8 15
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Visit 8 15's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)
Visit 8 15's page at Knotilus! Visit 8 15's page at the original Knot Atlas! |
Two trefoil knots along a closed loop, mutually interlinked. (See also 10 120.) |
Knot presentations
Planar diagram presentation | X1425 X3849 X5,12,6,13 X13,16,14,1 X9,14,10,15 X15,10,16,11 X11,6,12,7 X7283 |
Gauss code | -1, 8, -2, 1, -3, 7, -8, 2, -5, 6, -7, 3, -4, 5, -6, 4 |
Dowker-Thistlethwaite code | 4 8 12 2 14 6 16 10 |
Conway Notation | [21,21,2] |
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["8 15"];
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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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{ 33, -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: | (4, -7) |
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 8 15. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.
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-8 | -7 | -6 | -5 | -4 | -3 | -2 | -1 | 0 | χ | |||||||||
-3 | 1 | 1 | |||||||||||||||||
-5 | 2 | 1 | -1 | ||||||||||||||||
-7 | 3 | 3 | |||||||||||||||||
-9 | 2 | 2 | 0 | ||||||||||||||||
-11 | 4 | 3 | 1 | ||||||||||||||||
-13 | 2 | 2 | 0 | ||||||||||||||||
-15 | 2 | 4 | -2 | ||||||||||||||||
-17 | 1 | 2 | 1 | ||||||||||||||||
-19 | 2 | -2 | |||||||||||||||||
-21 | 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[8, 15]] |
Out[2]= | 8 |
In[3]:= | PD[Knot[8, 15]] |
Out[3]= | PD[X[1, 4, 2, 5], X[3, 8, 4, 9], X[5, 12, 6, 13], X[13, 16, 14, 1], X[9, 14, 10, 15], X[15, 10, 16, 11], X[11, 6, 12, 7], X[7, 2, 8, 3]] |
In[4]:= | GaussCode[Knot[8, 15]] |
Out[4]= | GaussCode[-1, 8, -2, 1, -3, 7, -8, 2, -5, 6, -7, 3, -4, 5, -6, 4] |
In[5]:= | BR[Knot[8, 15]] |
Out[5]= | BR[4, {-1, -1, 2, -1, -3, -2, -2, -2, -3}] |
In[6]:= | alex = Alexander[Knot[8, 15]][t] |
Out[6]= | 3 8 2 |
In[7]:= | Conway[Knot[8, 15]][z] |
Out[7]= | 2 4 1 + 4 z + 3 z |
In[8]:= | Select[AllKnots[], (alex === Alexander[#][t])&] |
Out[8]= | {Knot[8, 15], Knot[11, NonAlternating, 65]} |
In[9]:= | {KnotDet[Knot[8, 15]], KnotSignature[Knot[8, 15]]} |
Out[9]= | {33, -4} |
In[10]:= | J=Jones[Knot[8, 15]][q] |
Out[10]= | -10 3 4 6 6 5 5 2 -2 |
In[11]:= | Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&] |
Out[11]= | {Knot[8, 15]} |
In[12]:= | A2Invariant[Knot[8, 15]][q] |
Out[12]= | -32 -30 2 -26 2 2 -20 3 -14 -12 2 |
In[13]:= | Kauffman[Knot[8, 15]][a, z] |
Out[13]= | 4 6 8 10 7 9 11 4 2 |
In[14]:= | {Vassiliev[2][Knot[8, 15]], Vassiliev[3][Knot[8, 15]]} |
Out[14]= | {0, -7} |
In[15]:= | Kh[Knot[8, 15]][q, t] |
Out[15]= | -5 -3 1 2 1 2 2 4 |