http://www.textmonracboc.com
Knot presentations
Planar diagram presentation
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X1627 X7,18,8,19 X3948 X17,3,18,2 X5,15,6,14 X9,17,10,16 X15,11,16,10 X11,5,12,4 X13,20,14,1 X19,12,20,13
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Gauss code
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-1, 4, -3, 8, -5, 1, -2, 3, -6, 7, -8, 10, -9, 5, -7, 6, -4, 2, -10, 9
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Dowker-Thistlethwaite code
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6 8 14 18 16 4 20 10 2 12
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Conway Notation
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[8*2:.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 11, width is 4,
Braid index is 4
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![10 119 AP.gif](/images/f/f1/10_119_AP.gif) [{12, 3}, {1, 5}, {6, 4}, {5, 2}, {3, 7}, {11, 6}, {8, 12}, {7, 9}, {2, 8}, {4, 10}, {9, 11}, {10, 1}]
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[edit Notes on presentations of 10 119]
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[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
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In[3]:=
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K = Knot["10 119"];
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KnotTheory::loading: Loading precomputed data in PD4Knots`.
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Out[4]=
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X1627 X7,18,8,19 X3948 X17,3,18,2 X5,15,6,14 X9,17,10,16 X15,11,16,10 X11,5,12,4 X13,20,14,1 X19,12,20,13
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Out[5]=
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-1, 4, -3, 8, -5, 1, -2, 3, -6, 7, -8, 10, -9, 5, -7, 6, -4, 2, -10, 9
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Out[6]=
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6 8 14 18 16 4 20 10 2 12
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(The path below may be different on your system)
In[7]:=
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AppendTo[$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[{12, 3}, {1, 5}, {6, 4}, {5, 2}, {3, 7}, {11, 6}, {8, 12}, {7, 9}, {2, 8}, {4, 10}, {9, 11}, {10, 1}]
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Four dimensional invariants
Polynomial invariants
Alexander polynomial |
![{\displaystyle -2t^{3}+10t^{2}-23t+31-23t^{-1}+10t^{-2}-2t^{-3}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8a983db6b7d396aa30315ee57ecefb29faaacafc) |
Conway polynomial |
![{\displaystyle -2z^{6}-2z^{4}-z^{2}+1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4374946debd771830fd3de437b4f0392596c3918) |
2nd Alexander ideal (db, data sources) |
![{\displaystyle \{1\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5acdcac635f883f8b4f0a01aa03b16b22f23b124) |
Determinant and Signature |
{ 101, 0 } |
Jones polynomial |
![{\displaystyle q^{6}-4q^{5}+8q^{4}-12q^{3}+16q^{2}-17q+16-13q^{-1}+9q^{-2}-4q^{-3}+q^{-4}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5c412167569e729ef6882e10c465569cdeee5750) |
HOMFLY-PT polynomial (db, data sources) |
![{\displaystyle -z^{6}a^{-2}-z^{6}+a^{2}z^{4}-2z^{4}a^{-2}+z^{4}a^{-4}-2z^{4}+a^{2}z^{2}-z^{2}a^{-2}+z^{2}a^{-4}-2z^{2}+a^{2}+a^{-2}-1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/273bc136f1c54ba004d73b7a50667dd680ff75c2) |
Kauffman polynomial (db, data sources) |
![{\displaystyle 3z^{9}a^{-1}+3z^{9}a^{-3}+15z^{8}a^{-2}+6z^{8}a^{-4}+9z^{8}+12az^{7}+13z^{7}a^{-1}+5z^{7}a^{-3}+4z^{7}a^{-5}+9a^{2}z^{6}-31z^{6}a^{-2}-14z^{6}a^{-4}+z^{6}a^{-6}-7z^{6}+4a^{3}z^{5}-17az^{5}-37z^{5}a^{-1}-26z^{5}a^{-3}-10z^{5}a^{-5}+a^{4}z^{4}-9a^{2}z^{4}+13z^{4}a^{-2}+8z^{4}a^{-4}-2z^{4}a^{-6}-7z^{4}-a^{3}z^{3}+9az^{3}+22z^{3}a^{-1}+19z^{3}a^{-3}+7z^{3}a^{-5}+4a^{2}z^{2}+z^{2}a^{-2}+z^{2}a^{-6}+6z^{2}-2az-4za^{-1}-3za^{-3}-za^{-5}-a^{2}-a^{-2}-1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/58ed7156da8d5aedb217b74b4f0db47b640da624) |
The A2 invariant |
![{\displaystyle q^{12}-2q^{10}+2q^{8}+2q^{6}-3q^{4}+3q^{2}-3+q^{-2}+q^{-4}-q^{-6}+4q^{-8}-3q^{-10}+q^{-12}+q^{-14}-2q^{-16}+q^{-18}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/857e2f806f63a46e77c1b3f7dffec26226ad93b7) |
The G2 invariant |
![{\displaystyle q^{66}-3q^{64}+6q^{62}-10q^{60}+11q^{58}-10q^{56}+4q^{54}+14q^{52}-35q^{50}+60q^{48}-78q^{46}+71q^{44}-44q^{42}-17q^{40}+105q^{38}-184q^{36}+238q^{34}-223q^{32}+127q^{30}+33q^{28}-222q^{26}+371q^{24}-410q^{22}+305q^{20}-82q^{18}-176q^{16}+370q^{14}-400q^{12}+264q^{10}-15q^{8}-242q^{6}+363q^{4}-299q^{2}+59+250q^{-2}-475q^{-4}+516q^{-6}-332q^{-8}-q^{-10}+350q^{-12}-598q^{-14}+642q^{-16}-473q^{-18}+155q^{-20}+204q^{-22}-467q^{-24}+561q^{-26}-441q^{-28}+178q^{-30}+115q^{-32}-337q^{-34}+382q^{-36}-246q^{-38}-7q^{-40}+274q^{-42}-413q^{-44}+359q^{-46}-130q^{-48}-174q^{-50}+415q^{-52}-496q^{-54}+387q^{-56}-150q^{-58}-116q^{-60}+311q^{-62}-370q^{-64}+302q^{-66}-149q^{-68}-7q^{-70}+108q^{-72}-148q^{-74}+125q^{-76}-73q^{-78}+27q^{-80}+8q^{-82}-22q^{-84}+21q^{-86}-16q^{-88}+8q^{-90}-3q^{-92}+q^{-94}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/08130b5ab05514e9c4dc930826169eb851885a11) |
Further Quantum Invariants
Further quantum knot invariants for 10_119.
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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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["10 119"];
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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:
{K11a84,}
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["10 119"];
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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 10 119. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.
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-4 | -3 | -2 | -1 | 0 | 1 | 2 | 3 | 4 | 5 | 6 | χ |
13 | | | | | | | | | | | 1 | 1 |
11 | | | | | | | | | | 3 | | -3 |
9 | | | | | | | | | 5 | 1 | | 4 |
7 | | | | | | | | 7 | 3 | | | -4 |
5 | | | | | | | 9 | 5 | | | | 4 |
3 | | | | | | 8 | 7 | | | | | -1 |
1 | | | | | 8 | 9 | | | | | | -1 |
-1 | | | | 6 | 9 | | | | | | | 3 |
-3 | | | 3 | 7 | | | | | | | | -4 |
-5 | | 1 | 6 | | | | | | | | | 5 |
-7 | | 3 | | | | | | | | | | -3 |
-9 | 1 | | | | | | | | | | | 1 |
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The Coloured Jones Polynomials