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L11a451

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Acknowledgement

L11a451 as Morse Link
DrawMorseLink

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

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

Jones Polynomial: q-4 - 2q-3 + 5q-2 - 7q-1 + 9 - 10q + 11q2 - 8q3 + 7q4 - 4q5 + 3q6 - q7

A2 (sl(3)) Invariant: q-14 + q-12 - q-10 + 2q-8 + q-6 - 2q-4 + 2q-2 + 2q2 + 4q4 + 4q6 + 6q8 + 2q10 + 2q12 + 3q14 - q16 + q18 + q20 - q22

HOMFLY-PT Polynomial: - a-6z2 + a-4z-2 + 2a-4 + a-4z2 + a-4z4 - 2a-2z-2 - 3a-2 - 2a-2z2 + a-2z4 + z-2 - z2 + z4 - 2a2z2 + a4

Kauffman Polynomial: 3a-7z3 - 4a-7z5 + a-7z7 - 7a-6z2 + 18a-6z4 - 14a-6z6 + 3a-6z8 - a-5z3 + 11a-5z5 - 12a-5z7 + 3a-5z9 - a-4z-2 + 8a-4 - 23a-4z2 + 35a-4z4 - 22a-4z6 + 2a-4z8 + a-4z10 + 2a-3z-1 - 11a-3z + 8a-3z3 + 12a-3z5 - 17a-3z7 + 5a-3z9 - 2a-2z-2 + 13a-2 - 28a-2z2 + 29a-2z4 - 16a-2z6 + 2a-2z8 + a-2z10 + 2a-1z-1 - 11a-1z + 14a-1z3 - 7a-1z5 - a-1z7 + 2a-1z9 - z-2 + 5 - 9z2 + 8z4 - 5z6 + 3z8 - 2az5 + 3az7 + a2z2 - 3a2z4 + 3a2z6 - 2a3z3 + 2a3z5 + a4 - 2a4z2 + a4z4

Khovanov Homology:
trqj r = -4r = -3r = -2r = -1r = 0r = 1r = 2r = 3r = 4r = 5r = 6r = 7
j = 15           1
j = 13          2 
j = 11         21 
j = 9        52  
j = 7       54   
j = 5      63    
j = 3     45     
j = 1    56      
j = -1   35       
j = -3  24        
j = -5  3         
j = -712          
j = -91           


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, 451]]]
Out[3]=   
 -Graphics- 
In[4]:=
PD[L = Link[11, Alternating, 451]]
Out[4]=   
PD[X[6, 1, 7, 2], X[14, 3, 15, 4], X[20, 10, 21, 9], X[16, 8, 17, 7], 
 
>   X[18, 12, 19, 11], X[22, 16, 13, 15], X[12, 18, 5, 17], X[10, 20, 11, 19], 
 
>   X[8, 22, 9, 21], X[2, 5, 3, 6], X[4, 13, 1, 14]]
In[5]:=
GaussCode[L]
Out[5]=   
GaussCode[{1, -10, 2, -11}, {10, -1, 4, -9, 3, -8, 5, -7}, 
 
>   {11, -2, 6, -4, 7, -5, 8, -3, 9, -6}]
In[6]:=
Jones[L][q]
Out[6]=   
     -4   2    5    7              2      3      4      5      6    7
9 + q   - -- + -- - - - 10 q + 11 q  - 8 q  + 7 q  - 4 q  + 3 q  - q
           3    2   q
          q    q
In[7]:=
A2Invariant[L][q]
Out[7]=   
 -14    -12    -10   2     -6   2    2       2      4      6      8      10
q    + q    - q    + -- + q   - -- + -- + 2 q  + 4 q  + 4 q  + 6 q  + 2 q   + 
                      8          4    2
                     q          q    q
 
       12      14    16    18    20    22
>   2 q   + 3 q   - q   + q   + q   - q
In[8]:=
HOMFLYPT[Link[11, Alternating, 451]][a, z]
Out[8]=   
                                           2    2      2                   4
2    3     4    -2     1       2      2   z    z    2 z       2  2    4   z
-- - -- + a  + z   + ----- - ----- - z  - -- + -- - ---- - 2 a  z  + z  + -- + 
 4    2               4  2    2  2         6    4     2                    4
a    a               a  z    a  z         a    a     a                    a
 
     4
    z
>   --
     2
    a
In[9]:=
Kauffman[Link[11, Alternating, 451]][a, z]
Out[9]=   
    8    13    4    -2     1       2      2      2    11 z   11 z      2
5 + -- + -- + a  - z   - ----- - ----- + ---- + --- - ---- - ---- - 9 z  - 
     4    2               4  2    2  2    3     a z     3     a
    a    a               a  z    a  z    a  z          a
 
       2       2       2                        3    3      3       3
    7 z    23 z    28 z     2  2      4  2   3 z    z    8 z    14 z
>   ---- - ----- - ----- + a  z  - 2 a  z  + ---- - -- + ---- + ----- - 
      6      4       2                         7     5     3      a
     a      a       a                         a     a     a
 
                         4       4       4                        5       5
       3  3      4   18 z    35 z    29 z       2  4    4  4   4 z    11 z
>   2 a  z  + 8 z  + ----- + ----- + ----- - 3 a  z  + a  z  - ---- + ----- + 
                       6       4       2                         7      5
                      a       a       a                         a      a
 
        5      5                                 6       6       6
    12 z    7 z         5      3  5      6   14 z    22 z    16 z       2  6
>   ----- - ---- - 2 a z  + 2 a  z  - 5 z  - ----- - ----- - ----- + 3 a  z  + 
      3      a                                 6       4       2
     a                                        a       a       a
 
     7       7       7    7                      8      8      8      9
    z    12 z    17 z    z         7      8   3 z    2 z    2 z    3 z
>   -- - ----- - ----- - -- + 3 a z  + 3 z  + ---- + ---- + ---- + ---- + 
     7     5       3     a                      6      4      2      5
    a     a       a                            a      a      a      a
 
       9      9    10    10
    5 z    2 z    z     z
>   ---- + ---- + --- + ---
      3     a      4     2
     a            a     a
In[10]:=
Kh[L][q, t]
Out[10]=   
5           1       1       2       3       2      4      3               3
- + 5 q + ----- + ----- + ----- + ----- + ----- + ---- + --- + 6 q t + 4 q  t + 
q          9  4    7  4    7  3    5  2    3  2    3     q t
          q  t    q  t    q  t    q  t    q  t    q  t
 
       3  2      5  2      5  3      7  3      7  4      9  4      9  5
>   5 q  t  + 6 q  t  + 3 q  t  + 5 q  t  + 4 q  t  + 5 q  t  + 2 q  t  + 
 
       11  5    11  6      13  6    15  7
>   2 q   t  + q   t  + 2 q   t  + q   t


Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a451
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