10 3
From Knot Atlas
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 (KnotPlot image) |
See the full Rolfsen Knot Table. Visit 10 3's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) |
Knot presentations
| Planar diagram presentation | X1627 X11,16,12,17 X5,13,6,12 X3,15,4,14 X13,5,14,4 X15,3,16,2 X7,20,8,1 X9,18,10,19 X17,10,18,11 X19,8,20,9 |
| Gauss code | -1, 6, -4, 5, -3, 1, -7, 10, -8, 9, -2, 3, -5, 4, -6, 2, -9, 8, -10, 7 |
| Dowker-Thistlethwaite code | 6 14 12 20 18 16 4 2 10 8 |
| Conway Notation | [64] |
| Minimum Braid Representative | A Morse Link Presentation | An Arc Presentation | ||||||
Length is 13, width is 6, Braid index is 6 |
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 [{12, 7}, {6, 8}, {7, 5}, {4, 6}, {5, 3}, {2, 4}, {3, 1}, {9, 2}, {8, 10}, {11, 9}, {10, 12}, {1, 11}] |
[edit Notes on presentations of 10 3]
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[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
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Loading KnotTheory` version of May 31, 2006, 14:15:20.091.
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In[3]:=
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K = Knot["10 3"];
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In[4]:=
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PD[K]
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KnotTheory::loading: Loading precomputed data in PD4Knots`.
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Out[4]=
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X1627 X11,16,12,17 X5,13,6,12 X3,15,4,14 X13,5,14,4 X15,3,16,2 X7,20,8,1 X9,18,10,19 X17,10,18,11 X19,8,20,9 |
In[5]:=
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GaussCode[K]
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Out[5]=
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-1, 6, -4, 5, -3, 1, -7, 10, -8, 9, -2, 3, -5, 4, -6, 2, -9, 8, -10, 7 |
In[6]:=
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DTCode[K]
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Out[6]=
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6 14 12 20 18 16 4 2 10 8 |
(The path below may be different on your system)
In[7]:=
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AppendTo[$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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[64] |
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}(6,\{-1,-1,-2,1,-2,-3,2,4,-3,4,5,-4,5\})} |
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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{ 6, 13, 6 } |
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[{12, 7}, {6, 8}, {7, 5}, {4, 6}, {5, 3}, {2, 4}, {3, 1}, {9, 2}, {8, 10}, {11, 9}, {10, 12}, {1, 11}] |
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 -6 t+13-6 t^{-1} } |
| 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 1-6 z^2} |
| 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 | { 25, 0 } |
| Jones polynomial | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^4-q^3+2 q^2-3 q+4-4 q^{-1} +3 q^{-2} -3 q^{-3} +2 q^{-4} - q^{-5} + q^{-6} } |
| HOMFLY-PT polynomial (db, data sources) | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a^6-z^2 a^4-2 z^2 a^2-a^2-2 z^2-z^2 a^{-2} + a^{-4} } |
| Kauffman polynomial (db, data sources) | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a^3 z^9+a z^9+a^4 z^8+2 a^2 z^8+z^8+a^5 z^7-6 a^3 z^7-6 a z^7+z^7 a^{-1} +a^6 z^6-4 a^4 z^6-10 a^2 z^6+z^6 a^{-2} -4 z^6-4 a^5 z^5+15 a^3 z^5+15 a z^5-3 z^5 a^{-1} +z^5 a^{-3} -5 a^6 z^4+4 a^4 z^4+18 a^2 z^4-2 z^4 a^{-2} +z^4 a^{-4} +6 z^4+3 a^5 z^3-18 a^3 z^3-15 a z^3+4 z^3 a^{-1} -2 z^3 a^{-3} +6 a^6 z^2-2 a^4 z^2-12 a^2 z^2+z^2 a^{-2} -3 z^2 a^{-4} +6 a^3 z+6 a z-a^6+a^2+ a^{-4} } |
| The A2 invariant | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{20}+q^{18}+q^{14}-q^{10}-q^6-q^4+ q^{-2} - q^{-4} + q^{-8} + q^{-12} + q^{-14} } |
| The G2 invariant | Data:10 3/QuantumInvariant/G2/1,0 |
Further Quantum Invariants
Further quantum knot invariants for 10_3.
