10 66
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![]() (KnotPlot image) |
See the full Rolfsen Knot Table. Visit 10 66's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) |
Knot presentations
Planar diagram presentation | X1425 X3,10,4,11 X5,14,6,15 X7,16,8,17 X15,6,16,7 X17,20,18,1 X11,18,12,19 X19,12,20,13 X13,8,14,9 X9,2,10,3 |
Gauss code | -1, 10, -2, 1, -3, 5, -4, 9, -10, 2, -7, 8, -9, 3, -5, 4, -6, 7, -8, 6 |
Dowker-Thistlethwaite code | 4 10 14 16 2 18 8 6 20 12 |
Conway Notation | [31,21,21] |
Minimum Braid Representative | A Morse Link Presentation | An Arc Presentation | ||||
Length is 11, width is 4, Braid index is 4 |
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![]() [{13, 3}, {2, 11}, {9, 12}, {11, 13}, {10, 4}, {3, 5}, {4, 6}, {5, 9}, {6, 1}, {7, 10}, {8, 2}, {12, 7}, {1, 8}] |
[edit Notes on presentations of 10 66]
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[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
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Loading KnotTheory` version of May 31, 2006, 14:15:20.091.
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In[3]:=
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K = Knot["10 66"];
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In[4]:=
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PD[K]
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KnotTheory::loading: Loading precomputed data in PD4Knots`.
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Out[4]=
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X1425 X3,10,4,11 X5,14,6,15 X7,16,8,17 X15,6,16,7 X17,20,18,1 X11,18,12,19 X19,12,20,13 X13,8,14,9 X9,2,10,3 |
In[5]:=
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GaussCode[K]
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Out[5]=
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-1, 10, -2, 1, -3, 5, -4, 9, -10, 2, -7, 8, -9, 3, -5, 4, -6, 7, -8, 6 |
In[6]:=
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DTCode[K]
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Out[6]=
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4 10 14 16 2 18 8 6 20 12 |
(The path below may be different on your system)
In[7]:=
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AppendTo[$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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[31,21,21] |
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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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[{13, 3}, {2, 11}, {9, 12}, {11, 13}, {10, 4}, {3, 5}, {4, 6}, {5, 9}, {6, 1}, {7, 10}, {8, 2}, {12, 7}, {1, 8}] |
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
A1 Invariants.
Weight | Invariant |
---|---|
1 | |
2 | |
3 | |
4 | |
5 |
A2 Invariants.
Weight | Invariant |
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1,0 | |
1,1 | |
2,0 |
A3 Invariants.
Weight | Invariant |
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0,1,0 | |
1,0,0 |
A4 Invariants.
Weight | Invariant |
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0,1,0,0 | |
1,0,0,0 |
B2 Invariants.
Weight | Invariant |
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0,1 | |
1,0 |
D4 Invariants.
Weight | Invariant |
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1,0,0,0 |
G2 Invariants.
Weight | Invariant |
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1,0 |
.
