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http://www.texttrdelcali.com |
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Revision as of 13:29, 22 May 2009
http://www.texttrdelcali.com
Knot presentations
Planar diagram presentation
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X6271 X8493 X14,6,15,5 X20,15,1,16 X16,9,17,10 X10,19,11,20 X18,11,19,12 X12,17,13,18 X2837 X4,14,5,13
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Gauss code
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1, -9, 2, -10, 3, -1, 9, -2, 5, -6, 7, -8, 10, -3, 4, -5, 8, -7, 6, -4
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Dowker-Thistlethwaite code
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6 8 14 2 16 18 4 20 12 10
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Conway Notation
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[311,3,2]
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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 11, width is 4,
Braid index is 4
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![10 52 AP.gif](/images/5/5b/10_52_AP.gif) [{7, 13}, {2, 12}, {13, 11}, {12, 8}, {1, 6}, {5, 7}, {6, 9}, {8, 4}, {3, 5}, {4, 10}, {9, 3}, {11, 2}, {10, 1}]
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[edit Notes on presentations of 10 52]
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[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
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In[3]:=
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K = Knot["10 52"];
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KnotTheory::loading: Loading precomputed data in PD4Knots`.
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Out[4]=
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X6271 X8493 X14,6,15,5 X20,15,1,16 X16,9,17,10 X10,19,11,20 X18,11,19,12 X12,17,13,18 X2837 X4,14,5,13
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Out[5]=
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1, -9, 2, -10, 3, -1, 9, -2, 5, -6, 7, -8, 10, -3, 4, -5, 8, -7, 6, -4
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Out[6]=
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6 8 14 2 16 18 4 20 12 10
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(The path below may be different on your system)
In[7]:=
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AppendTo[$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[{7, 13}, {2, 12}, {13, 11}, {12, 8}, {1, 6}, {5, 7}, {6, 9}, {8, 4}, {3, 5}, {4, 10}, {9, 3}, {11, 2}, {10, 1}]
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Four dimensional invariants
Polynomial invariants
Alexander polynomial |
![{\displaystyle 2t^{3}-7t^{2}+13t-15+13t^{-1}-7t^{-2}+2t^{-3}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d38bd836db923dad2ecc1b5fe6d59094375df5a1) |
Conway polynomial |
![{\displaystyle 2z^{6}+5z^{4}+3z^{2}+1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/374b5938316d29576d32e6f3b028222824f301d6) |
2nd Alexander ideal (db, data sources) |
![{\displaystyle \{1\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5acdcac635f883f8b4f0a01aa03b16b22f23b124) |
Determinant and Signature |
{ 59, 2 } |
Jones polynomial |
![{\displaystyle -q^{6}+3q^{5}-6q^{4}+8q^{3}-9q^{2}+10q-8+7q^{-1}-4q^{-2}+2q^{-3}-q^{-4}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/21d570b19bd84aac8f0368b02437dca416b28ac3) |
HOMFLY-PT polynomial (db, data sources) |
![{\displaystyle z^{6}a^{-2}+z^{6}-a^{2}z^{4}+3z^{4}a^{-2}-z^{4}a^{-4}+4z^{4}-3a^{2}z^{2}+2z^{2}a^{-2}-2z^{2}a^{-4}+6z^{2}-2a^{2}-a^{-4}+4}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4f2a3ead7b650d494016a75bdb19f3e86a124eb6) |
Kauffman polynomial (db, data sources) |
