http://www.textvarerracrace.com
Knot presentations
Planar diagram presentation
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X6271 X14,6,15,5 X20,15,1,16 X16,7,17,8 X8,19,9,20 X18,11,19,12 X10,4,11,3 X4,10,5,9 X12,17,13,18 X2,14,3,13
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Gauss code
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1, -10, 7, -8, 2, -1, 4, -5, 8, -7, 6, -9, 10, -2, 3, -4, 9, -6, 5, -3
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Dowker-Thistlethwaite code
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6 10 14 16 4 18 2 20 12 8
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Conway Notation
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[8*20.20]
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Minimum Braid Representative
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A Morse Link Presentation
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An Arc Presentation
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Length is 12, width is 5,
Braid index is 5
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![10 115 AP.gif](/images/3/3d/10_115_AP.gif) [{3, 11}, {2, 9}, {8, 10}, {9, 12}, {11, 4}, {5, 3}, {4, 7}, {6, 8}, {7, 13}, {12, 6}, {1, 5}, {13, 2}, {10, 1}]
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[edit Notes on presentations of 10 115]
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[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
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In[3]:=
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K = Knot["10 115"];
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KnotTheory::loading: Loading precomputed data in PD4Knots`.
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Out[4]=
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X6271 X14,6,15,5 X20,15,1,16 X16,7,17,8 X8,19,9,20 X18,11,19,12 X10,4,11,3 X4,10,5,9 X12,17,13,18 X2,14,3,13
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Out[5]=
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1, -10, 7, -8, 2, -1, 4, -5, 8, -7, 6, -9, 10, -2, 3, -4, 9, -6, 5, -3
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Out[6]=
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6 10 14 16 4 18 2 20 12 8
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(The path below may be different on your system)
In[7]:=
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AppendTo[$Path, "C:/bin/LinKnot/"];
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In[8]:=
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ConwayNotation[K]
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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::loading: Loading precomputed data in IndianaData`.
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In[11]:=
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Show[BraidPlot[br]]
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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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In[13]:=
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ap = ArcPresentation[K]
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Out[13]=
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ArcPresentation[{3, 11}, {2, 9}, {8, 10}, {9, 12}, {11, 4}, {5, 3}, {4, 7}, {6, 8}, {7, 13}, {12, 6}, {1, 5}, {13, 2}, {10, 1}]
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Four dimensional invariants
Polynomial invariants
Alexander polynomial |
![{\displaystyle -t^{3}+9t^{2}-26t+37-26t^{-1}+9t^{-2}-t^{-3}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/54fcf3a4d0a157a4b6d8e697857982c2a099eefe) |
Conway polynomial |
![{\displaystyle -z^{6}+3z^{4}+z^{2}+1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/91afae098bc1334227c6701858c4066024a462f8) |
2nd Alexander ideal (db, data sources) |
![{\displaystyle \left\{2,t^{2}+t+1\right\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/07849af6abb8bfeca48fd15c388feb618754ca61) |
Determinant and Signature |
{ 109, 0 } |
Jones polynomial |
![{\displaystyle -q^{5}+4q^{4}-9q^{3}+14q^{2}-17q+19-17q^{-1}+14q^{-2}-9q^{-3}+4q^{-4}-q^{-5}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/468b980a7e27e5fc20fa17006d5b63bc9565a0df) |
HOMFLY-PT polynomial (db, data sources) |
![{\displaystyle -z^{6}+2a^{2}z^{4}+2z^{4}a^{-2}-z^{4}-a^{4}z^{2}+a^{2}z^{2}+z^{2}a^{-2}-z^{2}a^{-4}+z^{2}-a^{2}-a^{-2}+3}](https://wikimedia.org/api/rest_v1/media/math/render/svg/59566fbddda6d47b0cf57e10d449016ea2d95ea7) |
Kauffman polynomial (db, data sources) |
![{\displaystyle 3az^{9}+3z^{9}a^{-1}+8a^{2}z^{8}+8z^{8}a^{-2}+16z^{8}+8a^{3}z^{7}+13az^{7}+13z^{7}a^{-1}+8z^{7}a^{-3}+4a^{4}z^{6}-9a^{2}z^{6}-9z^{6}a^{-2}+4z^{6}a^{-4}-26z^{6}+a^{5}z^{5}-13a^{3}z^{5}-34az^{5}-34z^{5}a^{-1}-13z^{5}a^{-3}+z^{5}a^{-5}-5a^{4}z^{4}+a^{2}z^{4}+z^{4}a^{-2}-5z^{4}a^{-4}+12z^{4}-a^{5}z^{3}+8a^{3}z^{3}+22az^{3}+22z^{3}a^{-1}+8z^{3}a^{-3}-z^{3}a^{-5}+2a^{4}z^{2}-a^{2}z^{2}-z^{2}a^{-2}+2z^{2}a^{-4}-6z^{2}-2a^{3}z-5az-5za^{-1}-2za^{-3}+a^{2}+a^{-2}+3}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3b213d8154e9e35cea32bd1cd5ff7716224e98f0) |
The A2 invariant |
![{\displaystyle -q^{16}+q^{14}+2q^{12}-4q^{10}+2q^{8}-q^{6}-2q^{4}+5q^{2}-1+5q^{-2}-2q^{-4}-q^{-6}+2q^{-8}-4q^{-10}+2q^{-12}+q^{-14}-q^{-16}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3c030b7f5899b98f20b538673a50bb617d9f546a) |
