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L10n39

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Acknowledgement

L10n39 as Morse Link
DrawMorseLink

PD Presentation: X6172 X12,4,13,3 X14,10,15,9 X16,12,17,11 X10,16,11,15 X17,5,18,20 X7,19,8,18 X19,9,20,8 X2536 X4,14,1,13

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

Jones Polynomial: - q1/2 + q3/2 - 4q5/2 + 3q7/2 - 4q9/2 + 4q11/2 - 3q13/2 + 3q15/2 - q17/2

A2 (sl(3)) Invariant: q2 + q4 + 3q6 + 4q8 + 4q10 + 4q12 - 4q18 - 3q20 - 2q22 - q24 + q26 + q30

HOMFLY-PT Polynomial: - a-9z-1 - a-9z + 5a-7z-1 + 8a-7z + 5a-7z3 + a-7z5 - 8a-5z-1 - 15a-5z - 13a-5z3 - 6a-5z5 - a-5z7 + 4a-3z-1 + 8a-3z + 5a-3z3 + a-3z5

Kauffman Polynomial: - a-11z + 2a-10 - 3a-10z2 - a-9z-1 - a-9z + 4a-9z3 - 2a-9z5 + 9a-8 - 23a-8z2 + 20a-8z4 - 5a-8z6 - 5a-7z-1 + 11a-7z - 18a-7z3 + 16a-7z5 - 4a-7z7 + 14a-6 - 33a-6z2 + 23a-6z4 - 2a-6z6 - a-6z8 - 8a-5z-1 + 23a-5z - 35a-5z3 + 24a-5z5 - 5a-5z7 + 8a-4 - 13a-4z2 + 3a-4z4 + 3a-4z6 - a-4z8 - 4a-3z-1 + 12a-3z - 13a-3z3 + 6a-3z5 - a-3z7

Khovanov Homology:
trqj r = -2r = -1r = 0r = 1r = 2r = 3r = 4r = 5r = 6
j = 18        1
j = 16       2 
j = 14      22 
j = 12     32  
j = 10    121  
j = 8   23    
j = 6  21     
j = 4 14      
j = 2         
j = 01        


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, NonAlternating, 39]]]
Out[3]=   
 -Graphics- 
In[4]:=
PD[L = Link[10, NonAlternating, 39]]
Out[4]=   
PD[X[6, 1, 7, 2], X[12, 4, 13, 3], X[14, 10, 15, 9], X[16, 12, 17, 11], 
 
>   X[10, 16, 11, 15], X[17, 5, 18, 20], X[7, 19, 8, 18], X[19, 9, 20, 8], 
 
>   X[2, 5, 3, 6], X[4, 14, 1, 13]]
In[5]:=
GaussCode[L]
Out[5]=   
GaussCode[{1, -9, 2, -10}, {9, -1, -7, 8, 3, -5, 4, -2, 10, -3, 5, -4, -6, 7, 
 
>    -8, 6}]
In[6]:=
Jones[L][q]
Out[6]=   
            3/2      5/2      7/2      9/2      11/2      13/2      15/2    17/2
-Sqrt[q] + q    - 4 q    + 3 q    - 4 q    + 4 q     - 3 q     + 3 q     - q
In[7]:=
A2Invariant[L][q]
Out[7]=   
 2    4      6      8      10      12      18      20      22    24    26    30
q  + q  + 3 q  + 4 q  + 4 q   + 4 q   - 4 q   - 3 q   - 2 q   - q   + q   + q
In[8]:=
HOMFLYPT[Link[10, NonAlternating, 39]][a, z]
Out[8]=   
                                                          3       3      3
   1       5      8      4     z    8 z   15 z   8 z   5 z    13 z    5 z
-(----) + ---- - ---- + ---- - -- + --- - ---- + --- + ---- - ----- + ---- + 
   9       7      5      3      9    7      5     3      7      5       3
  a  z    a  z   a  z   a  z   a    a      a     a      a      a       a
 
     5      5    5    7
    z    6 z    z    z
>   -- - ---- + -- - --
     7     5     3    5
    a     a     a    a
In[9]:=
Kauffman[Link[10, NonAlternating, 39]][a, z]
Out[9]=   
 2    9    14   8     1      5      8      4      z    z    11 z   23 z
--- + -- + -- + -- - ---- - ---- - ---- - ---- - --- - -- + ---- + ---- + 
 10    8    6    4    9      7      5      3      11    9     7      5
a     a    a    a    a  z   a  z   a  z   a  z   a     a     a      a
 
              2       2       2       2      3       3       3       3
    12 z   3 z    23 z    33 z    13 z    4 z    18 z    35 z    13 z
>   ---- - ---- - ----- - ----- - ----- + ---- - ----- - ----- - ----- + 
      3     10      8       6       4       9      7       5       3
     a     a       a       a       a       a      a       a       a
 
        4       4      4      5       5       5      5      6      6      6
    20 z    23 z    3 z    2 z    16 z    24 z    6 z    5 z    2 z    3 z
>   ----- + ----- + ---- - ---- + ----- + ----- + ---- - ---- - ---- + ---- - 
      8       6       4      9      7       5       3      8      6      4
     a       a       a      a      a       a       a      a      a      a
 
       7      7    7    8    8
    4 z    5 z    z    z    z
>   ---- - ---- - -- - -- - --
      7      5     3    6    4
     a      a     a    a    a
In[10]:=
Kh[L][q, t]
Out[10]=   
                     4
   4      6    -2   q     6        8        8  2    10  2      10  3
4 q  + 2 q  + t   + -- + q  t + 2 q  t + 3 q  t  + q   t  + 2 q   t  + 
                    t
 
       12  3    10  4      12  4      14  4      14  5      16  5    18  6
>   3 q   t  + q   t  + 2 q   t  + 2 q   t  + 2 q   t  + 2 q   t  + q   t


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