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L11n97

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Acknowledgement

L11n97 as Morse Link
DrawMorseLink

PD Presentation: X6172 X12,3,13,4 X16,8,17,7 X18,10,19,9 X8,18,9,17 X19,22,20,5 X13,20,14,21 X21,14,22,15 X10,16,11,15 X2536 X4,11,1,12

Gauss Code: {{1, -10, 2, -11}, {10, -1, 3, -5, 4, -9, 11, -2, -7, 8, 9, -3, 5, -4, -6, 7, -8, 6}}

Jones Polynomial: q-11/2 - 4q-9/2 + 5q-7/2 - 8q-5/2 + 9q-3/2 - 9q-1/2 + 7q1/2 - 5q3/2 + 3q5/2 - q7/2

A2 (sl(3)) Invariant: 3q-14 + q-12 + 3q-10 + 2q-8 - q-6 + 2q-4 - 2q-2 + 2 - q2 - q4 + q6 - q8 + q10

HOMFLY-PT Polynomial: - 2a-1z - 3a-1z3 - a-1z5 + 3az + 8az3 + 5az5 + az7 - a3z-1 - 3a3z - 3a3z3 - a3z5 + a5z-1

Kauffman Polynomial: 2a-3z3 - a-3z5 - 2a-2z2 + 7a-2z4 - 3a-2z6 + 2a-1z - 7a-1z3 + 10a-1z5 - 4a-1z7 + z2 - 4z4 + 6z6 - 3z8 + az - 9az3 + 8az5 - 2az7 - az9 + 4a2z2 - 13a2z4 + 10a2z6 - 4a2z8 - a3z-1 + 6a3z3 - 7a3z5 + 2a3z7 - a3z9 + a4 + 2a4z2 - 3a4z4 + a4z6 - a4z8 - a5z-1 + a5z + 6a5z3 - 4a5z5 + a6z2 - a6z4

Khovanov Homology:
trqj r = -5r = -4r = -3r = -2r = -1r = 0r = 1r = 2r = 3r = 4
j = 8         1
j = 6        2 
j = 4       31 
j = 2      42  
j = 0     53   
j = -2    55    
j = -4   34     
j = -6  25      
j = -8 23       
j = -10 3        
j = -121         


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, 97]]]
Out[3]=   
 -Graphics- 
In[4]:=
PD[L = Link[11, NonAlternating, 97]]
Out[4]=   
PD[X[6, 1, 7, 2], X[12, 3, 13, 4], X[16, 8, 17, 7], X[18, 10, 19, 9], 
 
>   X[8, 18, 9, 17], X[19, 22, 20, 5], X[13, 20, 14, 21], X[21, 14, 22, 15], 
 
>   X[10, 16, 11, 15], X[2, 5, 3, 6], X[4, 11, 1, 12]]
In[5]:=
GaussCode[L]
Out[5]=   
GaussCode[{1, -10, 2, -11}, {10, -1, 3, -5, 4, -9, 11, -2, -7, 8, 9, -3, 5, -4, 
 
>    -6, 7, -8, 6}]
In[6]:=
Jones[L][q]
Out[6]=   
 -(11/2)    4      5      8      9        9                     3/2      5/2
q        - ---- + ---- - ---- + ---- - ------- + 7 Sqrt[q] - 5 q    + 3 q    - 
            9/2    7/2    5/2    3/2   Sqrt[q]
           q      q      q      q
 
     7/2
>   q
In[7]:=
A2Invariant[L][q]
Out[7]=   
     3     -12    3    2     -6   2    2     2    4    6    8    10
2 + --- + q    + --- + -- - q   + -- - -- - q  - q  + q  - q  + q
     14           10    8          4    2
    q            q     q          q    q
In[8]:=
HOMFLYPT[Link[11, NonAlternating, 97]][a, z]
Out[8]=   
   3     5                             3                       5
  a     a    2 z              3     3 z         3      3  3   z         5
-(--) + -- - --- + 3 a z - 3 a  z - ---- + 8 a z  - 3 a  z  - -- + 5 a z  - 
  z     z     a                      a                        a
 
     3  5      7
>   a  z  + a z
In[9]:=
Kauffman[Link[11, NonAlternating, 97]][a, z]
Out[9]=   
      3    5                              2
 4   a    a    2 z          5      2   2 z       2  2      4  2    6  2
a  - -- - -- + --- + a z + a  z + z  - ---- + 4 a  z  + 2 a  z  + a  z  + 
     z    z     a                        2
                                        a
 
       3      3                                          4
    2 z    7 z         3      3  3      5  3      4   7 z        2  4
>   ---- - ---- - 9 a z  + 6 a  z  + 6 a  z  - 4 z  + ---- - 13 a  z  - 
      3     a                                           2
     a                                                 a
 
                       5       5                                          6
       4  4    6  4   z    10 z         5      3  5      5  5      6   3 z
>   3 a  z  - a  z  - -- + ----- + 8 a z  - 7 a  z  - 4 a  z  + 6 z  - ---- + 
                       3     a                                           2
                      a                                                 a
 
                          7
        2  6    4  6   4 z         7      3  7      8      2  8    4  8
>   10 a  z  + a  z  - ---- - 2 a z  + 2 a  z  - 3 z  - 4 a  z  - a  z  - 
                        a
 
       9    3  9
>   a z  - a  z
In[10]:=
Kh[L][q, t]
Out[10]=   
    5      1        3        2       3       2       5       3      4
5 + -- + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ---- + 
     2    12  5    10  4    8  4    8  3    6  3    6  2    4  2    4
    q    q   t    q   t    q  t    q  t    q  t    q  t    q  t    q  t
 
     5              2        2  2      4  2    4  3      6  3    8  4
>   ---- + 3 t + 4 q  t + 2 q  t  + 3 q  t  + q  t  + 2 q  t  + q  t
     2
    q  t


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