Edit T(7,2) Further Notes and Views
Knot presentations
Planar diagram presentation
|
X5,13,6,12 X13,7,14,6 X7,1,8,14 X1928 X9,3,10,2 X3,11,4,10 X11,5,12,4
|
Gauss code
|
-4, 5, -6, 7, -1, 2, -3, 4, -5, 6, -7, 1, -2, 3
|
Dowker-Thistlethwaite code
|
8 10 12 14 2 4 6
|
Polynomial invariants
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]:=
|
AppendTo[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
|
In[3]:=
|
K = Knot["T(7,2)"];
|
|
KnotTheory::loading: Loading precomputed data in PD4Knots`.
|
Out[4]=
|
|
Out[5]=
|
|
In[6]:=
|
Alexander[K, 2][t]
|
|
KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.
|
Out[6]=
|
|
In[7]:=
|
{KnotDet[K], KnotSignature[K]}
|
|
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
|
Out[8]=
|
|
In[9]:=
|
HOMFLYPT[K][a, z]
|
|
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
|
Out[9]=
|
|
In[10]:=
|
Kauffman[K][a, z]
|
|
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
|
Out[10]=
|
|
"Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial:
{7_1,}
Same Jones Polynomial (up to mirroring, ):
{7_1,}
Computer Talk
The above data is available with the
Mathematica package
KnotTheory`
. Your input (in
red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.
(The path below may be different on your system, and possibly also the KnotTheory` date)
In[1]:=
|
AppendTo[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
|
In[3]:=
|
K = Knot["T(7,2)"];
|
In[4]:=
|
{A = Alexander[K][t], J = Jones[K][q]}
|
|
KnotTheory::loading: Loading precomputed data in PD4Knots`.
|
|
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
|
Out[4]=
|
{ , }
|
In[5]:=
|
DeleteCases[Select[AllKnots[], (A === Alexander[#][t]) &], K]
|
|
KnotTheory::loading: Loading precomputed data in DTCode4KnotsTo11`.
|
|
KnotTheory::credits: The GaussCode to PD conversion was written by Siddarth Sankaran at the University of Toronto in the summer of 2005.
|
In[6]:=
|
DeleteCases[
Select[
AllKnots[],
(J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) &
],
K
]
|
|
KnotTheory::loading: Loading precomputed data in Jones4Knots11`.
|
V2,1 through V6,9:
|
V2,1
|
V3,1
|
V4,1
|
V4,2
|
V4,3
|
V5,1
|
V5,2
|
V5,3
|
V5,4
|
V6,1
|
V6,2
|
V6,3
|
V6,4
|
V6,5
|
V6,6
|
V6,7
|
V6,8
|
V6,9
|
Data:T(7,2)/V 2,1
|
Data:T(7,2)/V 3,1
|
Data:T(7,2)/V 4,1
|
Data:T(7,2)/V 4,2
|
Data:T(7,2)/V 4,3
|
Data:T(7,2)/V 5,1
|
Data:T(7,2)/V 5,2
|
Data:T(7,2)/V 5,3
|
Data:T(7,2)/V 5,4
|
Data:T(7,2)/V 6,1
|
Data:T(7,2)/V 6,2
|
Data:T(7,2)/V 6,3
|
Data:T(7,2)/V 6,4
|
Data:T(7,2)/V 6,5
|
Data:T(7,2)/V 6,6
|
Data:T(7,2)/V 6,7
|
Data:T(7,2)/V 6,8
|
Data:T(7,2)/V 6,9
|
|
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 6 is the signature of T(7,2). Nonzero entries off the critical diagonals (if any exist) are highlighted in red.
|
|
|
0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | χ |
21 | | | | | | | | 1 | -1 |
19 | | | | | | | | | 0 |
17 | | | | | | 1 | 1 | | 0 |
15 | | | | | | | | | 0 |
13 | | | | 1 | 1 | | | | 0 |
11 | | | | | | | | | 0 |
9 | | | 1 | | | | | | 1 |
7 | 1 | | | | | | | | 1 |
5 | 1 | | | | | | | | 1 |
|