9 49
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Visit 9 49's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)
Visit 9 49's page at Knotilus! Visit 9 49's page at the original Knot Atlas! |
9 49 Quick Notes |
Knot presentations
Planar diagram presentation | X6271 X12,8,13,7 X5,15,6,14 X3,11,4,10 X11,3,12,2 X15,5,16,4 X17,9,18,8 X9,17,10,16 X18,14,1,13 |
Gauss code | 1, 5, -4, 6, -3, -1, 2, 7, -8, 4, -5, -2, 9, 3, -6, 8, -7, -9 |
Dowker-Thistlethwaite code | 6 -10 -14 12 -16 -2 18 -4 -8 |
Conway Notation | [-20:-20:-20] |
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 |
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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["9 49"];
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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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{ 25, 4 } |
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: | (6, 14) |
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 4 is the signature of 9 49. 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 | |||||||||
19 | 2 | -2 | ||||||||||||||||
17 | 1 | 1 | ||||||||||||||||
15 | 3 | 2 | -1 | |||||||||||||||
13 | 2 | 1 | 1 | |||||||||||||||
11 | 2 | 3 | 1 | |||||||||||||||
9 | 2 | 2 | 0 | |||||||||||||||
7 | 2 | 2 | ||||||||||||||||
5 | 1 | 2 | -1 | |||||||||||||||
3 | 1 | 1 |
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[9, 49]] |
Out[2]= | 9 |
In[3]:= | PD[Knot[9, 49]] |
Out[3]= | PD[X[6, 2, 7, 1], X[12, 8, 13, 7], X[5, 15, 6, 14], X[3, 11, 4, 10],X[11, 3, 12, 2], X[15, 5, 16, 4], X[17, 9, 18, 8], X[9, 17, 10, 16],X[18, 14, 1, 13]] |
In[4]:= | GaussCode[Knot[9, 49]] |
Out[4]= | GaussCode[1, 5, -4, 6, -3, -1, 2, 7, -8, 4, -5, -2, 9, 3, -6, 8, -7, -9] |
In[5]:= | BR[Knot[9, 49]] |
Out[5]= | BR[4, {1, 1, 2, 1, 1, -3, 2, -1, 2, 3, 3}] |
In[6]:= | alex = Alexander[Knot[9, 49]][t] |
Out[6]= | 3 6 2 |
In[7]:= | Conway[Knot[9, 49]][z] |
Out[7]= | 2 4 1 + 6 z + 3 z |
In[8]:= | Select[AllKnots[], (alex === Alexander[#][t])&] |
Out[8]= | {Knot[9, 49]} |
In[9]:= | {KnotDet[Knot[9, 49]], KnotSignature[Knot[9, 49]]} |
Out[9]= | {25, 4} |
In[10]:= | J=Jones[Knot[9, 49]][q] |
Out[10]= | 2 3 4 5 6 7 8 9 q - 2 q + 4 q - 4 q + 5 q - 4 q + 3 q - 2 q |
In[11]:= | Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&] |
Out[11]= | {Knot[9, 49]} |
In[12]:= | A2Invariant[Knot[9, 49]][q] |
Out[12]= | 6 8 10 14 16 18 20 22 24 26 28 q - q + q + q + 3 q + q + 2 q - q - q - q - 2 q |
In[13]:= | Kauffman[Knot[9, 49]][a, z] |
Out[13]= | 2 2 2 2 3 3 |
In[14]:= | {Vassiliev[2][Knot[9, 49]], Vassiliev[3][Knot[9, 49]]} |
Out[14]= | {0, 14} |
In[15]:= | Kh[Knot[9, 49]][q, t] |
Out[15]= | 3 5 5 7 2 9 2 9 3 11 3 11 4 |