10 22: Difference between revisions
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{{Knot Navigation Links|ext=gif}} |
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{{Vassiliev Invariants}} |
{{Vassiliev Invariants}} |
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{{Khovanov Homology|table=<table border=1> |
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The coefficients of the monomials <math>t^rq^j</math> are shown, along with their alternating sums <math>\chi</math> (fixed <math>j</math>, alternation over <math>r</math>). The squares with <font class=HLYellow>yellow</font> highlighting are those on the "critical diagonals", where <math>j-2r=s+1</math> or <math>j-2r=s+1</math>, where <math>s=</math>{{Data:{{PAGENAME}}/Signature}} is the signature of {{PAGENAME}}. Nonzero entries off the critical diagonals (if any exist) are highlighted in <font class=HLRed>red</font>. |
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<tr align=center><td>-7</td><td bgcolor=yellow> </td><td bgcolor=yellow>1</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>-1</td></tr> |
<tr align=center><td>-7</td><td bgcolor=yellow> </td><td bgcolor=yellow>1</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>-1</td></tr> |
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<tr align=center><td>-9</td><td bgcolor=yellow>1</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>1</td></tr> |
<tr align=center><td>-9</td><td bgcolor=yellow>1</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>1</td></tr> |
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{{Computer Talk Header}} |
{{Computer Talk Header}} |
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q t + q t + q t</nowiki></pre></td></tr> |
q t + q t + q t</nowiki></pre></td></tr> |
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[[Category:Knot Page]] |
Revision as of 19:07, 28 August 2005
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Visit 10 22's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)
Visit 10 22's page at Knotilus! Visit 10 22's page at the original Knot Atlas! |
Knot presentations
Planar diagram presentation | X6271 X16,12,17,11 X12,3,13,4 X2,15,3,16 X14,5,15,6 X18,8,19,7 X20,10,1,9 X8,20,9,19 X4,13,5,14 X10,18,11,17 |
Gauss code | 1, -4, 3, -9, 5, -1, 6, -8, 7, -10, 2, -3, 9, -5, 4, -2, 10, -6, 8, -7 |
Dowker-Thistlethwaite code | 6 12 14 18 20 16 4 2 10 8 |
Conway Notation | [3313] |
Three dimensional invariants
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Four dimensional invariants
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Polynomial invariants
A1 Invariants.
Weight | Invariant |
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1 | |
2 | |
3 | |
4 | |
5 |
A2 Invariants.
Weight | Invariant |
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1,0 | |
1,1 | |
2,0 |
A3 Invariants.
Weight | Invariant |
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0,1,0 | |
1,0,0 |
A4 Invariants.
Weight | Invariant |
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0,1,0,0 | |
1,0,0,0 |
B2 Invariants.
Weight | Invariant |
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0,1 | |
1,0 |
D4 Invariants.
Weight | Invariant |
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1,0,0,0 |
G2 Invariants.
Weight | Invariant |
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1,0 |
.
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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Loading KnotTheory` version of August 31, 2006, 11:25:27.5625.
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In[3]:=
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K = Knot["10 22"];
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In[4]:=
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Alexander[K][t]
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KnotTheory::loading: Loading precomputed data in PD4Knots`.
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Out[4]=
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In[5]:=
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Conway[K][z]
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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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Out[7]=
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{ 49, 0 } |
In[8]:=
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Jones[K][q]
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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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Vassiliev invariants
V2 and V3: | (-4, -2) |
V2,1 through 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.
Khovanov Homology
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 22. Nonzero entries off the critical diagonals (if any exist) are highlighted in red. |
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Integral Khovanov Homology
(db, data source) |
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Computer Talk
Much of the above data can be recomputed by Mathematica using the package KnotTheory`
. See A Sample KnotTheory` Session.
In[1]:= |
<< KnotTheory` |
Loading KnotTheory` (version of August 17, 2005, 14:44:34)... | |
In[2]:= | Crossings[Knot[10, 22]] |
Out[2]= | 10 |
In[3]:= | PD[Knot[10, 22]] |
Out[3]= | PD[X[6, 2, 7, 1], X[16, 12, 17, 11], X[12, 3, 13, 4], X[2, 15, 3, 16],X[14, 5, 15, 6], X[18, 8, 19, 7], X[20, 10, 1, 9], X[8, 20, 9, 19],X[4, 13, 5, 14], X[10, 18, 11, 17]] |
In[4]:= | GaussCode[Knot[10, 22]] |
Out[4]= | GaussCode[1, -4, 3, -9, 5, -1, 6, -8, 7, -10, 2, -3, 9, -5, 4, -2, 10, -6, 8, -7] |
In[5]:= | BR[Knot[10, 22]] |
Out[5]= | BR[4, {1, 1, 1, 1, 2, -1, -3, 2, -3, -3, -3}] |
In[6]:= | alex = Alexander[Knot[10, 22]][t] |
Out[6]= | 2 6 10 2 3 |
In[7]:= | Conway[Knot[10, 22]][z] |
Out[7]= | 2 4 6 1 - 4 z - 6 z - 2 z |
In[8]:= | Select[AllKnots[], (alex === Alexander[#][t])&] |
Out[8]= | {Knot[10, 22]} |
In[9]:= | {KnotDet[Knot[10, 22]], KnotSignature[Knot[10, 22]]} |
Out[9]= | {49, 0} |
In[10]:= | J=Jones[Knot[10, 22]][q] |
Out[10]= | -4 2 4 6 2 3 4 5 6 |
In[11]:= | Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&] |
Out[11]= | {Knot[10, 22], Knot[10, 35]} |
In[12]:= | A2Invariant[Knot[10, 22]][q] |
Out[12]= | -12 -8 -6 -4 2 4 6 8 10 12 14 |
In[13]:= | Kauffman[Knot[10, 22]][a, z] |
Out[13]= | 2 2 22 2 2 z z z 2 4 z 6 z 12 z |
In[14]:= | {Vassiliev[2][Knot[10, 22]], Vassiliev[3][Knot[10, 22]]} |
Out[14]= | {0, -2} |
In[15]:= | Kh[Knot[10, 22]][q, t] |
Out[15]= | 5 1 1 1 3 1 3 3 |