![{\displaystyle az^{9}+z^{9}a^{-1}+2a^{2}z^{8}+4z^{8}a^{-2}+6z^{8}+a^{3}z^{7}+az^{7}+7z^{7}a^{-1}+7z^{7}a^{-3}-9a^{2}z^{6}-3z^{6}a^{-2}+8z^{6}a^{-4}-20z^{6}-5a^{3}z^{5}-16az^{5}-28z^{5}a^{-1}-11z^{5}a^{-3}+6z^{5}a^{-5}+13a^{2}z^{4}-9z^{4}a^{-2}-12z^{4}a^{-4}+3z^{4}a^{-6}+19z^{4}+8a^{3}z^{3}+24az^{3}+24z^{3}a^{-1}+2z^{3}a^{-3}-5z^{3}a^{-5}+z^{3}a^{-7}-7a^{2}z^{2}+4z^{2}a^{-2}+6z^{2}a^{-4}-9z^{2}-4a^{3}z-9az-7za^{-1}+2za^{-5}+2a^{2}-a^{-4}+4}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a0ffef3e7007c928c190eb14dd0d620a39ee7d8c) |
The A2 invariant |
![{\displaystyle -q^{12}-q^{8}-q^{6}+2q^{4}+3+2q^{-2}+2q^{-6}-2q^{-8}+q^{-10}-q^{-12}-q^{-14}+q^{-16}-q^{-18}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1296e97434ff1cb148657c34d97d0dee5eef0b2c) |
The G2 invariant |
![{\displaystyle q^{60}-q^{58}+4q^{56}-6q^{54}+6q^{52}-6q^{50}-q^{48}+12q^{46}-25q^{44}+33q^{42}-31q^{40}+12q^{38}+14q^{36}-46q^{34}+64q^{32}-64q^{30}+38q^{28}-46q^{24}+71q^{22}-71q^{20}+46q^{18}-4q^{16}-32q^{14}+53q^{12}-47q^{10}+21q^{8}+18q^{6}-42q^{4}+56q^{2}-36+2q^{-2}+45q^{-4}-75q^{-6}+88q^{-8}-62q^{-10}+17q^{-12}+40q^{-14}-83q^{-16}+100q^{-18}-82q^{-20}+39q^{-22}+15q^{-24}-58q^{-26}+72q^{-28}-57q^{-30}+21q^{-32}+17q^{-34}-41q^{-36}+39q^{-38}-19q^{-40}-13q^{-42}+38q^{-44}-48q^{-46}+40q^{-48}-15q^{-50}-17q^{-52}+41q^{-54}-55q^{-56}+50q^{-58}-32q^{-60}+8q^{-62}+15q^{-64}-33q^{-66}+39q^{-68}-36q^{-70}+26q^{-72}-9q^{-74}-5q^{-76}+12q^{-78}-18q^{-80}+16q^{-82}-12q^{-84}+8q^{-86}-q^{-88}-2q^{-90}+3q^{-92}-4q^{-94}+3q^{-96}-2q^{-98}+q^{-100}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c6e9a7fede68b6eeec490d0a6644cd8e64eec50c) |
Further Quantum Invariants
Further quantum knot invariants for 10_52.
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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4
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5
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A2 Invariants.
Weight
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Invariant
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1,0
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1,1
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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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A4 Invariants.
Weight
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Invariant
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0,1,0,0
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1,0,0,0
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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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D4 Invariants.
Weight
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Invariant
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1,0,0,0
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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 52"];
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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:
{10_23,}
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`
|
In[3]:=
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K = Knot["10 52"];
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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 2 is the signature of 10 52. 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 |
13 | | | | | | | | | | | 1 | -1 |
11 | | | | | | | | | | 2 | | 2 |
9 | | | | | | | | | 4 | 1 | | -3 |
7 | | | | | | | | 4 | 2 | | | 2 |
5 | | | | | | | 5 | 4 | | | | -1 |
3 | | | | | | 5 | 4 | | | | | 1 |
1 | | | | | 4 | 6 | | | | | | 2 |
-1 | | | | 3 | 4 | | | | | | | -1 |
-3 | | | 1 | 4 | | | | | | | | 3 |
-5 | | 1 | 3 | | | | | | | | | -2 |
-7 | | 1 | | | | | | | | | | 1 |
-9 | 1 | | | | | | | | | | | -1 |
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The Coloured Jones Polynomials