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The 7-Crossing Torus Knot T(7,2)Visit T(7,2)'s page at Knotilus! |
PD Presentation: | 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 |
Gauss Code: | {-4, 5, -6, 7, -1, 2, -3, 4, -5, 6, -7, 1, -2, 3} |
Braid Representative: |
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Alexander Polynomial: | t-3 - t-2 + t-1 - 1 + t - t2 + t3 |
Conway Polynomial: | 1 + 6z2 + 5z4 + z6 |
Other knots with the same Alexander/Conway Polynomial: | {71, ...} |
Determinant and Signature: | {7, 6} |
Jones Polynomial: | q3 + q5 - q6 + q7 - q8 + q9 - q10 |
Other knots (up to mirrors) with the same Jones Polynomial: | {71, ...} |
Coloured Jones Polynomial (in the 3-dimensional representation of sl(2); n=2): | q6 + q9 - q11 + q12 - q14 + q15 - q17 + q18 - q20 - q23 + q24 - q26 + q27 |
A2 (sl(3)) Invariant: | q10 + q12 + 2q14 + q16 + q18 - q26 - q28 - q30 |
Kauffman Polynomial: | a-13z + a-12z2 - a-11z + a-11z3 - 2a-10z2 + a-10z4 + a-9z - 3a-9z3 + a-9z5 - 3a-8 + 7a-8z2 - 5a-8z4 + a-8z6 + 3a-7z - 4a-7z3 + a-7z5 - 4a-6 + 10a-6z2 - 6a-6z4 + a-6z6 |
V2 and V3, the type 2 and 3 Vassiliev invariants: | {6, 14} |
Khovanov Homology.
The coefficients of the monomials trqj
are shown, along with their alternating sums χ (fixed j,
alternation over r).
The squares with yellow highlighting
are those on the "critical diagonals", where j-2r=s+1 or
j-2r=s+1, where s=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 |
Computer Talk. The data above can be recomputed by Mathematica using the package KnotTheory`. Following setup, the sample Mathematica session below reproduces most of the above data (Mathematica system prompts in blue, human input in red, Mathematica output in black):
In[1]:= |
<< KnotTheory` |
Loading KnotTheory` (version of August 30, 2005, 10:15:35)... | |
In[2]:= | TubePlot[TorusKnot[7, 2]] |
![]() | |
Out[2]= | -Graphics- |
In[3]:= | Crossings[TorusKnot[7, 2]] |
Out[3]= | 7 |
In[4]:= | PD[TorusKnot[7, 2]] |
Out[4]= | PD[X[5, 13, 6, 12], X[13, 7, 14, 6], X[7, 1, 8, 14], X[1, 9, 2, 8], > X[9, 3, 10, 2], X[3, 11, 4, 10], X[11, 5, 12, 4]] |
In[5]:= | GaussCode[TorusKnot[7, 2]] |
Out[5]= | GaussCode[-4, 5, -6, 7, -1, 2, -3, 4, -5, 6, -7, 1, -2, 3] |
In[6]:= | BR[TorusKnot[7, 2]] |
Out[6]= | BR[2, {1, 1, 1, 1, 1, 1, 1}] |
In[7]:= | alex = Alexander[TorusKnot[7, 2]][t] |
Out[7]= | -3 -2 1 2 3 -1 + t - t + - + t - t + t t |
In[8]:= | Conway[TorusKnot[7, 2]][z] |
Out[8]= | 2 4 6 1 + 6 z + 5 z + z |
In[9]:= | Select[AllKnots[], (alex === Alexander[#][t])&] |
Out[9]= | {Knot[7, 1]} |
In[10]:= | {KnotDet[TorusKnot[7, 2]], KnotSignature[TorusKnot[7, 2]]} |
Out[10]= | {7, 6} |
In[11]:= | J=Jones[TorusKnot[7, 2]][q] |
Out[11]= | 3 5 6 7 8 9 10 q + q - q + q - q + q - q |
In[12]:= | Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&] |
Out[12]= | {Knot[7, 1]} |
In[13]:= | ColouredJones[TorusKnot[7, 2], 2][q] |
Out[13]= | 6 9 11 12 14 15 17 18 20 23 24 26 27 q + q - q + q - q + q - q + q - q - q + q - q + q |
In[14]:= | A2Invariant[TorusKnot[7, 2]][q] |
Out[14]= | 10 12 14 16 18 26 28 30 q + q + 2 q + q + q - q - q - q |
In[15]:= | Kauffman[TorusKnot[7, 2]][a, z] |
Out[15]= | 2 2 2 2 3 3 -3 4 z z z 3 z z 2 z 7 z 10 z z 3 z -- - -- + --- - --- + -- + --- + --- - ---- + ---- + ----- + --- - ---- - 8 6 13 11 9 7 12 10 8 6 11 9 a a a a a a a a a a a a 3 4 4 4 5 5 6 6 4 z z 5 z 6 z z z z z > ---- + --- - ---- - ---- + -- + -- + -- + -- 7 10 8 6 9 7 8 6 a a a a a a a a |
In[16]:= | {Vassiliev[2][TorusKnot[7, 2]], Vassiliev[3][TorusKnot[7, 2]]} |
Out[16]= | {6, 14} |
In[17]:= | Kh[TorusKnot[7, 2]][q, t] |
Out[17]= | 5 7 9 2 13 3 13 4 17 5 17 6 21 7 q + q + q t + q t + q t + q t + q t + q t |
Dror Bar-Natan: The Knot Atlas: Torus Knots: The Torus Knot T(7,2) |
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