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L11a407

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Acknowledgement

L11a407 as Morse Link
DrawMorseLink

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

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

Jones Polynomial: q-3 - 2q-2 + 7q-1 - 10 + 16q - 18q2 + 19q3 - 17q4 + 13q5 - 8q6 + 4q7 - q8

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

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

Kauffman Polynomial: - a-9z3 + a-9z5 + 2a-8z2 - 6a-8z4 + 4a-8z6 + 5a-7z3 - 11a-7z5 + 7a-7z7 - a-6 + 4a-6z2 - 2a-6z4 - 7a-6z6 + 7a-6z8 - 2a-5z + 8a-5z3 - 14a-5z5 + 4a-5z7 + 4a-5z9 - a-4 + 9a-4z2 - 5a-4z4 - 13a-4z6 + 10a-4z8 + a-4z10 - 6a-3z + 16a-3z3 - 17a-3z5 + 6a-3z9 + a-2z-2 - 3a-2 + 9a-2z2 - 8a-2z4 - 7a-2z6 + 6a-2z8 + a-2z10 - 2a-1z-1 + 14a-1z3 - 19a-1z5 + 5a-1z7 + 2a-1z9 + 2z-2 - 6 + 8z2 - 3z4 - 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         51 
j = 11        83  
j = 9       95   
j = 7      108    
j = 5     1011     
j = 3    68      
j = 1   511       
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, 407]]]
Out[3]=   
 -Graphics- 
In[4]:=
PD[L = Link[11, Alternating, 407]]
Out[4]=   
PD[X[6, 1, 7, 2], X[12, 3, 13, 4], X[22, 14, 11, 13], X[8, 16, 9, 15], 
 
>   X[14, 8, 15, 7], X[18, 10, 19, 9], X[20, 18, 21, 17], X[16, 22, 17, 21], 
 
>   X[10, 20, 5, 19], X[2, 5, 3, 6], X[4, 11, 1, 12]]
In[5]:=
GaussCode[L]
Out[5]=   
GaussCode[{1, -10, 2, -11}, {10, -1, 5, -4, 6, -9}, 
 
>   {11, -2, 3, -5, 4, -8, 7, -6, 9, -7, 8, -3}]
In[6]:=
Jones[L][q]
Out[6]=   
       -3   2    7              2       3       4       5      6      7    8
-10 + q   - -- + - + 16 q - 18 q  + 19 q  - 17 q  + 13 q  - 8 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
4 + q    + q   + q   + -- + -- + 7 q  - q  + 4 q  - q  + 2 q   - 3 q   + 
                        4    2
                       q    q
 
       16    18    20      22    24
>   4 q   - q   - q   + 2 q   - q
In[8]:=
HOMFLYPT[Link[11, Alternating, 407]][a, z]
Out[8]=   
                                2           2      2    2                   4
      -4      2   2      1     a       2   z    2 z    z     2  2      4   z
-3 + a   + 2 a  - -- + ----- + -- - 4 z  - -- + ---- - -- + a  z  - 2 z  - -- + 
                   2    2  2    2           6     4     2                   6
                  z    a  z    z           a     a     a                   a
 
       4    4    6    6
    2 z    z    z    z
>   ---- + -- + -- + --
      4     2    4    2
     a     a    a    a
In[9]:=
Kauffman[Link[11, Alternating, 407]][a, z]
Out[9]=   
                                           2
      -6    -4   3       2   2      1     a     2    2 a   2 z   6 z
-6 - a   - a   - -- - 4 a  + -- + ----- + -- - --- - --- - --- - --- + 4 a z + 
                  2           2    2  2    2   a z    z     5     3
                 a           z    a  z    z                a     a
 
              2      2      2      2              3      3      3       3
       2   2 z    4 z    9 z    9 z       2  2   z    5 z    8 z    16 z
>   8 z  + ---- + ---- + ---- + ---- + 6 a  z  - -- + ---- + ---- + ----- + 
             8      6      4      2               9     7      5      3
            a      a      a      a               a     a      a      a
 
        3             4      4      4      4              5       5       5
    14 z       4   6 z    2 z    5 z    8 z       2  4   z    11 z    14 z
>   ----- - 3 z  - ---- - ---- - ---- - ---- - 4 a  z  + -- - ----- - ----- - 
      a              8      6      4      2               9     7       5
                    a      a      a      a               a     a       a
 
        5       5                      6      6       6      6              7
    17 z    19 z         5      6   4 z    7 z    13 z    7 z     2  6   7 z
>   ----- - ----- - 4 a z  - 4 z  + ---- - ---- - ----- - ---- + a  z  + ---- + 
      3       a                       8      6      4       2              7
     a                               a      a      a       a              a
 
       7      7                      8       8      8      9      9      9
    4 z    5 z         7      8   7 z    10 z    6 z    4 z    6 z    2 z
>   ---- + ---- + 2 a z  + 3 z  + ---- + ----- + ---- + ---- + ---- + ---- + 
      5     a                       6      4       2      5      3     a
     a                             a      a       a      a      a
 
     10    10
    z     z
>   --- + ---
     4     2
    a     a
In[10]:=
Kh[L][q, t]
Out[10]=   
          3     1       1       2       5      2      5    5 q      3
11 q + 6 q  + ----- + ----- + ----- + ----- + ---- + --- + --- + 8 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
>   10 q  t + 11 q  t  + 10 q  t  + 8 q  t  + 9 q  t  + 5 q  t  + 8 q   t  + 
 
       11  5      13  5    13  6      15  6    17  7
>   3 q   t  + 5 q   t  + q   t  + 3 q   t  + q   t


Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a407
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