 L11n81 |
 L11n83 |
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 DrawMorseLink |
| PD Presentation: | X6172 X20,7,21,8 X4,21,1,22 X5,14,6,15 X3,10,4,11 X11,16,12,17 X15,12,16,13 X13,22,14,5 X18,9,19,10 X17,2,18,3 X8,19,9,20 |
| Gauss Code: | {{1, 10, -5, -3}, {-4, -1, 2, -11, 9, 5, -6, 7, -8, 4, -7, 6, -10, -9, 11, -2, 3, 8}} |
| Jones Polynomial: | 2q-23/2 - 4q-21/2 + 7q-19/2 - 10q-17/2 + 10q-15/2 - 10q-13/2 + 8q-11/2 - 6q-9/2 + 2q-7/2 - q-5/2 |
| A2 (sl(3)) Invariant: | - q-40 - q-38 - 2q-36 - q-34 + 2q-32 - q-30 + 2q-28 + q-26 + 3q-22 - q-20 + 3q-18 + q-16 + q-14 + 3q-12 - q-10 + q-8 |
| HOMFLY-PT Polynomial: | - a5z-1 - 3a5z - 3a5z3 - a5z5 + a7z-1 - a7z - 5a7z3 - 2a7z5 - a9z-1 - a9z - 2a9z3 - a9z5 + 2a11z-1 + 3a11z + a11z3 - a13z-1 |
| Kauffman Polynomial: | a5z-1 - 3a5z + 3a5z3 - a5z5 - a6 + 3a6z4 - 2a6z6 + a7z-1 - 2a7z + 4a7z5 - 3a7z7 - 3a8z2 + 8a8z4 - 2a8z6 - 2a8z8 + a9z-1 - 5a9z + 8a9z3 - a9z5 - 2a9z7 - a9z9 - 4a10 + 11a10z2 - 8a10z4 + 5a10z6 - 4a10z8 + 2a11z-1 - 9a11z + 14a11z3 - 7a11z5 - a11z9 - 7a12 + 20a12z2 - 16a12z4 + 5a12z6 - 2a12z8 + a13z-1 - 3a13z + 3a13z3 - a13z5 - a13z7 - 3a14 + 6a14z2 - 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, 82]]] |
| |
Out[3]= | -Graphics- |
In[4]:= | PD[L = Link[11, NonAlternating, 82]] |
Out[4]= | PD[X[6, 1, 7, 2], X[20, 7, 21, 8], X[4, 21, 1, 22], X[5, 14, 6, 15], > X[3, 10, 4, 11], X[11, 16, 12, 17], X[15, 12, 16, 13], X[13, 22, 14, 5], > X[18, 9, 19, 10], X[17, 2, 18, 3], X[8, 19, 9, 20]] |
In[5]:= | GaussCode[L] |
Out[5]= | GaussCode[{1, 10, -5, -3}, {-4, -1, 2, -11, 9, 5, -6, 7, -8, 4, -7, 6, -10, -9,
> 11, -2, 3, 8}] |
In[6]:= | Jones[L][q] |
Out[6]= | 2 4 7 10 10 10 8 6 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 -34 2 -30 2 -26 3 -20 3 -16
-q - q - --- - q + --- - q + --- + q + --- - q + --- + q +
36 32 28 22 18
q q q q q
-14 3 -10 -8
> q + --- - q + q
12
q |
In[8]:= | HOMFLYPT[Link[11, NonAlternating, 82]][a, z] |
Out[8]= | 5 7 9 11 13
a a a 2 a a 5 7 9 11 5 3
-(--) + -- - -- + ----- - --- - 3 a z - a z - a z + 3 a z - 3 a z -
z z z z z
7 3 9 3 11 3 5 5 7 5 9 5
> 5 a z - 2 a z + a z - a z - 2 a z - a z |
In[9]:= | Kauffman[Link[11, NonAlternating, 82]][a, z] |
Out[9]= | 5 7 9 11 13
6 10 12 14 a a a 2 a a 5 7
-a - 4 a - 7 a - 3 a + -- + -- + -- + ----- + --- - 3 a z - 2 a z -
z z z z z
9 11 13 8 2 10 2 12 2 14 2
> 5 a z - 9 a z - 3 a z - 3 a z + 11 a z + 20 a z + 6 a z +
5 3 9 3 11 3 13 3 6 4 8 4 10 4
> 3 a z + 8 a z + 14 a z + 3 a z + 3 a z + 8 a z - 8 a z -
12 4 14 4 5 5 7 5 9 5 11 5 13 5
> 16 a z - 3 a z - a z + 4 a z - a z - 7 a z - a z -
6 6 8 6 10 6 12 6 7 7 9 7 13 7
> 2 a z - 2 a z + 5 a z + 5 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 2 2 5 2 5 5
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
5 5 5 6 4 4 2 4 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 L11n82 |
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