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L11a4

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Acknowledgement

L11a4 as Morse Link
DrawMorseLink

PD Presentation: X6172 X16,7,17,8 X4,17,1,18 X10,6,11,5 X8493 X22,14,5,13 X20,12,21,11 X12,22,13,21 X14,20,15,19 X18,10,19,9 X2,16,3,15

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

Jones Polynomial: q-3/2 - 4q-1/2 + 7q1/2 - 14q3/2 + 18q5/2 - 22q7/2 + 22q9/2 - 19q11/2 + 15q13/2 - 9q15/2 + 4q17/2 - q19/2

A2 (sl(3)) Invariant: - q-4 + 2q-2 + 2q2 + 6q4 - q6 + 6q8 - q10 - q12 + q14 - 5q16 + 3q18 - 3q20 + 2q24 - 2q26 + q28

HOMFLY-PT Polynomial: - a-7z - 2a-7z3 - a-7z5 + a-5z-1 + 2a-5z + 3a-5z3 + 3a-5z5 + a-5z7 - 3a-3z-1 - 2a-3z + 2a-3z3 + 3a-3z5 + a-3z7 + 2a-1z-1 + a-1z - 2a-1z3 - a-1z5

Kauffman Polynomial: a-11z3 - a-11z5 - 2a-10z2 + 5a-10z4 - 4a-10z6 - 6a-9z3 + 12a-9z5 - 8a-9z7 - 3a-8z4 + 11a-8z6 - 9a-8z8 - 2a-7z + 8a-7z5 - 6a-7z9 - a-6 + 5a-6z2 - 10a-6z4 + 21a-6z6 - 11a-6z8 - 2a-6z10 + a-5z-1 - 4a-5z + 9a-5z3 - 10a-5z5 + 17a-5z7 - 11a-5z9 - 3a-4 + 5a-4z2 - 13a-4z4 + 21a-4z6 - 8a-4z8 - 2a-4z10 + 3a-3z-1 - 3a-3z - 6a-3z3 + 6a-3z5 + 5a-3z7 - 5a-3z9 - 3a-2 + 2a-2z2 - 9a-2z4 + 14a-2z6 - 6a-2z8 + 2a-1z-1 - a-1z - 8a-1z3 + 11a-1z5 - 4a-1z7 + 2z4 - z6

Khovanov Homology:
trqj r = -3r = -2r = -1r = 0r = 1r = 2r = 3r = 4r = 5r = 6r = 7r = 8
j = 20           1
j = 18          3 
j = 16         61 
j = 14        93  
j = 12       106   
j = 10      129    
j = 8     1010     
j = 6    812      
j = 4   610       
j = 2  310        
j = 0 14         
j = -2 3          
j = -41           


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, Alternating, 4]]]
Out[3]=   
 -Graphics- 
In[4]:=
PD[L = Link[11, Alternating, 4]]
Out[4]=   
PD[X[6, 1, 7, 2], X[16, 7, 17, 8], X[4, 17, 1, 18], X[10, 6, 11, 5], 
 
>   X[8, 4, 9, 3], X[22, 14, 5, 13], X[20, 12, 21, 11], X[12, 22, 13, 21], 
 
>   X[14, 20, 15, 19], X[18, 10, 19, 9], X[2, 16, 3, 15]]
In[5]:=
GaussCode[L]
Out[5]=   
GaussCode[{1, -11, 5, -3}, {4, -1, 2, -5, 10, -4, 7, -8, 6, -9, 11, -2, 3, -10, 
 
>    9, -7, 8, -6}]
In[6]:=
Jones[L][q]
Out[6]=   
 -(3/2)      4                      3/2       5/2       7/2       9/2
q       - ------- + 7 Sqrt[q] - 14 q    + 18 q    - 22 q    + 22 q    - 
          Sqrt[q]
 
        11/2       13/2      15/2      17/2    19/2
>   19 q     + 15 q     - 9 q     + 4 q     - q
In[7]:=
A2Invariant[L][q]
Out[7]=   
  -4   2       2      4    6      8    10    12    14      16      18      20
-q   + -- + 2 q  + 6 q  - q  + 6 q  - q   - q   + q   - 5 q   + 3 q   - 3 q   + 
        2
       q
 
       24      26    28
>   2 q   - 2 q   + q
In[8]:=
HOMFLYPT[Link[11, Alternating, 4]][a, z]
Out[8]=   
                                            3      3      3      3    5
 1      3      2    z    2 z   2 z   z   2 z    3 z    2 z    2 z    z
---- - ---- + --- - -- + --- - --- + - - ---- + ---- + ---- - ---- - -- + 
 5      3     a z    7    5     3    a     7      5      3     a      7
a  z   a  z         a    a     a          a      a      a            a
 
       5      5    5    7    7
    3 z    3 z    z    z    z
>   ---- + ---- - -- + -- + --
      5      3    a     5    3
     a      a          a    a
In[9]:=
Kauffman[Link[11, Alternating, 4]][a, z]
Out[9]=   
                                                              2      2      2
  -6   3    3     1      3      2    2 z   4 z   3 z   z   2 z    5 z    5 z
-a   - -- - -- + ---- + ---- + --- - --- - --- - --- - - - ---- + ---- + ---- + 
        4    2    5      3     a z    7     5     3    a    10      6      4
       a    a    a  z   a  z         a     a     a         a       a      a
 
       2    3       3      3      3      3             4      4       4
    2 z    z     6 z    9 z    6 z    8 z       4   5 z    3 z    10 z
>   ---- + --- - ---- + ---- - ---- - ---- + 2 z  + ---- - ---- - ----- - 
      2     11     9      5      3     a             10      8      6
     a     a      a      a      a                   a       a      a
 
        4      4    5        5      5       5      5       5           6
    13 z    9 z    z     12 z    8 z    10 z    6 z    11 z     6   4 z
>   ----- - ---- - --- + ----- + ---- - ----- + ---- + ----- - z  - ---- + 
      4       2     11     9       7      5       3      a           10
     a       a     a      a       a      a       a                  a
 
        6       6       6       6      7       7      7      7      8       8
    11 z    21 z    21 z    14 z    8 z    17 z    5 z    4 z    9 z    11 z
>   ----- + ----- + ----- + ----- - ---- + ----- + ---- - ---- - ---- - ----- - 
      8       6       4       2       9      5       3     a       8      6
     a       a       a       a       a      a       a             a      a
 
       8      8      9       9      9      10      10
    8 z    6 z    6 z    11 z    5 z    2 z     2 z
>   ---- - ---- - ---- - ----- - ---- - ----- - -----
      4      2      7      5       3      6       4
     a      a      a      a       a      a       a
In[10]:=
Kh[L][q, t]
Out[10]=   
                                            2
    2      4     1      -2     3     4   3 q        4        6         6  2
10 q  + 6 q  + ----- + t   + ----- + - + ---- + 10 q  t + 8 q  t + 12 q  t  + 
                4  3          2  2   t    t
               q  t          q  t
 
        8  2       8  3       10  3      10  4       12  4      12  5
>   10 q  t  + 10 q  t  + 12 q   t  + 9 q   t  + 10 q   t  + 6 q   t  + 
 
       14  5      14  6      16  6    16  7      18  7    20  8
>   9 q   t  + 3 q   t  + 6 q   t  + q   t  + 3 q   t  + q   t


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