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L11n110

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Acknowledgement

L11n110 as Morse Link
DrawMorseLink

PD Presentation: X6172 X3,12,4,13 X7,18,8,19 X19,22,20,5 X9,21,10,20 X21,9,22,8 X11,17,12,16 X17,15,18,14 X15,11,16,10 X2536 X13,4,14,1

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

Jones Polynomial: - 3q-9/2 + 6q-7/2 - 10q-5/2 + 12q-3/2 - 13q-1/2 + 12q1/2 - 10q3/2 + 6q5/2 - 3q7/2 + q9/2

A2 (sl(3)) Invariant: q-18 + q-16 + 4q-14 + q-10 + q-8 - 4q-6 + 2q-4 - 2q-2 + 3 + 2q2 + 3q6 - 2q8 - q14

HOMFLY-PT Polynomial: a-3z-1 + 2a-3z + a-3z3 - 4a-1z-1 - 9a-1z - 7a-1z3 - 2a-1z5 + 6az-1 + 12az + 11az3 + 5az5 + az7 - 5a3z-1 - 8a3z - 4a3z3 - a3z5 + 2a5z-1 + a5z

Kauffman Polynomial: a-4 - 3a-4z2 + 3a-4z4 - a-4z6 - a-3z-1 + 4a-3z - 9a-3z3 + 9a-3z5 - 3a-3z7 + 3a-2 - 11a-2z2 + 9a-2z4 + 3a-2z6 - 3a-2z8 - 4a-1z-1 + 19a-1z - 37a-1z3 + 36a-1z5 - 10a-1z7 - a-1z9 + 3 - 10z2 + 6z4 + 12z6 - 8z8 - 6az-1 + 29az - 51az3 + 44az5 - 14az7 - az9 + a2 - 3a2z2 + 5a2z6 - 5a2z8 - 5a3z-1 + 21a3z - 29a3z3 + 17a3z5 - 7a3z7 + a4 - a4z2 - 3a4z6 - 2a5z-1 + 7a5z - 6a5z3

Khovanov Homology:
trqj r = -4r = -3r = -2r = -1r = 0r = 1r = 2r = 3r = 4r = 5
j = 10         1
j = 8        2 
j = 6       41 
j = 4      62  
j = 2     64   
j = 0    76    
j = -2   67     
j = -4  46      
j = -6 26       
j = -814        
j = -103         


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, NonAlternating, 110]]]
Out[3]=   
 -Graphics- 
In[4]:=
PD[L = Link[11, NonAlternating, 110]]
Out[4]=   
PD[X[6, 1, 7, 2], X[3, 12, 4, 13], X[7, 18, 8, 19], X[19, 22, 20, 5], 
 
>   X[9, 21, 10, 20], X[21, 9, 22, 8], X[11, 17, 12, 16], X[17, 15, 18, 14], 
 
>   X[15, 11, 16, 10], X[2, 5, 3, 6], X[13, 4, 14, 1]]
In[5]:=
GaussCode[L]
Out[5]=   
GaussCode[{1, -10, -2, 11}, {10, -1, -3, 6, -5, 9, -7, 2, -11, 8, -9, 7, -8, 3, 
 
>    -4, 5, -6, 4}]
In[6]:=
Jones[L][q]
Out[6]=   
 -3     6      10     12      13                       3/2      5/2      7/2
---- + ---- - ---- + ---- - ------- + 12 Sqrt[q] - 10 q    + 6 q    - 3 q    + 
 9/2    7/2    5/2    3/2   Sqrt[q]
q      q      q      q
 
     9/2
>   q
In[7]:=
A2Invariant[L][q]
Out[7]=   
     -18    -16    4     -10    -8   4    2    2       2      6      8    14
3 + q    + q    + --- + q    + q   - -- + -- - -- + 2 q  + 3 q  - 2 q  - q
                   14                 6    4    2
                  q                  q    q    q
In[8]:=
HOMFLYPT[Link[11, NonAlternating, 110]][a, z]
Out[8]=   
                      3      5                                         3
 1      4    6 a   5 a    2 a    2 z   9 z               3      5     z
---- - --- + --- - ---- + ---- + --- - --- + 12 a z - 8 a  z + a  z + -- - 
 3     a z    z     z      z      3     a                              3
a  z                             a                                    a
 
       3                          5
    7 z          3      3  3   2 z         5    3  5      7
>   ---- + 11 a z  - 4 a  z  - ---- + 5 a z  - a  z  + a z
     a                          a
In[9]:=
Kauffman[Link[11, NonAlternating, 110]][a, z]
Out[9]=   
                                               3      5
     -4   3     2    4    1      4    6 a   5 a    2 a    4 z   19 z
3 + a   + -- + a  + a  - ---- - --- - --- - ---- - ---- + --- + ---- + 29 a z + 
           2              3     a z    z     z      z      3     a
          a              a  z                             a
 
                                  2       2                        3       3
        3        5         2   3 z    11 z       2  2    4  2   9 z    37 z
>   21 a  z + 7 a  z - 10 z  - ---- - ----- - 3 a  z  - a  z  - ---- - ----- - 
                                 4      2                         3      a
                                a      a                         a
 
                                             4      4      5       5
          3       3  3      5  3      4   3 z    9 z    9 z    36 z
>   51 a z  - 29 a  z  - 6 a  z  + 6 z  + ---- + ---- + ---- + ----- + 
                                            4      2      3      a
                                           a      a      a
 
                                  6      6                          7       7
          5       3  5       6   z    3 z       2  6      4  6   3 z    10 z
>   44 a z  + 17 a  z  + 12 z  - -- + ---- + 5 a  z  - 3 a  z  - ---- - ----- - 
                                  4     2                          3      a
                                 a     a                          a
 
                                  8              9
          7      3  7      8   3 z       2  8   z       9
>   14 a z  - 7 a  z  - 8 z  - ---- - 5 a  z  - -- - a z
                                 2              a
                                a
In[10]:=
Kh[L][q, t]
Out[10]=   
    7      3        1       4       2       6       4      6      6
7 + -- + ------ + ----- + ----- + ----- + ----- + ----- + ---- + ---- + 6 t + 
     2    10  4    8  4    8  3    6  3    6  2    4  2    4      2
    q    q   t    q  t    q  t    q  t    q  t    q  t    q  t   q  t
 
       2        2  2      4  2      4  3      6  3    6  4      8  4    10  5
>   6 q  t + 4 q  t  + 6 q  t  + 2 q  t  + 4 q  t  + q  t  + 2 q  t  + q   t


Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n110
L11n109
L11n109
L11n111
L11n111