10 48: Difference between revisions
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{{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>-9</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>-9</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>-11</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>-11</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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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 48's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)
Visit 10 48's page at Knotilus! Visit 10 48's page at the original Knot Atlas! |
Knot presentations
Planar diagram presentation | X6271 X8493 X14,6,15,5 X20,15,1,16 X16,9,17,10 X18,11,19,12 X10,17,11,18 X12,19,13,20 X2837 X4,14,5,13 |
Gauss code | 1, -9, 2, -10, 3, -1, 9, -2, 5, -7, 6, -8, 10, -3, 4, -5, 7, -6, 8, -4 |
Dowker-Thistlethwaite code | 6 8 14 2 16 18 4 20 10 12 |
Conway Notation | [41,3,2] |
Three dimensional invariants
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Four dimensional invariants
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Polynomial invariants
Vassiliev invariants
V2 and V3: | (4, 0) |
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 48. 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, 48]] |
Out[2]= | 10 |
In[3]:= | PD[Knot[10, 48]] |
Out[3]= | PD[X[6, 2, 7, 1], X[8, 4, 9, 3], X[14, 6, 15, 5], X[20, 15, 1, 16],X[16, 9, 17, 10], X[18, 11, 19, 12], X[10, 17, 11, 18],X[12, 19, 13, 20], X[2, 8, 3, 7], X[4, 14, 5, 13]] |
In[4]:= | GaussCode[Knot[10, 48]] |
Out[4]= | GaussCode[1, -9, 2, -10, 3, -1, 9, -2, 5, -7, 6, -8, 10, -3, 4, -5, 7, -6, 8, -4] |
In[5]:= | BR[Knot[10, 48]] |
Out[5]= | BR[3, {-1, -1, -1, -1, 2, 2, -1, 2, 2, 2}] |
In[6]:= | alex = Alexander[Knot[10, 48]][t] |
Out[6]= | -4 3 6 9 2 3 4 |
In[7]:= | Conway[Knot[10, 48]][z] |
Out[7]= | 2 4 6 8 1 + 4 z + 8 z + 5 z + z |
In[8]:= | Select[AllKnots[], (alex === Alexander[#][t])&] |
Out[8]= | {Knot[10, 48]} |
In[9]:= | {KnotDet[Knot[10, 48]], KnotSignature[Knot[10, 48]]} |
Out[9]= | {49, 0} |
In[10]:= | J=Jones[Knot[10, 48]][q] |
Out[10]= | -5 2 4 6 7 2 3 4 5 |
In[11]:= | Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&] |
Out[11]= | {Knot[10, 48]} |
In[12]:= | A2Invariant[Knot[10, 48]][q] |
Out[12]= | -14 2 4 2 10 14 |
In[13]:= | Kauffman[Knot[10, 48]][a, z] |
Out[13]= | 2 24 2 z 3 z 9 z 5 2 z 13 z |
In[14]:= | {Vassiliev[2][Knot[10, 48]], Vassiliev[3][Knot[10, 48]]} |
Out[14]= | {0, 0} |
In[15]:= | Kh[Knot[10, 48]][q, t] |
Out[15]= | 5 1 1 1 3 1 3 3 |