 z^3 a^{-9} +28 z^3 a^{-11} +8 z^3 a^{-13} -z^2 a^{-4} +7 z^2 a^{-6} -z^2 a^{-8} -9 z^2 a^{-10} +z^2 a^{-12} +z^2 a^{-14} -8 z a^{-9} -9 z a^{-11} -z a^{-13} -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^{-6} -2 q^{-8} +2 q^{-10} + q^{-12} -2 q^{-14} +4 q^{-16} +2 q^{-20} +2 q^{-22} - q^{-24} +2 q^{-26} -4 q^{-28} - q^{-30} -3 q^{-34} + q^{-36} + 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^{-30} -2 q^{-32} +4 q^{-34} -6 q^{-36} +6 q^{-38} -5 q^{-40} +11 q^{-44} -21 q^{-46} +33 q^{-48} -38 q^{-50} +31 q^{-52} -14 q^{-54} -15 q^{-56} +52 q^{-58} -84 q^{-60} +107 q^{-62} -105 q^{-64} +68 q^{-66} +3 q^{-68} -87 q^{-70} +169 q^{-72} -206 q^{-74} +183 q^{-76} -94 q^{-78} -40 q^{-80} +166 q^{-82} -226 q^{-84} +203 q^{-86} -85 q^{-88} -58 q^{-90} +169 q^{-92} -188 q^{-94} +107 q^{-96} +45 q^{-98} -192 q^{-100} +265 q^{-102} -220 q^{-104} +73 q^{-106} +125 q^{-108} -289 q^{-110} +359 q^{-112} -307 q^{-114} +147 q^{-116} +50 q^{-118} -229 q^{-120} +322 q^{-122} -304 q^{-124} +183 q^{-126} -14 q^{-128} -146 q^{-130} +224 q^{-132} -204 q^{-134} +85 q^{-136} +65 q^{-138} -188 q^{-140} +214 q^{-142} -138 q^{-144} -16 q^{-146} +177 q^{-148} -270 q^{-150} +259 q^{-152} -149 q^{-154} -14 q^{-156} +158 q^{-158} -232 q^{-160} +224 q^{-162} -139 q^{-164} +32 q^{-166} +59 q^{-168} -106 q^{-170} +105 q^{-172} -71 q^{-174} +33 q^{-176} + q^{-178} -19 q^{-180} +20 q^{-182} -16 q^{-184} +8 q^{-186} -3 q^{-188} + q^{-190} } |
Further Quantum Invariants
Further quantum knot invariants for 10_101.
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^{-3} -2 q^{-5} +4 q^{-7} -3 q^{-9} +4 q^{-11} - q^{-15} +2 q^{-17} -4 q^{-19} +3 q^{-21} -3 q^{-23} + q^{-25} } |
| 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^{-6} -2 q^{-8} + q^{-10} +6 q^{-12} -9 q^{-14} +3 q^{-16} +17 q^{-18} -24 q^{-20} +35 q^{-24} -26 q^{-26} -15 q^{-28} +34 q^{-30} -7 q^{-32} -22 q^{-34} +11 q^{-36} +16 q^{-38} -16 q^{-40} -15 q^{-42} +27 q^{-44} -2 q^{-46} -33 q^{-48} +25 q^{-50} +16 q^{-52} -35 q^{-54} +9 q^{-56} +23 q^{-58} -18 q^{-60} -5 q^{-62} +12 q^{-64} -2 q^{-66} -3 q^{-68} + q^{-70} } |
| 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^{-9} -2 q^{-11} + q^{-13} +3 q^{-15} -5 q^{-19} +3 q^{-21} +11 q^{-23} -11 q^{-25} -19 q^{-27} +30 q^{-29} +42 q^{-31} -44 q^{-33} -90 q^{-35} +59 q^{-37} +150 q^{-39} -41 q^{-41} -214 q^{-43} -7 q^{-45} +255 q^{-47} +78 q^{-49} -255 q^{-51} -147 q^{-53} +201 q^{-55} +200 q^{-57} -121 q^{-59} -219 q^{-61} +32 q^{-63} +201 q^{-65} +58 q^{-67} -176 q^{-69} -125 q^{-71} +130 q^{-73} +181 q^{-75} -95 q^{-77} -218 q^{-79} +44 q^{-81} +250 q^{-83} +12 q^{-85} -260 q^{-87} -79 q^{-89} +248 q^{-91} +146 q^{-93} -198 q^{-95} -201 q^{-97} +124 q^{-99} +226 q^{-101} -41 q^{-103} -202 q^{-105} -36 q^{-107} +151 q^{-109} +78 q^{-111} -86 q^{-113} -82 q^{-115} +29 q^{-117} +59 q^{-119} +4 q^{-121} -34 q^{-123} -9 q^{-125} +11 q^{-127} +7 q^{-129} -2 q^{-131} -3 q^{-133} + q^{-135} } |
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} -2 q^{-8} +2 q^{-10} + q^{-12} -2 q^{-14} +4 q^{-16} +2 q^{-20} +2 q^{-22} - q^{-24} +2 q^{-26} -4 q^{-28} - q^{-30} -3 q^{-34} + q^{-36} + q^{-38} } |
