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L11a385

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Acknowledgement

L11a385 as Morse Link
DrawMorseLink

PD Presentation: X6172 X10,3,11,4 X16,12,17,11 X18,22,19,21 X20,14,21,13 X12,20,13,19 X22,18,9,17 X8,16,5,15 X14,8,15,7 X2536 X4,9,1,10

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

Jones Polynomial: q-3 - 2q-2 + 7q-1 - 10 + 17q - 19q2 + 21q3 - 18q4 + 14q5 - 10q6 + 4q7 - q8

A2 (sl(3)) Invariant: q-10 + q-8 + q-6 + 6q-4 + 3q-2 + 5 + 10q2 + 2q4 + 7q6 + q8 + q12 - 6q14 + q16 - 4q18 - 3q20 + 2q22 - q24

HOMFLY-PT Polynomial: - a-6z-2 - 2a-6 - a-6z2 - a-6z4 + 3a-4z-2 + 6a-4 + 4a-4z2 + 2a-4z4 + a-4z6 - 2a-2z-2 - 3a-2 - a-2z2 + a-2z4 + a-2z6 - z-2 - 3 - 4z2 - 2z4 + a2z-2 + 2a2 + a2z2

Kauffman Polynomial: - a-9z3 + a-9z5 - 4a-8z4 + 4a-8z6 + 2a-7z-1 - 7a-7z + 11a-7z3 - 16a-7z5 + 9a-7z7 - a-6z-2 + a-6 - 2a-6z2 + 10a-6z4 - 17a-6z6 + 10a-6z8 + 8a-5z-1 - 27a-5z + 39a-5z3 - 31a-5z5 + 6a-5z7 + 5a-5z9 - 3a-4z-2 + 5a-4 - 4a-4z2 + 18a-4z4 - 31a-4z6 + 15a-4z8 + a-4z10 + 10a-3z-1 - 34a-3z + 50a-3z3 - 33a-3z5 + a-3z7 + 7a-3z9 - 2a-2z-2 + 4a-2 - 3a-2z2 + 6a-2z4 - 15a-2z6 + 8a-2z8 + a-2z10 + 2a-1z-1 - 10a-1z + 23a-1z3 - 23a-1z5 + 6a-1z7 + 2a-1z9 + z-2 - 3 + 5z2 - 2z4 - 4z6 + 3z8 - 2az-1 + 4az - 4az5 + 2az7 + a2z-2 - 4a2 + 6a2z2 - 4a2z4 + a2z6

Khovanov Homology:
trqj r = -4r = -3r = -2r = -1r = 0r = 1r = 2r = 3r = 4r = 5r = 6r = 7
j = 17           1
j = 15          3 
j = 13         71 
j = 11        73  
j = 9       117   
j = 7      107    
j = 5     911     
j = 3    810      
j = 1   512       
j = -1  25        
j = -3  5         
j = -512          
j = -71           


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]=   
3
In[3]:=
Show[DrawMorseLink[Link[11, Alternating, 385]]]
Out[3]=   
 -Graphics- 
In[4]:=
PD[L = Link[11, Alternating, 385]]
Out[4]=   
PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[16, 12, 17, 11], X[18, 22, 19, 21], 
 
>   X[20, 14, 21, 13], X[12, 20, 13, 19], X[22, 18, 9, 17], X[8, 16, 5, 15], 
 
>   X[14, 8, 15, 7], X[2, 5, 3, 6], X[4, 9, 1, 10]]
In[5]:=
GaussCode[L]
Out[5]=   
GaussCode[{1, -10, 2, -11}, {10, -1, 9, -8}, 
 