The G2 invariant |
![{\displaystyle q^{80}-3q^{78}+7q^{76}-13q^{74}+16q^{72}-17q^{70}+8q^{68}+17q^{66}-53q^{64}+98q^{62}-130q^{60}+121q^{58}-62q^{56}-61q^{54}+225q^{52}-360q^{50}+410q^{48}-311q^{46}+62q^{44}+258q^{42}-536q^{40}+646q^{38}-522q^{36}+193q^{34}+206q^{32}-514q^{30}+589q^{28}-396q^{26}+28q^{24}+339q^{22}-530q^{20}+436q^{18}-110q^{16}-314q^{14}+652q^{12}-743q^{10}+555q^{8}-133q^{6}-361q^{4}+759q^{2}-907+759q^{-2}-361q^{-4}-133q^{-6}+555q^{-8}-743q^{-10}+652q^{-12}-314q^{-14}-110q^{-16}+436q^{-18}-530q^{-20}+339q^{-22}+28q^{-24}-396q^{-26}+589q^{-28}-514q^{-30}+206q^{-32}+193q^{-34}-522q^{-36}+646q^{-38}-536q^{-40}+258q^{-42}+62q^{-44}-311q^{-46}+410q^{-48}-360q^{-50}+225q^{-52}-61q^{-54}-62q^{-56}+121q^{-58}-130q^{-60}+98q^{-62}-53q^{-64}+17q^{-66}+8q^{-68}-17q^{-70}+16q^{-72}-13q^{-74}+7q^{-76}-3q^{-78}+q^{-80}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a4f8b8a10a2e4914964557dad61564fd4faaf7dc) |
Further Quantum Invariants
Further quantum knot invariants for 10_115.
A1 Invariants.
Weight
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Invariant
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1
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2
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3
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A2 Invariants.
Weight
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Invariant
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1,0
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2,0
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A3 Invariants.
Weight
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Invariant
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0,1,0
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1,0,0
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B2 Invariants.
Weight
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Invariant
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0,1
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1,0
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G2 Invariants.
Weight
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Invariant
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1,0
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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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In[3]:=
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K = Knot["10 115"];
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KnotTheory::loading: Loading precomputed data in PD4Knots`.
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Out[4]=
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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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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:
{}
Same Jones Polynomial (up to mirroring,
):
{}
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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In[3]:=
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K = Knot["10 115"];
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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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{ , }
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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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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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V2,1 through V6,9:
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V2,1
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V3,1
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V4,1
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V4,2
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V4,3
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V5,1
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V5,2
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V5,3
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V5,4
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V6,1
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V6,2
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V6,3
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V6,4
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V6,5
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V6,6
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V6,7
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V6,8
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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.
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 0 is the signature of 10 115. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.
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-5 | -4 | -3 | -2 | -1 | 0 | 1 | 2 | 3 | 4 | 5 | χ |
11 | | | | | | | | | | | 1 | -1 |
9 | | | | | | | | | | 3 | | 3 |
7 | | | | | | | | | 6 | 1 | | -5 |
5 | | | | | | | | 8 | 3 | | | 5 |
3 | | | | | | | 9 | 6 | | | | -3 |
1 | | | | | | 10 | 8 | | | | | 2 |
-1 | | | | | 8 | 10 | | | | | | 2 |
-3 | | | | 6 | 9 | | | | | | | -3 |
-5 | | | 3 | 8 | | | | | | | | 5 |
-7 | | 1 | 6 | | | | | | | | | -5 |
-9 | | 3 | | | | | | | | | | 3 |
-11 | 1 | | | | | | | | | | | -1 |
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The Coloured Jones Polynomials