Edit T(7,2) Further Notes and Views
Knot presentations
| Planar diagram presentation
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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
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| Gauss code
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-4, 5, -6, 7, -1, 2, -3, 4, -5, 6, -7, 1, -2, 3
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| Dowker-Thistlethwaite code
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8 10 12 14 2 4 6
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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]:=
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AppendTo[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
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In[3]:=
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K = Knot["T(7,2)"];
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KnotTheory::loading: Loading precomputed data in PD4Knots`.
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Out[4]=
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[math]\displaystyle{ t^3-t^2+t-1+ t^{-1} - t^{-2} + t^{-3} }[/math]
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Out[5]=
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[math]\displaystyle{ z^6+5 z^4+6 z^2+1 }[/math]
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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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[math]\displaystyle{ \{1\} }[/math]
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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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[math]\displaystyle{ -q^{10}+q^9-q^8+q^7-q^6+q^5+q^3 }[/math]
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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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[math]\displaystyle{ z^6 a^{-6} +6 z^4 a^{-6} -z^4 a^{-8} +10 z^2 a^{-6} -4 z^2 a^{-8} +4 a^{-6} -3 a^{-8} }[/math]
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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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[math]\displaystyle{ z^6 a^{-6} +z^6 a^{-8} +z^5 a^{-7} +z^5 a^{-9} -6 z^4 a^{-6} -5 z^4 a^{-8} +z^4 a^{-10} -4 z^3 a^{-7} -3 z^3 a^{-9} +z^3 a^{-11} +10 z^2 a^{-6} +7 z^2 a^{-8} -2 z^2 a^{-10} +z^2 a^{-12} +3 z a^{-7} +z a^{-9} -z a^{-11} +z a^{-13} -4 a^{-6} -3 a^{-8} }[/math]
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"Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial:
{7_1,}
Same Jones Polynomial (up to mirroring, [math]\displaystyle{ q\leftrightarrow q^{-1} }[/math]):
{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]:=
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AppendTo[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
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In[3]:=
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K = Knot["T(7,2)"];
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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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{ [math]\displaystyle{ t^3-t^2+t-1+ t^{-1} - t^{-2} + t^{-3} }[/math], [math]\displaystyle{ -q^{10}+q^9-q^8+q^7-q^6+q^5+q^3 }[/math] }
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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(7,2)/V 2,1
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Data:T(7,2)/V 3,1
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Data:T(7,2)/V 4,1
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Data:T(7,2)/V 4,2
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Data:T(7,2)/V 4,3
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Data:T(7,2)/V 5,1
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Data:T(7,2)/V 5,2
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Data:T(7,2)/V 5,3
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Data:T(7,2)/V 5,4
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Data:T(7,2)/V 6,1
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Data:T(7,2)/V 6,2
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Data:T(7,2)/V 6,3
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Data:T(7,2)/V 6,4
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Data:T(7,2)/V 6,5
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Data:T(7,2)/V 6,6
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Data:T(7,2)/V 6,7
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Data:T(7,2)/V 6,8
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Data:T(7,2)/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 [math]\displaystyle{ t^rq^j }[/math] are shown, along with their alternating sums [math]\displaystyle{ \chi }[/math] (fixed [math]\displaystyle{ j }[/math], alternation over [math]\displaystyle{ r }[/math]). The squares with yellow highlighting are those on the "critical diagonals", where [math]\displaystyle{ j-2r=s+1 }[/math] or [math]\displaystyle{ j-2r=s-1 }[/math], where [math]\displaystyle{ s= }[/math]6 is the signature of T(7,2). 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 | 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 |
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| Integral Khovanov Homology
(db, data source)
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| [math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math]
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[math]\displaystyle{ i=5 }[/math]
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[math]\displaystyle{ i=7 }[/math]
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| [math]\displaystyle{ r=0 }[/math]
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[math]\displaystyle{ {\mathbb Z} }[/math]
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[math]\displaystyle{ {\mathbb Z} }[/math]
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| [math]\displaystyle{ r=1 }[/math]
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| [math]\displaystyle{ r=2 }[/math]
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[math]\displaystyle{ {\mathbb Z} }[/math]
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| [math]\displaystyle{ r=3 }[/math]
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[math]\displaystyle{ {\mathbb Z}_2 }[/math]
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[math]\displaystyle{ {\mathbb Z} }[/math]
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| [math]\displaystyle{ r=4 }[/math]
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[math]\displaystyle{ {\mathbb Z} }[/math]
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| [math]\displaystyle{ r=5 }[/math]
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[math]\displaystyle{ {\mathbb Z}_2 }[/math]
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[math]\displaystyle{ {\mathbb Z} }[/math]
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| [math]\displaystyle{ r=6 }[/math]
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[math]\displaystyle{ {\mathbb Z} }[/math]
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| [math]\displaystyle{ r=7 }[/math]
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[math]\displaystyle{ {\mathbb Z}_2 }[/math]
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[math]\displaystyle{ {\mathbb Z} }[/math]
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