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L10a10

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Acknowledgement

L10a10 as Morse Link
DrawMorseLink

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

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

Jones Polynomial: - q-7/2 + 2q-5/2 - 5q-3/2 + 7q-1/2 - 10q1/2 + 11q3/2 - 10q5/2 + 8q7/2 - 6q9/2 + 3q11/2 - q13/2

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

HOMFLY-PT Polynomial: - a-5z-1 - a-5z - a-5z3 + 2a-3z-1 + 3a-3z + 2a-3z3 + a-3z5 - a-1z-1 + a-1z3 + a-1z5 - az-1 - 3az - 2az3 + a3z-1 + a3z

Kauffman Polynomial: 2a-7z3 - a-7z5 - a-6z2 + 6a-6z4 - 3a-6z6 - a-5z-1 + 5a-5z - 11a-5z3 + 13a-5z5 - 5a-5z7 - a-4 + 4a-4z2 - 9a-4z4 + 9a-4z6 - 4a-4z8 - 2a-3z-1 + 13a-3z - 27a-3z3 + 21a-3z5 - 6a-3z7 - a-3z9 - 3a-2 + 11a-2z2 - 20a-2z4 + 15a-2z6 - 6a-2z8 - a-1z-1 + 6a-1z - 10a-1z3 + 8a-1z5 - 3a-1z7 - a-1z9 - 2 + 5z2 - z4 + z6 - 2z8 + az-1 - 5az + 7az3 - 2az7 - a2 - a2z2 + 4a2z4 - 2a2z6 + a3z-1 - 3a3z + 3a3z3 - a3z5

Khovanov Homology:
trqj r = -4r = -3r = -2r = -1r = 0r = 1r = 2r = 3r = 4r = 5r = 6
j = 14          1
j = 12         2 
j = 10        41 
j = 8       42  
j = 6      64   
j = 4     54    
j = 2    56     
j = 0   47      
j = -2  13       
j = -4 14        
j = -6 1         
j = -81          


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


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