10 1
From Knot Atlas
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 (KnotPlot image) |
See the full Rolfsen Knot Table. Visit 10 1's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) |
Knot presentations
| Planar diagram presentation | X1425 X11,14,12,15 X3,13,4,12 X13,3,14,2 X5,20,6,1 X7,18,8,19 X9,16,10,17 X15,10,16,11 X17,8,18,9 X19,6,20,7 |
| Gauss code | -1, 4, -3, 1, -5, 10, -6, 9, -7, 8, -2, 3, -4, 2, -8, 7, -9, 6, -10, 5 |
| Dowker-Thistlethwaite code | 4 12 20 18 16 14 2 10 8 6 |
| Conway Notation | [82] |
| 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, 9}, {8, 10}, {9, 7}, {6, 8}, {7, 5}, {4, 6}, {5, 3}, {2, 4}, {3, 1}, {11, 2}, {10, 12}, {1, 11}] |
[edit Notes on presentations of 10 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[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
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Loading KnotTheory` version of May 31, 2006, 14:15:20.091.
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In[3]:=
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K = Knot["10 1"];
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In[4]:=
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PD[K]
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KnotTheory::loading: Loading precomputed data in PD4Knots`.
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Out[4]=
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X1425 X11,14,12,15 X3,13,4,12 X13,3,14,2 X5,20,6,1 X7,18,8,19 X9,16,10,17 X15,10,16,11 X17,8,18,9 X19,6,20,7 |
In[5]:=
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GaussCode[K]
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Out[5]=
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-1, 4, -3, 1, -5, 10, -6, 9, -7, 8, -2, 3, -4, 2, -8, 7, -9, 6, -10, 5 |
In[6]:=
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DTCode[K]
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Out[6]=
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4 12 20 18 16 14 2 10 8 6 |
(The path below may be different on your system)
In[7]:=
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AppendTo[$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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[82] |
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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[math]\displaystyle{ \textrm{BR}(6,\{-1,-1,-2,1,-2,-3,2,-3,-4,3,5,-4,5\}) }[/math] |
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, 9}, {8, 10}, {9, 7}, {6, 8}, {7, 5}, {4, 6}, {5, 3}, {2, 4}, {3, 1}, {11, 2}, {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 | [math]\displaystyle{ -4 t+9-4 t^{-1} }[/math] |
| Conway polynomial | [math]\displaystyle{ 1-4 z^2 }[/math] |
| 2nd Alexander ideal (db, data sources) | [math]\displaystyle{ \{1\} }[/math] |
| Determinant and Signature | { 17, 0 } |
| Jones polynomial | [math]\displaystyle{ q^2-q+2-2 q^{-1} +2 q^{-2} -2 q^{-3} +2 q^{-4} -2 q^{-5} + q^{-6} - q^{-7} + q^{-8} }[/math] |
| HOMFLY-PT polynomial (db, data sources) | [math]\displaystyle{ a^8-z^2 a^6-a^6-z^2 a^4-z^2 a^2-z^2+ a^{-2} }[/math] |
| Kauffman polynomial (db, data sources) | [math]\displaystyle{ a^7 z^9+a^5 z^9+a^8 z^8+2 a^6 z^8+a^4 z^8-7 a^7 z^7-6 a^5 z^7+a^3 z^7-7 a^8 z^6-12 a^6 z^6-4 a^4 z^6+a^2 z^6+16 a^7 z^5+12 a^5 z^5-3 a^3 z^5+a z^5+15 a^8 z^4+21 a^6 z^4+3 a^4 z^4-2 a^2 z^4+z^4-14 a^7 z^3-11 a^5 z^3+a^3 z^3-a z^3+z^3 a^{-1} -10 a^8 z^2-11 a^6 z^2+z^2 a^{-2} +4 a^7 z+4 a^5 z+a^8+a^6- a^{-2} }[/math] |
| The A2 invariant | [math]\displaystyle{ q^{26}+q^{24}-q^{18}-q^{16}+ q^{-2} + q^{-6} + q^{-8} }[/math] |
| The G2 invariant | Data:10 1/QuantumInvariant/G2/1,0 |
Further Quantum Invariants
Further quantum knot invariants for 10_1.
A1 Invariants.
| Weight | Invariant |
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| 1 | [math]\displaystyle{ q^{17}-q^{11}+ q^{-1} + q^{-5} }[/math] |
| 2 | [math]\displaystyle{ q^{50}-q^{46}-q^{40}+q^{36}+q^{16}+q^{14}-q^4-q^2+ q^{-2} + q^{-8} + q^{-14} }[/math] |
| 3 | [math]\displaystyle{ q^{99}-q^{95}-q^{93}+q^{89}-q^{85}+q^{81}+q^{79}-q^{75}+q^{49}+q^{47}-q^{43}-q^{37}-q^{35}-q^{29}+q^{25}+q^{23}+q^{15}+q^{13}+q^{11}-q^9+q^5+q^3-q- q^{-1} + q^{-3} + q^{-5} - q^{-7} - q^{-9} + q^{-19} + q^{-27} }[/math] |
A2 Invariants.