| 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^{-12} -2 q^{-14} - q^{-16} +7 q^{-18} - q^{-20} -11 q^{-22} +6 q^{-24} +16 q^{-26} -5 q^{-28} -18 q^{-30} +9 q^{-32} +23 q^{-34} -10 q^{-36} -16 q^{-38} +14 q^{-40} +13 q^{-42} -5 q^{-44} -5 q^{-46} +8 q^{-48} -2 q^{-50} -3 q^{-52} +4 q^{-54} -6 q^{-56} -14 q^{-58} +2 q^{-60} +11 q^{-62} -15 q^{-64} -15 q^{-66} +9 q^{-68} +15 q^{-70} -9 q^{-72} -14 q^{-74} +13 q^{-76} +14 q^{-78} -3 q^{-80} -8 q^{-82} +3 q^{-84} +8 q^{-86} -2 q^{-88} -5 q^{-90} -2 q^{-92} + q^{-94} + 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^{-12} -2 q^{-14} +6 q^{-18} -7 q^{-20} -3 q^{-22} +18 q^{-24} -10 q^{-26} -11 q^{-28} +28 q^{-30} -8 q^{-32} -16 q^{-34} +31 q^{-36} -3 q^{-38} -11 q^{-40} +14 q^{-42} +2 q^{-44} -9 q^{-46} -13 q^{-48} +4 q^{-50} + q^{-52} -25 q^{-54} +7 q^{-56} +19 q^{-58} -23 q^{-60} +8 q^{-62} +20 q^{-64} -20 q^{-66} +6 q^{-68} +10 q^{-70} -12 q^{-72} +4 q^{-74} +2 q^{-76} -3 q^{-78} + q^{-80} } |
| 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^{-9} -2 q^{-11} +2 q^{-13} - q^{-15} +2 q^{-17} -2 q^{-19} +4 q^{-21} +2 q^{-25} +2 q^{-27} +2 q^{-29} +2 q^{-31} - q^{-33} +2 q^{-35} -4 q^{-37} - q^{-39} -4 q^{-41} -3 q^{-45} + q^{-47} + q^{-49} + q^{-51} } |
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^{-12} -2 q^{-14} +4 q^{-16} -8 q^{-18} +13 q^{-20} -17 q^{-22} +24 q^{-24} -28 q^{-26} +33 q^{-28} -32 q^{-30} +28 q^{-32} -16 q^{-34} +5 q^{-36} +13 q^{-38} -27 q^{-40} +44 q^{-42} -56 q^{-44} +63 q^{-46} -65 q^{-48} +58 q^{-50} -49 q^{-52} +33 q^{-54} -17 q^{-56} - q^{-58} +15 q^{-60} -26 q^{-62} +32 q^{-64} -34 q^{-66} +32 q^{-68} -28 q^{-70} +20 q^{-72} -14 q^{-74} +8 q^{-76} -3 q^{-78} + q^{-80} } |
| 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^{-18} -2 q^{-22} -2 q^{-24} +2 q^{-26} +7 q^{-28} +2 q^{-30} -10 q^{-32} -10 q^{-34} +6 q^{-36} +21 q^{-38} +7 q^{-40} -20 q^{-42} -22 q^{-44} +11 q^{-46} +34 q^{-48} +7 q^{-50} -31 q^{-52} -18 q^{-54} +23 q^{-56} +31 q^{-58} -9 q^{-60} -28 q^{-62} + q^{-64} +29 q^{-66} +7 q^{-68} -24 q^{-70} -13 q^{-72} +16 q^{-74} +13 q^{-76} -17 q^{-78} -20 q^{-80} +9 q^{-82} +20 q^{-84} -9 q^{-86} -30 q^{-88} -4 q^{-90} +32 q^{-92} +18 q^{-94} -26 q^{-96} -31 q^{-98} +14 q^{-100} +37 q^{-102} +6 q^{-104} -29 q^{-106} -18 q^{-108} +18 q^{-110} +22 q^{-112} -5 q^{-114} -16 q^{-116} -4 q^{-118} +9 q^{-120} +5 q^{-122} -3 q^{-124} -3 q^{-126} + q^{-130} } |
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^{-30} -2 q^{-32} +4 q^{-34} -6 q^{-36} +6 q^{-38} -5 q^{-40} +11 q^{-44} -21 q^{-46} +33 q^{-48} -38 q^{-50} +31 q^{-52} -14 q^{-54} -15 q^{-56} +52 q^{-58} -84 q^{-60} +107 q^{-62} -105 q^{-64} +68 q^{-66} +3 q^{-68} -87 q^{-70} +169 q^{-72} -206 q^{-74} +183 q^{-76} -94 q^{-78} -40 q^{-80} +166 q^{-82} -226 q^{-84} +203 q^{-86} -85 q^{-88} -58 q^{-90} +169 q^{-92} -188 q^{-94} +107 q^{-96} +45 q^{-98} -192 q^{-100} +265 q^{-102} -220 q^{-104} +73 q^{-106} +125 q^{-108} -289 q^{-110} +359 q^{-112} -307 q^{-114} +147 q^{-116} +50 q^{-118} -229 q^{-120} +322 q^{-122} -304 q^{-124} +183 q^{-126} -14 q^{-128} -146 q^{-130} +224 q^{-132} -204 q^{-134} +85 q^{-136} +65 q^{-138} -188 q^{-140} +214 q^{-142} -138 q^{-144} -16 q^{-146} +177 q^{-148} -270 q^{-150} +259 q^{-152} -149 q^{-154} -14 q^{-156} +158 q^{-158} -232 q^{-160} +224 q^{-162} -139 q^{-164} +32 q^{-166} +59 q^{-168} -106 q^{-170} +105 q^{-172} -71 q^{-174} +33 q^{-176} + q^{-178} -19 q^{-180} +20 q^{-182} -16 q^{-184} +8 q^{-186} -3 q^{-188} + q^{-190} } |
.
