 L11n39 |
 L11n41 |
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 DrawMorseLink |
| PD Presentation: | X6172 X12,7,13,8 X4,13,1,14 X9,18,10,19 X3849 X5,14,6,15 X15,20,16,21 X17,22,18,5 X21,16,22,17 X19,10,20,11 X11,2,12,3 |
| Gauss Code: | {{1, 11, -5, -3}, {-6, -1, 2, 5, -4, 10, -11, -2, 3, 6, -7, 9, -8, 4, -10, 7, -9, 8}} |
| Jones Polynomial: | 2q-25/2 - 4q-23/2 + 6q-21/2 - 7q-19/2 + 7q-17/2 - 7q-15/2 + 5q-13/2 - 4q-11/2 + q-9/2 - q-7/2 |
| A2 (sl(3)) Invariant: | - q-42 - 2q-38 - q-36 - q-34 - q-32 + 2q-30 - q-28 + 3q-26 + q-24 + 2q-22 + 3q-20 + q-18 + 3q-16 + q-12 |
| HOMFLY-PT Polynomial: | - 2a7z-1 - 9a7z - 12a7z3 - 6a7z5 - a7z7 + 2a9z-1 + 4a9z - 2a9z3 - 4a9z5 - a9z7 + a11z-1 + 4a11z + 4a11z3 + a11z5 - a13z-1 - a13z |
| Kauffman Polynomial: | - 2a7z-1 + 9a7z - 12a7z3 + 6a7z5 - a7z7 + 3a8 - 7a8z2 + a8z4 + 3a8z6 - a8z8 - 2a9z-1 + 7a9z - 10a9z3 + 4a9z5 + 2a9z7 - a9z9 + a10z2 - 9a10z4 + 13a10z6 - 4a10z8 + a11z-1 - 4a11z + a11z3 + 7a11z5 - a11z7 - a11z9 - 3a12 + 9a12z2 - 7a12z4 + 7a12z6 - 3a12z8 + a13z-1 - a13z - 4a13z3 + 8a13z5 - 4a13z7 - 2a14z2 + 3a14z4 - 3a14z6 + a15z - 3a15z3 - a15z5 + a16 - 3a16z2 |
| Khovanov Homology: |
|
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, 40]]] |
| |
Out[3]= | -Graphics- |
In[4]:= | PD[L = Link[11, NonAlternating, 40]] |
Out[4]= | PD[X[6, 1, 7, 2], X[12, 7, 13, 8], X[4, 13, 1, 14], X[9, 18, 10, 19], > X[3, 8, 4, 9], X[5, 14, 6, 15], X[15, 20, 16, 21], X[17, 22, 18, 5], > X[21, 16, 22, 17], X[19, 10, 20, 11], X[11, 2, 12, 3]] |
In[5]:= | GaussCode[L] |
Out[5]= | GaussCode[{1, 11, -5, -3}, {-6, -1, 2, 5, -4, 10, -11, -2, 3, 6, -7, 9, -8, 4,
> -10, 7, -9, 8}] |
In[6]:= | Jones[L][q] |
Out[6]= | 2 4 6 7 7 7 5 4 -(9/2)
----- - ----- + ----- - ----- + ----- - ----- + ----- - ----- + q -
25/2 23/2 21/2 19/2 17/2 15/2 13/2 11/2
q q q q q q q q
-(7/2)
> q |
In[7]:= | A2Invariant[L][q] |
Out[7]= | -42 2 -36 -34 -32 2 -28 3 -24 2 3 -18
-q - --- - q - q - q + --- - q + --- + q + --- + --- + q +
38 30 26 22 20
q q q q q
3 -12
> --- + q
16
q |
In[8]:= | HOMFLYPT[Link[11, NonAlternating, 40]][a, z] |
Out[8]= | 7 9 11 13
-2 a 2 a a a 7 9 11 13 7 3
----- + ---- + --- - --- - 9 a z + 4 a z + 4 a z - a z - 12 a z -
z z z z
9 3 11 3 7 5 9 5 11 5 7 7 9 7
> 2 a z + 4 a z - 6 a z - 4 a z + a z - a z - a z |
In[9]:= | Kauffman[Link[11, NonAlternating, 40]][a, z] |
Out[9]= | 7 9 11 13
8 12 16 2 a 2 a a a 7 9 11
3 a - 3 a + a - ---- - ---- + --- + --- + 9 a z + 7 a z - 4 a z -
z z z z
13 15 8 2 10 2 12 2 14 2 16 2
> a z + a z - 7 a z + a z + 9 a z - 2 a z - 3 a z -
7 3 9 3 11 3 13 3 15 3 8 4 10 4
> 12 a z - 10 a z + a z - 4 a z - 3 a z + a z - 9 a z -
12 4 14 4 7 5 9 5 11 5 13 5 15 5
> 7 a z + 3 a z + 6 a z + 4 a z + 7 a z + 8 a z - a z +
8 6 10 6 12 6 14 6 7 7 9 7 11 7
> 3 a z + 13 a z + 7 a z - 3 a z - a z + 2 a z - a z -
13 7 8 8 10 8 12 8 9 9 11 9
> 4 a z - a z - 4 a z - 3 a z - a z - a z |
In[10]:= | Kh[L][q, t] |
Out[10]= | -8 -6 2 2 2 4 2 3 4
q + q + ------ + ------ + ------ + ------ + ------ + ------ + ------ +
26 9 24 8 22 8 22 7 20 7 20 6 18 6
q t q t q t q t q t q t q t
4 3 3 5 3 2 1 3 1
> ------ + ------ + ------ + ------ + ------ + ------ + ------ + ------ + ----
18 5 16 5 16 4 14 4 14 3 12 3 12 2 10 2 8
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 L11n40 |
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