A1 Invariants.
| Weight | Invariant |
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| 1 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{13}+q^9-q^7-q^3+ q^{-1} - q^{-3} + q^{-5} + q^{-9} } |
| 2 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{38}+q^{32}-q^{30}-2 q^{28}+q^{26}-2 q^{22}+q^{20}+q^{18}-q^{16}+2 q^{12}+q^6+2 q^{-2} - q^{-4} - q^{-6} +2 q^{-8} - q^{-10} - q^{-12} - q^{-16} + q^{-20} + q^{-26} } |
| 3 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{75}-q^{65}-q^{63}+q^{59}-q^{57}-2 q^{55}+3 q^{51}+2 q^{49}-2 q^{47}-2 q^{45}+q^{43}+3 q^{41}-q^{37}-q^{35}+q^{33}+2 q^{31}+q^{29}-3 q^{27}-q^{25}+2 q^{23}+2 q^{21}-2 q^{19}-2 q^{17}+q^{15}-q^{11}-q^9+q^7+q^3-q- q^{-1} +2 q^{-3} +2 q^{-5} -2 q^{-9} +3 q^{-13} + q^{-15} -2 q^{-19} +2 q^{-23} +2 q^{-25} - q^{-27} -2 q^{-29} + q^{-33} -2 q^{-37} - q^{-39} + q^{-43} + q^{-51} } |
A2 Invariants.
| Weight | Invariant |
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| 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^{18}+q^{14}-q^{10}-q^6-q^4+ q^{-2} - q^{-4} + q^{-8} + q^{-12} + q^{-14} } |
| 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^{52}+q^{50}+q^{48}+q^{44}+q^{42}-q^{40}-3 q^{38}-2 q^{36}-2 q^{30}-q^{28}+q^{26}+q^{24}-q^{20}+2 q^{18}+2 q^{16}+2 q^{10}+q^8+q^6+q^4+1+ q^{-2} + q^{-4} -2 q^{-6} -2 q^{-8} +2 q^{-10} -3 q^{-14} -2 q^{-16} - q^{-22} + q^{-26} + q^{-28} + q^{-32} + q^{-34} + q^{-36} } |
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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 3"];
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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 -6 t+13-6 t^{-1} } |
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 1-6 z^2} |
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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{ 25, 0 } |
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^4-q^3+2 q^2-3 q+4-4 q^{-1} +3 q^{-2} -3 q^{-3} +2 q^{-4} - q^{-5} + q^{-6} } |
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 a^6-z^2 a^4-2 z^2 a^2-a^2-2 z^2-z^2 a^{-2} + a^{-4} } |
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 a^3 z^9+a z^9+a^4 z^8+2 a^2 z^8+z^8+a^5 z^7-6 a^3 z^7-6 a z^7+z^7 a^{-1} +a^6 z^6-4 a^4 z^6-10 a^2 z^6+z^6 a^{-2} -4 z^6-4 a^5 z^5+15 a^3 z^5+15 a z^5-3 z^5 a^{-1} +z^5 a^{-3} -5 a^6 z^4+4 a^4 z^4+18 a^2 z^4-2 z^4 a^{-2} +z^4 a^{-4} +6 z^4+3 a^5 z^3-18 a^3 z^3-15 a z^3+4 z^3 a^{-1} -2 z^3 a^{-3} +6 a^6 z^2-2 a^4 z^2-12 a^2 z^2+z^2 a^{-2} -3 z^2 a^{-4} +6 a^3 z+6 a z-a^6+a^2+ a^{-4} } |
"Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {}
Same Jones Polynomial (up to mirroring, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q\leftrightarrow q^{-1}} ): {}
Computer Talk
The above data is available with the Mathematica package
KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.
(The path below may be different on your system, and possibly also the KnotTheory` date)
In[1]:=
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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 3"];
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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 -6 t+13-6 t^{-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^4-q^3+2 q^2-3 q+4-4 q^{-1} +3 q^{-2} -3 q^{-3} +2 q^{-4} - q^{-5} + q^{-6} } } |
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: | (-6, 3) |
| 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=} 0 is the signature of 10 3. 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^{12}-q^{11}+2 q^9-2 q^8-q^7+3 q^6-3 q^5-q^4+6 q^3-6 q^2-q+9-8 q^{-1} - q^{-2} +9 q^{-3} -7 q^{-4} -2 q^{-5} +9 q^{-6} -5 q^{-7} -4 q^{-8} +8 q^{-9} -3 q^{-10} -4 q^{-11} +5 q^{-12} - q^{-13} -3 q^{-14} +2 q^{-15} - q^{-17} + q^{-18} } |
| 3 | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{24}-q^{23}+2 q^{20}-2 q^{19}-q^{18}-q^{17}+4 q^{16}-q^{15}-2 q^{14}-3 q^{13}+5 q^{12}+2 q^{11}-2 q^{10}-5 q^9+3 q^8+4 q^7-q^6-3 q^5+2 q^3+q^2-q- q^{-1} + q^{-2} + q^{-3} - q^{-4} - q^{-6} + q^{-7} + q^{-9} -4 q^{-10} + q^{-11} +4 q^{-12} + q^{-13} -7 q^{-14} - q^{-15} +8 q^{-16} +2 q^{-17} -8 q^{-18} -3 q^{-19} +8 q^{-20} +3 q^{-21} -5 q^{-22} -5 q^{-23} +5 q^{-24} +3 q^{-25} - q^{-26} -4 q^{-27} +2 q^{-28} + q^{-29} -2 q^{-31} + q^{-32} - q^{-35} + q^{-36} } |
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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