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["10 101"];
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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 7 t^2-21 t+29-21 t^{-1} +7 t^{-2} } |
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 7 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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{ 85, 4 } |
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^{12}-4 q^{11}+7 q^{10}-11 q^9+13 q^8-14 q^7+14 q^6-10 q^5+7 q^4-3 q^3+q^2} |
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^4 a^{-4} +3 z^4 a^{-6} +3 z^4 a^{-8} +z^2 a^{-4} +5 z^2 a^{-6} +5 z^2 a^{-8} -4 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 2 z^9 a^{-9} +2 z^9 a^{-11} +6 z^8 a^{-8} +11 z^8 a^{-10} +5 z^8 a^{-12} +7 z^7 a^{-7} +10 z^7 a^{-9} +7 z^7 a^{-11} +4 z^7 a^{-13} +6 z^6 a^{-6} -6 z^6 a^{-8} -24 z^6 a^{-10} -11 z^6 a^{-12} +z^6 a^{-14} +3 z^5 a^{-5} -8 z^5 a^{-7} -31 z^5 a^{-9} -31 z^5 a^{-11} -11 z^5 a^{-13} +z^4 a^{-4} -8 z^4 a^{-6} +z^4 a^{-8} +15 z^4 a^{-10} +3 z^4 a^{-12} -2 z^4 a^{-14} -2 z^3 a^{-5} +4 z^3 a^{-7} +26 z^3 a^{-9} +28 z^3 a^{-11} +8 z^3 a^{-13} -z^2 a^{-4} +7 z^2 a^{-6} -z^2 a^{-8} -9 z^2 a^{-10} +z^2 a^{-12} +z^2 a^{-14} -8 z a^{-9} -9 z a^{-11} -z a^{-13} -2 a^{-6} +2 a^{-8} +4 a^{-10} + a^{-12} } |
"Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {K11a200,}
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]:=
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K = Knot["10 101"];
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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 7 t^2-21 t+29-21 t^{-1} +7 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^{12}-4 q^{11}+7 q^{10}-11 q^9+13 q^8-14 q^7+14 q^6-10 q^5+7 q^4-3 q^3+q^2} } |
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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{K11a200,} |
In[6]:=
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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: | (7, 17) |
| V2,1 through V6,9: |
|
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=} 4 is the signature of 10 101. Nonzero entries off the critical diagonals (if any exist) are highlighted in red. |
|
| Integral Khovanov Homology
(db, data source) |
|
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^{34}-4 q^{33}+q^{32}+15 q^{31}-21 q^{30}-12 q^{29}+56 q^{28}-35 q^{27}-56 q^{26}+107 q^{25}-26 q^{24}-114 q^{23}+138 q^{22}+3 q^{21}-156 q^{20}+137 q^{19}+35 q^{18}-161 q^{17}+104 q^{16}+50 q^{15}-120 q^{14}+55 q^{13}+39 q^{12}-59 q^{11}+20 q^{10}+15 q^9-18 q^8+6 q^7+3 q^6-3 q^5+q^4} |
| 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^{66}-4 q^{65}+q^{64}+9 q^{63}+5 q^{62}-24 q^{61}-24 q^{60}+47 q^{59}+60 q^{58}-54 q^{57}-135 q^{56}+43 q^{55}+224 q^{54}+19 q^{53}-322 q^{52}-123 q^{51}+385 q^{50}+286 q^{49}-424 q^{48}-448 q^{47}+388 q^{46}+630 q^{45}-322 q^{44}-775 q^{43}+207 q^{42}+902 q^{41}-84 q^{40}-981 q^{39}-55 q^{38}+1025 q^{37}+192 q^{36}-1032 q^{35}-310 q^{34}+974 q^{33}+426 q^{32}-889 q^{31}-479 q^{30}+723 q^{29}+524 q^{28}-568 q^{27}-478 q^{26}+375 q^{25}+416 q^{24}-235 q^{23}-301 q^{22}+113 q^{21}+209 q^{20}-62 q^{19}-110 q^{18}+22 q^{17}+60 q^{16}-16 q^{15}-24 q^{14}+10 q^{13}+11 q^{12}-8 q^{11}-2 q^{10}+2 q^9+3 q^8-3 q^7+q^6} |
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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