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L10a115

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Acknowledgement

L10a115 as Morse Link
DrawMorseLink

Further views:   Rich Schwartz' 25
Rich Schwartz' "25"

PD Presentation: X12,1,13,2 X20,9,11,10 X14,3,15,4 X16,5,17,6 X4,15,5,16 X8,17,9,18 X18,7,19,8 X6,19,7,20 X2,11,3,12 X10,13,1,14

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

Jones Polynomial: - q-25/2 + q-23/2 - 3q-21/2 + 5q-19/2 - 6q-17/2 + 7q-15/2 - 7q-13/2 + 5q-11/2 - 4q-9/2 + 2q-7/2 - q-5/2

A2 (sl(3)) Invariant: q-38 + q-36 + q-34 + 2q-32 + q-28 - q-24 + q-22 - q-20 + 2q-18 + q-16 + q-12 - q-10 + q-8

HOMFLY-PT Polynomial: - a5z - 3a5z3 - a5z5 - 5a7z - 7a7z3 - 2a7z5 - a9z-1 - 2a9z - 3a9z3 - a9z5 + a11z-1 + 3a11z + a11z3

Kauffman Polynomial: - a5z + 3a5z3 - a5z5 - a6z2 + 5a6z4 - 2a6z6 + 5a7z - 11a7z3 + 10a7z5 - 3a7z7 - 4a8z4 + 5a8z6 - 2a8z8 + a9z-1 - 4a9z + 2a9z3 - 3a9z5 + 2a9z7 - a9z9 - a10 + 7a10z2 - 16a10z4 + 10a10z6 - 3a10z8 + a11z-1 - 6a11z + 13a11z3 - 12a11z5 + 4a11z7 - a11z9 + 6a12z2 - 5a12z4 + 2a12z6 - a12z8 + a13z3 + a13z5 - a13z7 + 2a14z4 - a14z6 - 4a15z + 4a15z3 - a15z5

Khovanov Homology:
trqj r = -10r = -9r = -8r = -7r = -6r = -5r = -4r = -3r = -2r = -1r = 0
j = -4          1
j = -6         21
j = -8        2  
j = -10       32  
j = -12      42   
j = -14     33    
j = -16    34     
j = -18   23      
j = -20  13       
j = -22  2        
j = -2411         
j = -261          


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, 115]]]
Out[3]=   
 -Graphics- 
In[4]:=
PD[L = Link[10, Alternating, 115]]
Out[4]=   
PD[X[12, 1, 13, 2], X[20, 9, 11, 10], X[14, 3, 15, 4], X[16, 5, 17, 6], 
 
>   X[4, 15, 5, 16], X[8, 17, 9, 18], X[18, 7, 19, 8], X[6, 19, 7, 20], 
 
>   X[2, 11, 3, 12], X[10, 13, 1, 14]]
In[5]:=
GaussCode[L]
Out[5]=   
GaussCode[{1, -9, 3, -5, 4, -8, 7, -6, 2, -10}, 
 
>   {9, -1, 10, -3, 5, -4, 6, -7, 8, -2}]
In[6]:=
Jones[L][q]
Out[6]=   
  -(25/2)    -(23/2)     3       5       6       7       7       5      4
-q        + q        - ----- + ----- - ----- + ----- - ----- + ----- - ---- + 
                        21/2    19/2    17/2    15/2    13/2    11/2    9/2
                       q       q       q       q       q       q       q
 
     2      -(5/2)
>   ---- - q
     7/2
    q
In[7]:=
A2Invariant[L][q]
Out[7]=   
 -38    -36    -34    2     -28    -24    -22    -20    2     -16    -12
q    + q    + q    + --- + q    - q    + q    - q    + --- + q    + q    - 
                      32                                18
                     q                                 q
 
     -10    -8
>   q    + q
In[8]:=
HOMFLYPT[Link[10, Alternating, 115]][a, z]
Out[8]=   
   9     11
  a     a      5        7        9        11        5  3      7  3      9  3
-(--) + --- - a  z - 5 a  z - 2 a  z + 3 a   z - 3 a  z  - 7 a  z  - 3 a  z  + 
  z      z
 
     11  3    5  5      7  5    9  5
>   a   z  - a  z  - 2 a  z  - a  z
In[9]:=
Kauffman[Link[10, Alternating, 115]][a, z]
Out[9]=   
        9    11
  10   a    a      5        7        9        11        15      6  2
-a   + -- + --- - a  z + 5 a  z - 4 a  z - 6 a   z - 4 a   z - a  z  + 
       z     z
 
       10  2      12  2      5  3       7  3      9  3       11  3    13  3
>   7 a   z  + 6 a   z  + 3 a  z  - 11 a  z  + 2 a  z  + 13 a   z  + a   z  + 
 
       15  3      6  4      8  4       10  4      12  4      14  4    5  5
>   4 a   z  + 5 a  z  - 4 a  z  - 16 a   z  - 5 a   z  + 2 a   z  - a  z  + 
 
        7  5      9  5       11  5    13  5    15  5      6  6      8  6
>   10 a  z  - 3 a  z  - 12 a   z  + a   z  - a   z  - 2 a  z  + 5 a  z  + 
 
        10  6      12  6    14  6      7  7      9  7      11  7    13  7
>   10 a   z  + 2 a   z  - a   z  - 3 a  z  + 2 a  z  + 4 a   z  - a   z  - 
 
       8  8      10  8    12  8    9  9    11  9
>   2 a  z  - 3 a   z  - a   z  - a  z  - a   z
In[10]:=
Kh[L][q, t]
Out[10]=   
 -6    -4      1         1        1        2        1        3        2
q   + q   + ------- + ------- + ------ + ------ + ------ + ------ + ------ + 
             26  10    24  10    24  9    22  8    20  8    20  7    18  7
            q   t     q   t     q   t    q   t    q   t    q   t    q   t
 
      3        3        4        3        3        4        2        3
>   ------ + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 
     18  6    16  6    16  5    14  5    14  4    12  4    12  3    10  3
    q   t    q   t    q   t    q   t    q   t    q   t    q   t    q   t
 
      2        2      2
>   ------ + ----- + ----
     10  2    8  2    6
    q   t    q  t    q  t


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