Edit T(15,2) Further Notes and Views
Knot presentations
Planar diagram presentation
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X9,25,10,24 X25,11,26,10 X11,27,12,26 X27,13,28,12 X13,29,14,28 X29,15,30,14 X15,1,16,30 X1,17,2,16 X17,3,18,2 X3,19,4,18 X19,5,20,4 X5,21,6,20 X21,7,22,6 X7,23,8,22 X23,9,24,8
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Gauss code
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-8, 9, -10, 11, -12, 13, -14, 15, -1, 2, -3, 4, -5, 6, -7, 8, -9, 10, -11, 12, -13, 14, -15, 1, -2, 3, -4, 5, -6, 7
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Dowker-Thistlethwaite code
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16 18 20 22 24 26 28 30 2 4 6 8 10 12 14
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Polynomial invariants
Alexander polynomial |
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Conway polynomial |
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2nd Alexander ideal (db, data sources) |
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Determinant and Signature |
{ 15, 14 } |
Jones polynomial |
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HOMFLY-PT polynomial (db, data sources) |
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Kauffman polynomial (db, data sources) |
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The A2 invariant |
Data:T(15,2)/QuantumInvariant/A2/1,0 |
The G2 invariant |
Data:T(15,2)/QuantumInvariant/G2/1,0 |
Further Quantum Invariants
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["T(15,2)"];
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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]:=
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AppendTo[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
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In[3]:=
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K = Knot["T(15,2)"];
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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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Data:T(15,2)/V 2,1
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Data:T(15,2)/V 3,1
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Data:T(15,2)/V 4,1
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Data:T(15,2)/V 4,2
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Data:T(15,2)/V 4,3
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Data:T(15,2)/V 5,1
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Data:T(15,2)/V 5,2
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Data:T(15,2)/V 5,3
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Data:T(15,2)/V 5,4
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Data:T(15,2)/V 6,1
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Data:T(15,2)/V 6,2
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Data:T(15,2)/V 6,3
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Data:T(15,2)/V 6,4
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Data:T(15,2)/V 6,5
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Data:T(15,2)/V 6,6
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Data:T(15,2)/V 6,7
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Data:T(15,2)/V 6,8
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Data:T(15,2)/V 6,9
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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 14 is the signature of T(15,2). Nonzero entries off the critical diagonals (if any exist) are highlighted in red.
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0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | χ |
45 | | | | | | | | | | | | | | | | 1 | -1 |
43 | | | | | | | | | | | | | | | | | 0 |
41 | | | | | | | | | | | | | | 1 | 1 | | 0 |
39 | | | | | | | | | | | | | | | | | 0 |
37 | | | | | | | | | | | | 1 | 1 | | | | 0 |
35 | | | | | | | | | | | | | | | | | 0 |
33 | | | | | | | | | | 1 | 1 | | | | | | 0 |
31 | | | | | | | | | | | | | | | | | 0 |
29 | | | | | | | | 1 | 1 | | | | | | | | 0 |
27 | | | | | | | | | | | | | | | | | 0 |
25 | | | | | | 1 | 1 | | | | | | | | | | 0 |
23 | | | | | | | | | | | | | | | | | 0 |
21 | | | | 1 | 1 | | | | | | | | | | | | 0 |
19 | | | | | | | | | | | | | | | | | 0 |
17 | | | 1 | | | | | | | | | | | | | | 1 |
15 | 1 | | | | | | | | | | | | | | | | 1 |
13 | 1 | | | | | | | | | | | | | | | | 1 |
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