| Weight | Invariant |
|---|---|
| 1,0 | [math]\displaystyle{ q^{26}+q^{24}-q^{18}-q^{16}+ q^{-2} + q^{-6} + q^{-8} }[/math] |
| 2,0 | [math]\displaystyle{ q^{68}+q^{66}+q^{64}-q^{62}-q^{60}-q^{58}-q^{56}-q^{54}-q^{52}+q^{50}+q^{48}+q^{46}+q^{24}+2 q^{22}+q^{20}+q^{18}+q^{16}-q^{10}-2 q^8-2 q^6-q^4+ q^{-4} + q^{-10} + q^{-12} + q^{-16} + q^{-18} + q^{-20} }[/math] |
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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 1"];
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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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[math]\displaystyle{ -4 t+9-4 t^{-1} }[/math] |
In[5]:=
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Conway[K][z]
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Out[5]=
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[math]\displaystyle{ 1-4 z^2 }[/math] |
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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[math]\displaystyle{ \{1\} }[/math] |
In[7]:=
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{KnotDet[K], KnotSignature[K]}
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Out[7]=
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{ 17, 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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[math]\displaystyle{ q^2-q+2-2 q^{-1} +2 q^{-2} -2 q^{-3} +2 q^{-4} -2 q^{-5} + q^{-6} - q^{-7} + q^{-8} }[/math] |
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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[math]\displaystyle{ a^8-z^2 a^6-a^6-z^2 a^4-z^2 a^2-z^2+ a^{-2} }[/math] |
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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[math]\displaystyle{ a^7 z^9+a^5 z^9+a^8 z^8+2 a^6 z^8+a^4 z^8-7 a^7 z^7-6 a^5 z^7+a^3 z^7-7 a^8 z^6-12 a^6 z^6-4 a^4 z^6+a^2 z^6+16 a^7 z^5+12 a^5 z^5-3 a^3 z^5+a z^5+15 a^8 z^4+21 a^6 z^4+3 a^4 z^4-2 a^2 z^4+z^4-14 a^7 z^3-11 a^5 z^3+a^3 z^3-a z^3+z^3 a^{-1} -10 a^8 z^2-11 a^6 z^2+z^2 a^{-2} +4 a^7 z+4 a^5 z+a^8+a^6- a^{-2} }[/math] |
"Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {8_3,}
Same Jones Polynomial (up to mirroring, [math]\displaystyle{ q\leftrightarrow q^{-1} }[/math]): {}
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 1"];
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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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{ [math]\displaystyle{ -4 t+9-4 t^{-1} }[/math], [math]\displaystyle{ q^2-q+2-2 q^{-1} +2 q^{-2} -2 q^{-3} +2 q^{-4} -2 q^{-5} + q^{-6} - q^{-7} + q^{-8} }[/math] } |
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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{8_3,} |
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: | (-4, 6) |
| 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 [math]\displaystyle{ t^rq^j }[/math] are shown, along with their alternating sums [math]\displaystyle{ \chi }[/math] (fixed [math]\displaystyle{ j }[/math], alternation over [math]\displaystyle{ r }[/math]). The squares with yellow highlighting are those on the "critical diagonals", where [math]\displaystyle{ j-2r=s+1 }[/math] or [math]\displaystyle{ j-2r=s-1 }[/math], where [math]\displaystyle{ s= }[/math]0 is the signature of 10 1. 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 [math]\displaystyle{ n+1 }[/math]-dimensional representation of [math]\displaystyle{ sl(2) }[/math])
| [math]\displaystyle{ n }[/math] | [math]\displaystyle{ J_n }[/math] |
| 2 | [math]\displaystyle{ q^6-q^5+2 q^3-2 q^2+3-3 q^{-1} - q^{-2} +3 q^{-3} -2 q^{-4} - q^{-5} +3 q^{-6} -2 q^{-7} +3 q^{-9} -3 q^{-10} +3 q^{-12} -3 q^{-13} +3 q^{-15} -3 q^{-16} +3 q^{-18} -2 q^{-19} - q^{-20} +2 q^{-21} - q^{-22} - q^{-23} + q^{-24} }[/math] |
| 3 | [math]\displaystyle{ q^{12}-q^{11}+2 q^8-2 q^7+2 q^4-3 q^3+2 q+2-5 q^{-1} +4 q^{-3} +2 q^{-4} -6 q^{-5} - q^{-6} +6 q^{-7} +2 q^{-8} -6 q^{-9} -2 q^{-10} +6 q^{-11} +2 q^{-12} -5 q^{-13} -2 q^{-14} +5 q^{-15} + q^{-16} -4 q^{-17} -2 q^{-18} +4 q^{-19} + q^{-20} -3 q^{-21} -2 q^{-22} +3 q^{-23} +2 q^{-24} -2 q^{-25} -2 q^{-26} +2 q^{-27} +2 q^{-28} -2 q^{-29} -2 q^{-30} +2 q^{-31} +2 q^{-32} -2 q^{-33} -2 q^{-34} +2 q^{-35} +2 q^{-36} -2 q^{-37} -2 q^{-38} + q^{-39} +3 q^{-40} - q^{-41} -2 q^{-42} +2 q^{-44} - q^{-46} - q^{-47} + q^{-48} }[/math] |
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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