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L11a3

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Acknowledgement

L11a3 as Morse Link
DrawMorseLink

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

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

Jones Polynomial: q-9/2 - 4q-7/2 + 9q-5/2 - 15q-3/2 + 20q-1/2 - 25q1/2 + 24q3/2 - 22q5/2 + 16q7/2 - 10q9/2 + 5q11/2 - q13/2

A2 (sl(3)) Invariant: - q-14 + q-12 + q-10 - 3q-8 + 4q-6 - 2q-4 - q-2 + 5 - 2q2 + 7q4 + 3q8 + 3q10 - 5q12 + 2q14 - 2q16 - 2q18 + q20

HOMFLY-PT Polynomial: a-5z-1 - a-5z3 - 3a-3z-1 - a-3z + 3a-3z3 + 2a-3z5 + 2a-1z-1 - 4a-1z3 - 3a-1z5 - a-1z7 + 2az + 4az3 + 2az5 - a3z - a3z3

Kauffman Polynomial: - a-7z5 - a-6 + 5a-6z4 - 5a-6z6 + a-5z-1 - 5a-5z3 + 14a-5z5 - 10a-5z7 - 3a-4 + 3a-4z2 - a-4z4 + 12a-4z6 - 11a-4z8 + 3a-3z-1 - a-3z - 15a-3z3 + 26a-3z5 - 5a-3z7 - 7a-3z9 - 3a-2 + 7a-2z2 - 27a-2z4 + 43a-2z6 - 19a-2z8 - 2a-2z10 + 2a-1z-1 + a-1z - 20a-1z3 + 16a-1z5 + 13a-1z7 - 13a-1z9 + 7z2 - 32z4 + 42z6 - 15z8 - 2z10 + 4az - 16az3 + 14az5 + 4az7 - 6az9 + 2a2z2 - 9a2z4 + 15a2z6 - 7a2z8 + 2a3z - 6a3z3 + 9a3z5 - 4a3z7 - a4z2 + 2a4z4 - a4z6

Khovanov Homology:
trqj r = -5r = -4r = -3r = -2r = -1r = 0r = 1r = 2r = 3r = 4r = 5r = 6
j = 14           1
j = 12          4 
j = 10         61 
j = 8        104  
j = 6       126   
j = 4      1210    
j = 2     1312     
j = 0    914      
j = -2   611       
j = -4  39        
j = -6 16         
j = -8 3          
j = -101           


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, 3]]]
Out[3]=   
 -Graphics- 
In[4]:=
PD[L = Link[11, Alternating, 3]]
Out[4]=   
PD[X[6, 1, 7, 2], X[16, 7, 17, 8], X[4, 17, 1, 18], X[14, 10, 15, 9], 
 
>   X[8, 4, 9, 3], X[10, 5, 11, 6], X[20, 12, 21, 11], X[22, 20, 5, 19], 
 
>   X[18, 13, 19, 14], X[12, 22, 13, 21], X[2, 16, 3, 15]]
In[5]:=
GaussCode[L]
Out[5]=   
GaussCode[{1, -11, 5, -3}, {6, -1, 2, -5, 4, -6, 7, -10, 9, -4, 11, -2, 3, -9, 
 
>    8, -7, 10, -8}]
In[6]:=
Jones[L][q]
Out[6]=   
 -(9/2)    4      9      15      20                       3/2       5/2
q       - ---- + ---- - ---- + ------- - 25 Sqrt[q] + 24 q    - 22 q    + 
           7/2    5/2    3/2   Sqrt[q]
          q      q      q
 
        7/2       9/2      11/2    13/2
>   16 q    - 10 q    + 5 q     - q
In[7]:=
A2Invariant[L][q]
Out[7]=   
     -14    -12    -10   3    4    2     -2      2      4      8      10
5 - q    + q    + q    - -- + -- - -- - q   - 2 q  + 7 q  + 3 q  + 3 q   - 
                          8    6    4
                         q    q    q
 
       12      14      16      18    20
>   5 q   + 2 q   - 2 q   - 2 q   + q
In[8]:=
HOMFLYPT[Link[11, Alternating, 3]][a, z]
Out[8]=   
                                         3      3      3
 1      3      2    z             3     z    3 z    4 z         3    3  3
---- - ---- + --- - -- + 2 a z - a  z - -- + ---- - ---- + 4 a z  - a  z  + 
 5      3     a z    3                   5     3     a
a  z   a  z         a                   a     a
 
       5      5             7
    2 z    3 z         5   z
>   ---- - ---- + 2 a z  - --
      3     a              a
     a
In[9]:=
Kauffman[Link[11, Alternating, 3]][a, z]
Out[9]=   
                                                                         2
  -6   3    3     1      3      2    z    z              3        2   3 z
-a   - -- - -- + ---- + ---- + --- - -- + - + 4 a z + 2 a  z + 7 z  + ---- + 
        4    2    5      3     a z    3   a                             4
       a    a    a  z   a  z         a                                 a
 
       2                        3       3       3
    7 z       2  2    4  2   5 z    15 z    20 z          3      3  3       4
>   ---- + 2 a  z  - a  z  - ---- - ----- - ----- - 16 a z  - 6 a  z  - 32 z  + 
      2                        5      3       a
     a                        a      a
 
       4    4       4                        5       5       5       5
    5 z    z    27 z       2  4      4  4   z    14 z    26 z    16 z
>   ---- - -- - ----- - 9 a  z  + 2 a  z  - -- + ----- + ----- + ----- + 
      6     4     2                          7     5       3       a
     a     a     a                          a     a       a
 
                                   6       6       6
          5      3  5       6   5 z    12 z    43 z        2  6    4  6
>   14 a z  + 9 a  z  + 42 z  - ---- + ----- + ----- + 15 a  z  - a  z  - 
                                  6      4       2
                                 a      a       a
 
        7      7       7                                  8       8
    10 z    5 z    13 z         7      3  7       8   11 z    19 z       2  8
>   ----- - ---- + ----- + 4 a z  - 4 a  z  - 15 z  - ----- - ----- - 7 a  z  - 
      5       3      a                                  4       2
     a       a                                         a       a
 
       9       9                       10
    7 z    13 z         9      10   2 z
>   ---- - ----- - 6 a z  - 2 z   - -----
      3      a                        2
     a                               a
In[10]:=
Kh[L][q, t]
Out[10]=   
         2     1        3       1       6       3       9       6     9
14 + 13 q  + ------ + ----- + ----- + ----- + ----- + ----- + ----- + - + 
              10  5    8  4    6  4    6  3    4  3    4  2    2  2   t
             q   t    q  t    q  t    q  t    q  t    q  t    q  t
 
     11        2         4         4  2       6  2      6  3       8  3
>   ---- + 12 q  t + 12 q  t + 10 q  t  + 12 q  t  + 6 q  t  + 10 q  t  + 
     2
    q  t
 
       8  4      10  4    10  5      12  5    14  6
>   4 q  t  + 6 q   t  + q   t  + 4 q   t  + q   t


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