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 66"];
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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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In[5]:=
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Conway[K][z]
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Out[5]=
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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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In[7]:=
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{KnotDet[K], KnotSignature[K]}
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Out[7]=
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{ 75, -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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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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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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"Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {K11a245,}
Same Jones Polynomial (up to mirroring, ): {}
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 66"];
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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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{ , } |
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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{K11a245,} |
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 are shown, along with their alternating sums (fixed , alternation over ). The squares with yellow highlighting are those on the "critical diagonals", where or , where -6 is the signature of 10 66. 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
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^{-6} -2 q^{-7} + q^{-8} +8 q^{-9} -12 q^{-10} -4 q^{-11} +33 q^{-12} -28 q^{-13} -27 q^{-14} +73 q^{-15} -36 q^{-16} -67 q^{-17} +113 q^{-18} -28 q^{-19} -105 q^{-20} +131 q^{-21} -10 q^{-22} -122 q^{-23} +116 q^{-24} +8 q^{-25} -104 q^{-26} +77 q^{-27} +15 q^{-28} -62 q^{-29} +36 q^{-30} +10 q^{-31} -24 q^{-32} +10 q^{-33} +3 q^{-34} -4 q^{-35} + q^{-36} } |
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^{-10} + q^{-11} +3 q^{-12} +3 q^{-13} -12 q^{-14} -4 q^{-15} +21 q^{-16} +23 q^{-17} -41 q^{-18} -45 q^{-19} +43 q^{-20} +105 q^{-21} -56 q^{-22} -155 q^{-23} +18 q^{-24} +243 q^{-25} +16 q^{-26} -301 q^{-27} -107 q^{-28} +378 q^{-29} +187 q^{-30} -405 q^{-31} -307 q^{-32} +436 q^{-33} +407 q^{-34} -426 q^{-35} -514 q^{-36} +403 q^{-37} +601 q^{-38} -365 q^{-39} -660 q^{-40} +301 q^{-41} +701 q^{-42} -237 q^{-43} -696 q^{-44} +160 q^{-45} +661 q^{-46} -92 q^{-47} -584 q^{-48} +28 q^{-49} +490 q^{-50} +11 q^{-51} -378 q^{-52} -37 q^{-53} +277 q^{-54} +39 q^{-55} -183 q^{-56} -38 q^{-57} +118 q^{-58} +24 q^{-59} -65 q^{-60} -20 q^{-61} +39 q^{-62} +7 q^{-63} -16 q^{-64} -4 q^{-65} +6 q^{-66} +3 q^{-67} -4 q^{-68} + q^{-69} } |