>   {11, -2, 3, -6, 5, -9, 8, -3, 7, -4, 6, -5, 4, -7}]
In[6]:=
Jones[L][q]
Out[6]=   
       -3   2    7              2       3       4       5       6      7    8
-10 + q   - -- + - + 17 q - 19 q  + 21 q  - 18 q  + 14 q  - 10 q  + 4 q  - q
             2   q
            q
In[7]:=
A2Invariant[L][q]
Out[7]=   
     -10    -8    -6   6    3        2      4      6    8    12      14    16
5 + q    + q   + q   + -- + -- + 10 q  + 2 q  + 7 q  + q  + q   - 6 q   + q   - 
                        4    2
                       q    q
 
       18      20      22    24
>   4 q   - 3 q   + 2 q   - q
In[8]:=
HOMFLYPT[Link[11, Alternating, 385]][a, z]
Out[8]=   
                                                          2           2
     2    6    3       2    -2     1       3       2     a       2   z
-3 - -- + -- - -- + 2 a  - z   - ----- + ----- - ----- + -- - 4 z  - -- + 
      6    4    2                 6  2    4  2    2  2    2           6
     a    a    a                 a  z    a  z    a  z    z           a
 
       2    2                   4      4    4    6    6
    4 z    z     2  2      4   z    2 z    z    z    z
>   ---- - -- + a  z  - 2 z  - -- + ---- + -- + -- + --
      4     2                   6     4     2    4    2
     a     a                   a     a     a    a    a
In[9]:=
Kauffman[Link[11, Alternating, 385]][a, z]
Out[9]=   
                                                           2
      -6   5    4       2    -2     1       3       2     a     2      8
-3 + a   + -- + -- - 4 a  + z   - ----- - ----- - ----- + -- + ---- + ---- + 
            4    2                 6  2    4  2    2  2    2    7      5
           a    a                 a  z    a  z    a  z    z    a  z   a  z
 
                                                                    2      2
     10     2    2 a   7 z   27 z   34 z   10 z              2   2 z    4 z
>   ---- + --- - --- - --- - ---- - ---- - ---- + 4 a z + 5 z  - ---- - ---- - 
     3     a z    z     7      5      3     a                      6      4
    a  z               a      a      a                            a      a
 
       2              3       3       3       3       3             4       4
    3 z       2  2   z    11 z    39 z    50 z    23 z       4   4 z    10 z
>   ---- + 6 a  z  - -- + ----- + ----- + ----- + ----- - 2 z  - ---- + ----- + 
      2               9     7       5       3       a              8      6
     a               a     a       a       a                      a      a
 
        4      4              5       5       5       5       5
    18 z    6 z       2  4   z    16 z    31 z    33 z    23 z         5
>   ----- + ---- - 4 a  z  + -- - ----- - ----- - ----- - ----- - 4 a z  - 
      4       2               9     7       5       3       a
     a       a               a     a       a       a
 
              6       6       6       6              7      7    7      7
       6   4 z    17 z    31 z    15 z     2  6   9 z    6 z    z    6 z
>   4 z  + ---- - ----- - ----- - ----- + a  z  + ---- + ---- + -- + ---- + 
             8      6       4       2               7      5     3    a
            a      a       a       a               a      a     a
 
                        8       8      8      9      9      9    10    10
         7      8   10 z    15 z    8 z    5 z    7 z    2 z    z     z
>   2 a z  + 3 z  + ----- + ----- + ---- + ---- + ---- + ---- + --- + ---
                      6       4       2      5      3     a      4     2
                     a       a       a      a      a            a     a
In[10]:=
Kh[L][q, t]
Out[10]=   
          3     1       1       2       5      2      5    5 q       3
12 q + 8 q  + ----- + ----- + ----- + ----- + ---- + --- + --- + 10 q  t + 
               7  4    5  4    5  3    3  2      2   q t    t
              q  t    q  t    q  t    q  t    q t
 
       5         5  2       7  2      7  3       9  3      9  4      11  4
>   9 q  t + 11 q  t  + 10 q  t  + 7 q  t  + 11 q  t  + 7 q  t  + 7 q   t  + 
 
       11  5      13  5    13  6      15  6    17  7
>   3 q   t  + 7 q   t  + q   t  + 3 q   t  + q   t


Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a385
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