 L11n28 |
 L11n30 |
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 DrawMorseLink |
| PD Presentation: | X6172 X18,7,19,8 X4,19,1,20 X5,12,6,13 X3849 X9,16,10,17 X13,22,14,5 X15,10,16,11 X21,14,22,15 X11,20,12,21 X17,2,18,3 |
| Gauss Code: | {{1, 11, -5, -3}, {-4, -1, 2, 5, -6, 8, -10, 4, -7, 9, -8, 6, -11, -2, 3, 10, -9, 7}} |
| Jones Polynomial: | 2q-23/2 - 3q-21/2 + 5q-19/2 - 7q-17/2 + 7q-15/2 - 8q-13/2 + 5q-11/2 - 4q-9/2 + 2q-7/2 - q-5/2 |
| A2 (sl(3)) Invariant: | - q-40 - q-38 - 2q-36 - 2q-34 - q-30 + 2q-28 + 2q-26 + 2q-24 + 4q-22 + q-20 + 3q-18 + q-16 + q-12 - q-10 + q-8 |
| HOMFLY-PT Polynomial: | - a5z - 3a5z3 - a5z5 - 2a7z-1 - 6a7z - 7a7z3 - 2a7z5 + 2a9z-1 + 2a9z - 2a9z3 - a9z5 + a11z-1 + 3a11z + a11z3 - a13z-1 |
| Kauffman Polynomial: | - a5z + 3a5z3 - a5z5 - a6z2 + 5a6z4 - 2a6z6 - 2a7z-1 + 8a7z - 12a7z3 + 10a7z5 - 3a7z7 + 2a8 - 5a8z2 + 4a8z6 - 2a8z8 - 2a9z-1 + 11a9z - 19a9z3 + 8a9z5 - a9z9 - 4a10 + 14a10z2 - 25a10z4 + 15a10z6 - 4a10z8 + a11z-1 + 2a11z - 3a11z3 - 2a11z5 + 2a11z7 - a11z9 - 9a12 + 26a12z2 - 23a12z4 + 9a12z6 - 2a12z8 + a13z-1 + a13z3 + a13z5 - a13z7 - 4a14 + 8a14z2 - 3a14z4 |
| 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]= | 2 |
In[3]:= | Show[DrawMorseLink[Link[11, NonAlternating, 29]]] |
| |
Out[3]= | -Graphics- |
In[4]:= | PD[L = Link[11, NonAlternating, 29]] |
Out[4]= | PD[X[6, 1, 7, 2], X[18, 7, 19, 8], X[4, 19, 1, 20], X[5, 12, 6, 13], > X[3, 8, 4, 9], X[9, 16, 10, 17], X[13, 22, 14, 5], X[15, 10, 16, 11], > X[21, 14, 22, 15], X[11, 20, 12, 21], X[17, 2, 18, 3]] |
In[5]:= | GaussCode[L] |
Out[5]= | GaussCode[{1, 11, -5, -3}, {-4, -1, 2, 5, -6, 8, -10, 4, -7, 9, -8, 6, -11, -2,
> 3, 10, -9, 7}] |
In[6]:= | Jones[L][q] |
Out[6]= | 2 3 5 7 7 8 5 4 2 -(5/2) ----- - ----- + ----- - ----- + ----- - ----- + ----- - ---- + ---- - q 23/2 21/2 19/2 17/2 15/2 13/2 11/2 9/2 7/2 q q q q q q q q q |
In[7]:= | A2Invariant[L][q] |
Out[7]= | -40 -38 2 2 -30 2 2 2 4 -20 3 -16
-q - q - --- - --- - q + --- + --- + --- + --- + q + --- + q +
36 34 28 26 24 22 18
q q q q q q q
-12 -10 -8
> q - q + q |
In[8]:= | HOMFLYPT[Link[11, NonAlternating, 29]][a, z] |
Out[8]= | 7 9 11 13
-2 a 2 a a a 5 7 9 11 5 3
----- + ---- + --- - --- - a z - 6 a z + 2 a z + 3 a z - 3 a z -
z z z z
7 3 9 3 11 3 5 5 7 5 9 5
> 7 a z - 2 a z + a z - a z - 2 a z - a z |
In[9]:= | Kauffman[Link[11, NonAlternating, 29]][a, z] |
Out[9]= | 7 9 11 13
8 10 12 14 2 a 2 a a a 5 7
2 a - 4 a - 9 a - 4 a - ---- - ---- + --- + --- - a z + 8 a z +
z z z z
9 11 6 2 8 2 10 2 12 2 14 2
> 11 a z + 2 a z - a z - 5 a z + 14 a z + 26 a z + 8 a z +
5 3 7 3 9 3 11 3 13 3 6 4 10 4
> 3 a z - 12 a z - 19 a z - 3 a z + a z + 5 a z - 25 a z -
12 4 14 4 5 5 7 5 9 5 11 5 13 5
> 23 a z - 3 a z - a z + 10 a z + 8 a z - 2 a z + a z -
6 6 8 6 10 6 12 6 7 7 11 7 13 7
> 2 a z + 4 a z + 15 a z + 9 a z - 3 a z + 2 a z - a z -
8 8 10 8 12 8 9 9 11 9
> 2 a z - 4 a z - 2 a z - a z - a z |
In[10]:= | Kh[L][q, t] |
Out[10]= | -6 -4 2 1 2 4 1 3 4
q + q + ------ + ------ + ------ + ------ + ------ + ------ + ------ +
24 9 22 8 20 8 20 7 18 7 18 6 16 6
q t q t q t q t q t q t q t
4 3 4 5 2 3 2 2 2
> ------ + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----
16 5 14 5 14 4 12 4 12 3 10 3 10 2 8 2 6
q t q t q t q t q t q t q t q t q t |
| Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n29 |
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