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L11a113

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Acknowledgement

L11a113 as Morse Link
DrawMorseLink

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

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

Jones Polynomial: q-15/2 - 3q-13/2 + 7q-11/2 - 13q-9/2 + 16q-7/2 - 20q-5/2 + 20q-3/2 - 18q-1/2 + 13q1/2 - 8q3/2 + 4q5/2 - q7/2

A2 (sl(3)) Invariant: - q-24 - q-22 + q-20 - 2q-18 + q-16 + 5q-14 + 6q-10 + q-8 + 2q-4 - 4q-2 + 4 - 3q2 - q4 + 2q6 - 2q8 + q10

HOMFLY-PT Polynomial: - a-1z - 2a-1z3 - a-1z5 + az-1 + 4az + 7az3 + 4az5 + az7 - 4a3z-1 - 10a3z - 9a3z3 - 3a3z5 + 4a5z-1 + 6a5z + 3a5z3 - a7z-1 - a7z

Kauffman Polynomial: a-3z3 - a-3z5 - 2a-2z2 + 6a-2z4 - 4a-2z6 + a-1z - 6a-1z3 + 11a-1z5 - 7a-1z7 + 1 - 4z2 + 3z4 + 6z6 - 7z8 - az-1 + 4az - 17az3 + 23az5 - 7az7 - 4az9 + 4a2 - 10a2z2 - 3a2z4 + 22a2z6 - 13a2z8 - a2z10 - 4a3z-1 + 14a3z - 27a3z3 + 26a3z5 - a3z7 - 7a3z9 + 7a4 - 16a4z2 + 3a4z4 + 18a4z6 - 10a4z8 - a4z10 - 4a5z-1 + 15a5z - 25a5z3 + 23a5z5 - 4a5z7 - 3a5z9 + 4a6 - 11a6z2 + 6a6z4 + 5a6z6 - 4a6z8 - a7z-1 + 4a7z - 8a7z3 + 8a7z5 - 3a7z7 + a8 - 3a8z2 + 3a8z4 - a8z6

Khovanov Homology:
trqj r = -7r = -6r = -5r = -4r = -3r = -2r = -1r = 0r = 1r = 2r = 3r = 4
j = 8           1
j = 6          3 
j = 4         51 
j = 2        83  
j = 0       105   
j = -2      119    
j = -4     99     
j = -6    711      
j = -8   69       
j = -10  28        
j = -12 15         
j = -14 2          
j = -161           


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, 113]]]
Out[3]=   
 -Graphics- 
In[4]:=
PD[L = Link[11, Alternating, 113]]
Out[4]=   
PD[X[6, 1, 7, 2], X[14, 3, 15, 4], X[16, 8, 17, 7], X[22, 12, 5, 11], 
 
>   X[8, 22, 9, 21], X[18, 9, 19, 10], X[20, 17, 21, 18], X[10, 19, 11, 20], 
 
>   X[12, 16, 13, 15], X[2, 5, 3, 6], X[4, 13, 1, 14]]
In[5]:=
GaussCode[L]
Out[5]=   
GaussCode[{1, -10, 2, -11}, {10, -1, 3, -5, 6, -8, 4, -9, 11, -2, 9, -3, 7, -6, 
 
>    8, -7, 5, -4}]
In[6]:=
Jones[L][q]
Out[6]=   
 -(15/2)     3       7      13     16     20     20      18
q        - ----- + ----- - ---- + ---- - ---- + ---- - ------- + 13 Sqrt[q] - 
            13/2    11/2    9/2    7/2    5/2    3/2   Sqrt[q]
           q       q       q      q      q      q
 
       3/2      5/2    7/2
>   8 q    + 4 q    - q
In[7]:=
A2Invariant[L][q]
Out[7]=   
     -24    -22    -20    2     -16    5     6     -8   2    4       2    4
4 - q    - q    + q    - --- + q    + --- + --- + q   + -- - -- - 3 q  - q  + 
                          18           14    10          4    2
                         q            q     q           q    q
 
       6      8    10
>   2 q  - 2 q  + q
In[8]:=
HOMFLYPT[Link[11, Alternating, 113]][a, z]
Out[8]=   
       3      5    7                                            3
a   4 a    4 a    a    z               3        5      7     2 z         3
- - ---- + ---- - -- - - + 4 a z - 10 a  z + 6 a  z - a  z - ---- + 7 a z  - 
z    z      z     z    a                                      a
 
                         5
       3  3      5  3   z         5      3  5      7
>   9 a  z  + 3 a  z  - -- + 4 a z  - 3 a  z  + a z
                        a
In[9]:=
Kauffman[Link[11, Alternating, 113]][a, z]
Out[9]=   
                                     3      5    7
       2      4      6    8   a   4 a    4 a    a    z               3
1 + 4 a  + 7 a  + 4 a  + a  - - - ---- - ---- - -- + - + 4 a z + 14 a  z + 
                              z    z      z     z    a
 
                                 2
        5        7        2   2 z        2  2       4  2       6  2      8  2
>   15 a  z + 4 a  z - 4 z  - ---- - 10 a  z  - 16 a  z  - 11 a  z  - 3 a  z  + 
                                2
                               a
 
     3      3                                                       4
    z    6 z          3       3  3       5  3      7  3      4   6 z
>   -- - ---- - 17 a z  - 27 a  z  - 25 a  z  - 8 a  z  + 3 z  + ---- - 
     3    a                                                        2
    a                                                             a
 
                                             5       5
       2  4      4  4      6  4      8  4   z    11 z          5       3  5
>   3 a  z  + 3 a  z  + 6 a  z  + 3 a  z  - -- + ----- + 23 a z  + 26 a  z  + 
                                             3     a
                                            a
 
                                   6
        5  5      7  5      6   4 z        2  6       4  6      6  6    8  6
>   23 a  z  + 8 a  z  + 6 z  - ---- + 22 a  z  + 18 a  z  + 5 a  z  - a  z  - 
                                  2
                                 a
 
       7
    7 z         7    3  7      5  7      7  7      8       2  8       4  8
>   ---- - 7 a z  - a  z  - 4 a  z  - 3 a  z  - 7 z  - 13 a  z  - 10 a  z  - 
     a
 
       6  8        9      3  9      5  9    2  10    4  10
>   4 a  z  - 4 a z  - 7 a  z  - 3 a  z  - a  z   - a  z
In[10]:=
Kh[L][q, t]
Out[10]=   
     9      1        2        1        5        2        8        6       9
10 + -- + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 
      2    16  7    14  6    12  6    12  5    10  5    10  4    8  4    8  3
     q    q   t    q   t    q   t    q   t    q   t    q   t    q  t    q  t
 
      7      11       9      9      11             2        2  2      4  2
>   ----- + ----- + ----- + ---- + ---- + 5 t + 8 q  t + 3 q  t  + 5 q  t  + 
     6  3    6  2    4  2    4      2
    q  t    q  t    q  t    q  t   q  t
 
     4  3      6  3    8  4
>   q  t  + 3 q  t  + q  t


Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a113
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