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^{-12} -2 q^{-13} + q^{-14} +3 q^{-15} -2 q^{-16} +3 q^{-17} -13 q^{-18} +2 q^{-19} +23 q^{-20} +2 q^{-21} +10 q^{-22} -67 q^{-23} -28 q^{-24} +73 q^{-25} +64 q^{-26} +92 q^{-27} -186 q^{-28} -194 q^{-29} +49 q^{-30} +197 q^{-31} +433 q^{-32} -218 q^{-33} -523 q^{-34} -279 q^{-35} +171 q^{-36} +1072 q^{-37} +138 q^{-38} -726 q^{-39} -962 q^{-40} -378 q^{-41} +1683 q^{-42} +940 q^{-43} -401 q^{-44} -1656 q^{-45} -1497 q^{-46} +1821 q^{-47} +1837 q^{-48} +539 q^{-49} -1946 q^{-50} -2839 q^{-51} +1378 q^{-52} +2444 q^{-53} +1786 q^{-54} -1738 q^{-55} -3999 q^{-56} +578 q^{-57} +2647 q^{-58} +2979 q^{-59} -1198 q^{-60} -4771 q^{-61} -330 q^{-62} +2500 q^{-63} +3893 q^{-64} -478 q^{-65} -5038 q^{-66} -1196 q^{-67} +2008 q^{-68} +4344 q^{-69} +331 q^{-70} -4647 q^{-71} -1809 q^{-72} +1172 q^{-73} +4094 q^{-74} +1037 q^{-75} -3575 q^{-76} -1907 q^{-77} +237 q^{-78} +3138 q^{-79} +1318 q^{-80} -2200 q^{-81} -1430 q^{-82} -368 q^{-83} +1881 q^{-84} +1081 q^{-85} -1070 q^{-86} -731 q^{-87} -479 q^{-88} +878 q^{-89} +616 q^{-90} -441 q^{-91} -230 q^{-92} -312 q^{-93} +333 q^{-94} +255 q^{-95} -175 q^{-96} -30 q^{-97} -135 q^{-98} +110 q^{-99} +80 q^{-100} -66 q^{-101} +9 q^{-102} -42 q^{-103} +29 q^{-104} +20 q^{-105} -19 q^{-106} +4 q^{-107} -8 q^{-108} +6 q^{-109} +3 q^{-110} -4 q^{-111} + q^{-112} } |
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^{-15} -2 q^{-16} + q^{-17} +3 q^{-18} -2 q^{-19} -2 q^{-20} +2 q^{-21} -7 q^{-22} +3 q^{-23} +20 q^{-24} +5 q^{-25} -17 q^{-26} -17 q^{-27} -40 q^{-28} +2 q^{-29} +81 q^{-30} +93 q^{-31} +6 q^{-32} -95 q^{-33} -217 q^{-34} -146 q^{-35} +145 q^{-36} +388 q^{-37} +358 q^{-38} +8 q^{-39} -589 q^{-40} -807 q^{-41} -294 q^{-42} +631 q^{-43} +1298 q^{-44} +1055 q^{-45} -422 q^{-46} -1881 q^{-47} -1953 q^{-48} -368 q^{-49} +2028 q^{-50} +3311 q^{-51} +1689 q^{-52} -1807 q^{-53} -4307 q^{-54} -3616 q^{-55} +510 q^{-56} +5178 q^{-57} +5833 q^{-58} +1377 q^{-59} -4932 q^{-60} -8010 q^{-61} -4336 q^{-62} +3966 q^{-63} +9713 q^{-64} +7475 q^{-65} -1645 q^{-66} -10611 q^{-67} -10992 q^{-68} -1276 q^{-69} +10504 q^{-70} +13934 q^{-71} +5094 q^{-72} -9441 q^{-73} -16569 q^{-74} -8880 q^{-75} +7555 q^{-76} +18254 q^{-77} +12832 q^{-78} -5102 q^{-79} -19456 q^{-80} -16321 q^{-81} +2389 q^{-82} +19912 q^{-83} +19501 q^{-84} +446 q^{-85} -19991 q^{-86} -22235 q^{-87} -3206 q^{-88} +19703 q^{-89} +24507 q^{-90} +5915 q^{-91} -19023 q^{-92} -26444 q^{-93} -8553 q^{-94} +18043 q^{-95} +27810 q^{-96} +11104 q^{-97} -16437 q^{-98} -28639 q^{-99} -13581 q^{-100} +14359 q^{-101} +28564 q^{-102} +15773 q^{-103} -11583 q^{-104} -27561 q^{-105} -17462 q^{-106} +8386 q^{-107} +25406 q^{-108} +18387 q^{-109} -4942 q^{-110} -22316 q^{-111} -18285 q^{-112} +1738 q^{-113} +18385 q^{-114} +17153 q^{-115} +965 q^{-116} -14217 q^{-117} -15075 q^{-118} -2753 q^{-119} +10117 q^{-120} +12370 q^{-121} +3698 q^{-122} -6640 q^{-123} -9456 q^{-124} -3764 q^{-125} +3899 q^{-126} +6721 q^{-127} +3315 q^{-128} -2072 q^{-129} -4411 q^{-130} -2537 q^{-131} +892 q^{-132} +2713 q^{-133} +1791 q^{-134} -368 q^{-135} -1525 q^{-136} -1078 q^{-137} +46 q^{-138} +817 q^{-139} +657 q^{-140} -31 q^{-141} -410 q^{-142} -300 q^{-143} -18 q^{-144} +184 q^{-145} +170 q^{-146} -5 q^{-147} -101 q^{-148} -61 q^{-149} +12 q^{-150} +39 q^{-151} +19 q^{-152} +5 q^{-153} -23 q^{-154} -20 q^{-155} +17 q^{-156} +10 q^{-157} -6 q^{-158} + q^{-159} -8 q^{-161} +6 q^{-162} +3 q^{-163} -4 q^{-164} + q^{-165} } |
