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L11n401

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Acknowledgement

L11n401 as Morse Link
DrawMorseLink

PD Presentation: X6172 X10,3,11,4 X22,16,19,15 X7,20,8,21 X19,8,20,9 X18,14,5,13 X14,12,15,11 X12,18,13,17 X16,22,17,21 X2536 X4,9,1,10

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

Jones Polynomial: 3q-4 - 5q-3 + 11q-2 - 12q-1 + 14 - 13q + 11q2 - 7q3 + 3q4 - q5

A2 (sl(3)) Invariant: 2q-14 + 5q-12 + 3q-10 + 9q-8 + 7q-6 + 3q-4 + 5q-2 - 3 + q2 - 3q4 - q6 + 2q8 - 3q10 + q12 - q16

HOMFLY-PT Polynomial: - a-4 - a-4z2 - a-2z-2 + a-2 + 3a-2z2 + 2a-2z4 + 4z-2 + 5 - 2z4 - z6 - 5a2z-2 - 7a2 - 2a2z2 + a2z4 + 2a4z-2 + 2a4

Kauffman Polynomial: a-5z - 2a-5z3 + a-5z5 - a-4 + 3a-4z2 - 5a-4z4 + 3a-4z6 + a-3z-1 - 3a-3z + 5a-3z3 - 8a-3z5 + 5a-3z7 - a-2z-2 + 6a-2z2 - 10a-2z4 + 4a-2z8 + 5a-1z-1 - 20a-1z + 36a-1z3 - 35a-1z5 + 13a-1z7 + a-1z9 - 4z-2 + 11 - 10z2 + 7z4 - 11z6 + 8z8 + 9az-1 - 32az + 43az3 - 32az5 + 11az7 + az9 - 5a2z-2 + 18a2 - 26a2z2 + 18a2z4 - 8a2z6 + 4a2z8 + 5a3z-1 - 16a3z + 14a3z3 - 6a3z5 + 3a3z7 - 2a4z-2 + 9a4 - 13a4z2 + 6a4z4

Khovanov Homology:
trqj r = -4r = -3r = -2r = -1r = 0r = 1r = 2r = 3r = 4r = 5
j = 11         1
j = 9        2 
j = 7       51 
j = 5      62  
j = 3     75   
j = 1    76    
j = -1   79     
j = -3  45      
j = -5 17       
j = -724        
j = -93         


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, NonAlternating, 401]]]
Out[3]=   
 -Graphics- 
In[4]:=
PD[L = Link[11, NonAlternating, 401]]
Out[4]=   
PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[22, 16, 19, 15], X[7, 20, 8, 21], 
 
>   X[19, 8, 20, 9], X[18, 14, 5, 13], X[14, 12, 15, 11], X[12, 18, 13, 17], 
 
>   X[16, 22, 17, 21], X[2, 5, 3, 6], X[4, 9, 1, 10]]
In[5]:=
GaussCode[L]
Out[5]=   
GaussCode[{1, -10, 2, -11}, {-5, 4, 9, -3}, 
 
>   {10, -1, -4, 5, 11, -2, 7, -8, 6, -7, 3, -9, 8, -6}]
In[6]:=
Jones[L][q]
Out[6]=   
     3    5    11   12              2      3      4    5
14 + -- - -- + -- - -- - 13 q + 11 q  - 7 q  + 3 q  - q
      4    3    2   q
     q    q    q
In[7]:=
A2Invariant[L][q]
Out[7]=   
      2     5     3    9    7    3    5     2      4    6      8      10
-3 + --- + --- + --- + -- + -- + -- + -- + q  - 3 q  - q  + 2 q  - 3 q   + 
      14    12    10    8    6    4    2
     q     q     q     q    q    q    q
 
     12    16
>   q   - q
In[8]:=
HOMFLYPT[Link[11, NonAlternating, 401]][a, z]
Out[8]=   
                                              2      4    2      2
     -4    -2      2      4   4      1     5 a    2 a    z    3 z       2  2
5 - a   + a   - 7 a  + 2 a  + -- - ----- - ---- + ---- - -- + ---- - 2 a  z  - 
                               2    2  2     2      2     4     2
                              z    a  z     z      z     a     a
 
              4
       4   2 z     2  4    6
>   2 z  + ---- + a  z  - z
             2
            a
In[9]:=
Kauffman[Link[11, NonAlternating, 401]][a, z]
Out[9]=   
                                          2      4                         3
      -4       2      4   4      1     5 a    2 a     1      5    9 a   5 a
11 - a   + 18 a  + 9 a  - -- - ----- - ---- - ---- + ---- + --- + --- + ---- + 
                           2    2  2     2      2     3     a z    z     z
                          z    a  z     z      z     a  z
 
                                                    2      2
    z    3 z   20 z                3         2   3 z    6 z        2  2
>   -- - --- - ---- - 32 a z - 16 a  z - 10 z  + ---- + ---- - 26 a  z  - 
     5    3     a                                  4      2
    a    a                                        a      a
 
                  3      3       3                                  4       4
        4  2   2 z    5 z    36 z          3       3  3      4   5 z    10 z
>   13 a  z  - ---- + ---- + ----- + 43 a z  + 14 a  z  + 7 z  - ---- - ----- + 
                 5      3      a                                   4      2
                a      a                                          a      a
 
                          5      5       5                                  6
        2  4      4  4   z    8 z    35 z          5      3  5       6   3 z
>   18 a  z  + 6 a  z  + -- - ---- - ----- - 32 a z  - 6 a  z  - 11 z  + ---- - 
                          5     3      a                                   4
                         a     a                                          a
 
                 7       7                                 8              9
       2  6   5 z    13 z          7      3  7      8   4 z       2  8   z
>   8 a  z  + ---- + ----- + 11 a z  + 3 a  z  + 8 z  + ---- + 4 a  z  + -- + 
                3      a                                  2              a
               a                                         a
 
       9
>   a z
In[10]:=
Kh[L][q, t]
Out[10]=   
9           3       2       4       1       7       4      5      7
- + 7 q + ----- + ----- + ----- + ----- + ----- + ----- + ---- + --- + 6 q t + 
q          9  4    7  4    7  3    5  3    5  2    3  2    3     q t
          q  t    q  t    q  t    q  t    q  t    q  t    q  t
 
       3        3  2      5  2      5  3      7  3    7  4      9  4    11  5
>   7 q  t + 5 q  t  + 6 q  t  + 2 q  t  + 5 q  t  + q  t  + 2 q  t  + q   t


Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n401
L11n400
L11n400
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L11n402