 L10a130 |
 L10a132 |
| © | Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: |
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 Knotscape |
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The 3-Component Link L10a131Visit L10a131's page at Knotilus! |
 DrawMorseLink |
| PD Presentation: | X6172 X12,4,13,3 X20,16,11,15 X14,8,15,7 X10,12,5,11 X8,18,9,17 X18,10,19,9 X16,20,17,19 X2536 X4,14,1,13 |
| Gauss Code: | {{1, -9, 2, -10}, {9, -1, 4, -6, 7, -5}, {5, -2, 10, -4, 3, -8, 6, -7, 8, -3}} |
| Jones Polynomial: | 1 - 2q + 6q2 - 7q3 + 11q4 - 11q5 + 11q6 - 9q7 + 6q8 - 3q9 + q10 |
| A2 (sl(3)) Invariant: | 1 + 2q4 + 4q6 + 2q8 + 7q10 + 3q12 + 4q14 + 3q16 - q18 + 2q20 - 2q22 + q24 + q26 - q28 + q30 |
| HOMFLY-PT Polynomial: | a-8 + 2a-8z2 + a-8z4 + a-6z-2 - a-6 - 3a-6z2 - 3a-6z4 - a-6z6 - 2a-4z-2 - 3a-4 - 3a-4z2 - 3a-4z4 - a-4z6 + a-2z-2 + 3a-2 + 3a-2z2 + a-2z4 |
| Kauffman Polynomial: | - a-12z2 + a-12z4 - 3a-11z3 + 3a-11z5 - a-10 + 3a-10z2 - 6a-10z4 + 5a-10z6 - 2a-9z + 9a-9z3 - 10a-9z5 + 6a-9z7 - a-8 + 4a-8z2 - 2a-8z4 - 3a-8z6 + 4a-8z8 - 6a-7z + 19a-7z3 - 22a-7z5 + 8a-7z7 + a-7z9 + a-6z-2 - 3a-6 + 6a-6z2 - a-6z4 - 10a-6z6 + 6a-6z8 - 2a-5z-1 + 8a-5z3 - 14a-5z5 + 4a-5z7 + a-5z9 + 2a-4z-2 - 6a-4 + 12a-4z2 - 10a-4z4 - a-4z6 + 2a-4z8 - 2a-3z-1 + 4a-3z + a-3z3 - 5a-3z5 + 2a-3z7 + a-2z-2 - 4a-2 + 6a-2z2 - 4a-2z4 + a-2z6 |
| Khovanov Homology: |
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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]:= | Length[Skeleton[L]] |
Out[2]= | 3 |
In[3]:= | Show[DrawMorseLink[Link[10, Alternating, 131]]] |
| |
Out[3]= | -Graphics- |
In[4]:= | PD[L = Link[10, Alternating, 131]] |
Out[4]= | PD[X[6, 1, 7, 2], X[12, 4, 13, 3], X[20, 16, 11, 15], X[14, 8, 15, 7], > X[10, 12, 5, 11], X[8, 18, 9, 17], X[18, 10, 19, 9], X[16, 20, 17, 19], > X[2, 5, 3, 6], X[4, 14, 1, 13]] |
In[5]:= | GaussCode[L] |
Out[5]= | GaussCode[{1, -9, 2, -10}, {9, -1, 4, -6, 7, -5},
> {5, -2, 10, -4, 3, -8, 6, -7, 8, -3}] |
In[6]:= | Jones[L][q] |
Out[6]= | 2 3 4 5 6 7 8 9 10 1 - 2 q + 6 q - 7 q + 11 q - 11 q + 11 q - 9 q + 6 q - 3 q + q |
In[7]:= | A2Invariant[L][q] |
Out[7]= | 4 6 8 10 12 14 16 18 20 22
1 + 2 q + 4 q + 2 q + 7 q + 3 q + 4 q + 3 q - q + 2 q - 2 q +
24 26 28 30
> q + q - q + q |
In[8]:= | HOMFLYPT[Link[10, Alternating, 131]][a, z] |
Out[8]= | 2 2 2 2 4
-8 -6 3 3 1 2 1 2 z 3 z 3 z 3 z z
a - a - -- + -- + ----- - ----- + ----- + ---- - ---- - ---- + ---- + -- -
4 2 6 2 4 2 2 2 8 6 4 2 8
a a a z a z a z a a a a a
4 4 4 6 6
3 z 3 z z z z
> ---- - ---- + -- - -- - --
6 4 2 6 4
a a a a a |
In[9]:= | Kauffman[Link[10, Alternating, 131]][a, z] |
Out[9]= | -10 -8 3 6 4 1 2 1 2 2 2 z 6 z
-a - a - -- - -- - -- + ----- + ----- + ----- - ---- - ---- - --- - --- +
6 4 2 6 2 4 2 2 2 5 3 9 7
a a a a z a z a z a z a z a a
2 2 2 2 2 2 3 3 3
4 z z 3 z 4 z 6 z 12 z 6 z 3 z 9 z 19 z
> --- - --- + ---- + ---- + ---- + ----- + ---- - ---- + ---- + ----- +
3 12 10 8 6 4 2 11 9 7
a a a a a a a a a a
3 3 4 4 4 4 4 4 5 5 5
8 z z z 6 z 2 z z 10 z 4 z 3 z 10 z 22 z
> ---- + -- + --- - ---- - ---- - -- - ----- - ---- + ---- - ----- - ----- -
5 3 12 10 8 6 4 2 11 9 7
a a a a a a a a a a a
5 5 6 6 6 6 6 7 7 7 7
14 z 5 z 5 z 3 z 10 z z z 6 z 8 z 4 z 2 z
> ----- - ---- + ---- - ---- - ----- - -- + -- + ---- + ---- + ---- + ---- +
5 3 10 8 6 4 2 9 7 5 3
a a a a a a a a a a a
8 8 8 9 9
4 z 6 z 2 z z z
> ---- + ---- + ---- + -- + --
8 6 4 7 5
a a a a a |
In[10]:= | Kh[L][q, t] |
Out[10]= | 3
3 5 1 q q 5 7 7 2 9 2 9 3
5 q + 3 q + ---- + - + -- + 4 q t + 3 q t + 7 q t + 5 q t + 5 q t +
2 t t
q t
11 3 11 4 13 4 13 5 15 5 15 6
> 6 q t + 6 q t + 5 q t + 3 q t + 6 q t + 3 q t +
17 6 17 7 19 7 21 8
> 4 q t + q t + 2 q t + q t |
| Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L10a131 |
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