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^{-18} -2 q^{-19} + q^{-20} +3 q^{-21} -2 q^{-22} -2 q^{-23} -3 q^{-24} +8 q^{-25} -6 q^{-26} +22 q^{-28} -4 q^{-29} -15 q^{-30} -33 q^{-31} +12 q^{-32} -11 q^{-33} +12 q^{-34} +109 q^{-35} +45 q^{-36} -30 q^{-37} -169 q^{-38} -92 q^{-39} -145 q^{-40} -8 q^{-41} +399 q^{-42} +417 q^{-43} +267 q^{-44} -312 q^{-45} -484 q^{-46} -945 q^{-47} -718 q^{-48} +517 q^{-49} +1363 q^{-50} +1806 q^{-51} +759 q^{-52} -232 q^{-53} -2577 q^{-54} -3504 q^{-55} -1710 q^{-56} +1087 q^{-57} +4331 q^{-58} +4838 q^{-59} +4099 q^{-60} -1819 q^{-61} -7223 q^{-62} -8412 q^{-63} -5186 q^{-64} +2619 q^{-65} +9264 q^{-66} +14519 q^{-67} +7614 q^{-68} -4181 q^{-69} -14862 q^{-70} -18731 q^{-71} -10973 q^{-72} +3647 q^{-73} +23531 q^{-74} +25770 q^{-75} +14209 q^{-76} -8018 q^{-77} -28920 q^{-78} -34705 q^{-79} -20922 q^{-80} +15622 q^{-81} +38976 q^{-82} +44006 q^{-83} +20800 q^{-84} -18491 q^{-85} -52186 q^{-86} -58073 q^{-87} -17154 q^{-88} +29079 q^{-89} +66699 q^{-90} +63094 q^{-91} +19104 q^{-92} -45268 q^{-93} -87954 q^{-94} -64499 q^{-95} -8764 q^{-96} +64897 q^{-97} +98738 q^{-98} +72032 q^{-99} -10940 q^{-100} -94527 q^{-101} -106650 q^{-102} -61777 q^{-103} +37275 q^{-104} +113524 q^{-105} +121253 q^{-106} +37707 q^{-107} -77814 q^{-108} -131152 q^{-109} -112182 q^{-110} -3359 q^{-111} +108395 q^{-112} +155711 q^{-113} +84670 q^{-114} -49577 q^{-115} -139215 q^{-116} -150494 q^{-117} -43178 q^{-118} +93488 q^{-119} +176291 q^{-120} +122336 q^{-121} -20779 q^{-122} -138757 q^{-123} -177616 q^{-124} -76882 q^{-125} +76464 q^{-126} +188492 q^{-127} +152178 q^{-128} +6344 q^{-129} -133783 q^{-130} -197392 q^{-131} -107215 q^{-132} +56166 q^{-133} +192865 q^{-134} +177073 q^{-135} +36403 q^{-136} -119619 q^{-137} -207712 q^{-138} -136562 q^{-139} +26139 q^{-140} +181743 q^{-141} +192656 q^{-142} +71194 q^{-143} -88400 q^{-144} -198592 q^{-145} -158247 q^{-146} -14279 q^{-147} +146792 q^{-148} +186918 q^{-149} +101207 q^{-150} -41454 q^{-151} -161908 q^{-152} -158190 q^{-153} -52386 q^{-154} +92208 q^{-155} +152282 q^{-156} +110215 q^{-157} +5020 q^{-158} -104830 q^{-159} -129424 q^{-160} -70153 q^{-161} +37579 q^{-162} +98405 q^{-163} +91783 q^{-164} +31675 q^{-165} -49065 q^{-166} -83258 q^{-167} -61888 q^{-168} +2693 q^{-169} +47459 q^{-170} +57844 q^{-171} +33653 q^{-172} -13203 q^{-173} -40753 q^{-174} -39411 q^{-175} -8703 q^{-176} +15411 q^{-177} +27114 q^{-178} +22137 q^{-179} +838 q^{-180} -14669 q^{-181} -18678 q^{-182} -6914 q^{-183} +2273 q^{-184} +9133 q^{-185} +10343 q^{-186} +2700 q^{-187} -3705 q^{-188} -6736 q^{-189} -2762 q^{-190} -656 q^{-191} +2006 q^{-192} +3625 q^{-193} +1353 q^{-194} -613 q^{-195} -1941 q^{-196} -506 q^{-197} -527 q^{-198} +158 q^{-199} +1014 q^{-200} +376 q^{-201} -74 q^{-202} -504 q^{-203} +89 q^{-204} -169 q^{-205} -79 q^{-206} +248 q^{-207} +64 q^{-208} -20 q^{-209} -140 q^{-210} +100 q^{-211} -35 q^{-212} -41 q^{-213} +57 q^{-214} +2 q^{-215} -4 q^{-216} -42 q^{-217} +39 q^{-218} -2 q^{-219} -16 q^{-220} +14 q^{-221} -3 q^{-222} -8 q^{-224} +6 q^{-225} +3 q^{-226} -4 q^{-227} + q^{-228} } |
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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