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L11n385

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Acknowledgement

L11n385 as Morse Link
DrawMorseLink

PD Presentation: X6172 X14,7,15,8 X4,15,1,16 X5,12,6,13 X3849 X9,16,10,17 X17,19,18,22 X11,20,12,21 X19,10,20,11 X21,5,22,18 X13,2,14,3

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

Jones Polynomial: - q-10 + q-9 - q-8 + 2q-7 - 2q-6 + 3q-5 - q-4 + 3q-3 - q-2 + q-1

A2 (sl(3)) Invariant: - q-34 - 2q-30 - 2q-28 - q-26 + 2q-22 + 2q-20 + 5q-18 + 5q-16 + 6q-14 + 5q-12 + 4q-10 + 2q-8 + q-6 + q-4

HOMFLY-PT Polynomial: 2a4z-2 + 8a4 + 11a4z2 + 6a4z4 + a4z6 - 5a6z-2 - 16a6 - 21a6z2 - 17a6z4 - 7a6z6 - a6z8 + 4a8z-2 + 10a8 + 11a8z2 + 6a8z4 + a8z6 - a10z-2 - 2a10 - a10z2

Kauffman Polynomial: - 2a4z-2 + 10a4 - 19a4z2 + 17a4z4 - 7a4z6 + a4z8 + 5a5z-1 - 15a5z + 12a5z3 + 3a5z5 - 5a5z7 + a5z9 - 5a6z-2 + 22a6 - 47a6z2 + 54a6z4 - 26a6z6 + 4a6z8 + 9a7z-1 - 29a7z + 30a7z3 - 8a7z5 - 3a7z7 + a7z9 - 4a8z-2 + 17a8 - 36a8z2 + 39a8z4 - 19a8z6 + 3a8z8 + 5a9z-1 - 17a9z + 19a9z3 - 11a9z5 + 2a9z7 - a10z-2 + 4a10 - 7a10z2 + 2a10z4 + a11z-1 - 2a11z + a11z3 + a12z2 + a13z

Khovanov Homology:
trqj r = -7r = -6r = -5r = -4r = -3r = -2r = -1r = 0r = 1r = 2
j = -1         1
j = -3          
j = -5       31 
j = -7     112  
j = -9     31   
j = -11   221    
j = -13  143     
j = -15 111      
j = -17 22       
j = -1911        
j = -211         


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, 385]]]
Out[3]=   
 -Graphics- 
In[4]:=
PD[L = Link[11, NonAlternating, 385]]
Out[4]=   
PD[X[6, 1, 7, 2], X[14, 7, 15, 8], X[4, 15, 1, 16], X[5, 12, 6, 13], 
 
>   X[3, 8, 4, 9], X[9, 16, 10, 17], X[17, 19, 18, 22], X[11, 20, 12, 21], 
 
>   X[19, 10, 20, 11], X[21, 5, 22, 18], X[13, 2, 14, 3]]
In[5]:=
GaussCode[L]
Out[5]=   
GaussCode[{1, 11, -5, -3}, {-9, 8, -10, 7}, 
 
>   {-4, -1, 2, 5, -6, 9, -8, 4, -11, -2, 3, 6, -7, 10}]
In[6]:=
Jones[L][q]
Out[6]=   
  -10    -9    -8   2    2    3     -4   3     -2   1
-q    + q   - q   + -- - -- + -- - q   + -- - q   + -
                     7    6    5          3         q
                    q    q    q          q
In[7]:=
A2Invariant[L][q]
Out[7]=   
  -34    2     2     -26    2     2     5     5     6     5     4    2     -6
-q    - --- - --- - q    + --- + --- + --- + --- + --- + --- + --- + -- + q   + 
         30    28           22    20    18    16    14    12    10    8
        q     q            q     q     q     q     q     q     q     q
 
     -4
>   q
In[8]:=
HOMFLYPT[Link[11, NonAlternating, 385]][a, z]
Out[8]=   
                                  4      6      8    10
   4       6       8      10   2 a    5 a    4 a    a         4  2       6  2
8 a  - 16 a  + 10 a  - 2 a   + ---- - ---- + ---- - --- + 11 a  z  - 21 a  z  + 
                                 2      2      2     2
                                z      z      z     z
 
        8  2    10  2      4  4       6  4      8  4    4  6      6  6
>   11 a  z  - a   z  + 6 a  z  - 17 a  z  + 6 a  z  + a  z  - 7 a  z  + 
 
     8  6    6  8
>   a  z  - a  z
In[9]:=
Kauffman[Link[11, NonAlternating, 385]][a, z]
Out[9]=   
                                   4      6      8    10      5      7      9
    4       6       8      10   2 a    5 a    4 a    a     5 a    9 a    5 a
10 a  + 22 a  + 17 a  + 4 a   - ---- - ---- - ---- - --- + ---- + ---- + ---- + 
                                  2      2      2     2     z      z      z
                                 z      z      z     z
 
     11
    a         5         7         9        11      13         4  2       6  2
>   --- - 15 a  z - 29 a  z - 17 a  z - 2 a   z + a   z - 19 a  z  - 47 a  z  - 
     z
 
        8  2      10  2    12  2       5  3       7  3       9  3    11  3
>   36 a  z  - 7 a   z  + a   z  + 12 a  z  + 30 a  z  + 19 a  z  + a   z  + 
 
        4  4       6  4       8  4      10  4      5  5      7  5       9  5
>   17 a  z  + 54 a  z  + 39 a  z  + 2 a   z  + 3 a  z  - 8 a  z  - 11 a  z  - 
 
       4  6       6  6       8  6      5  7      7  7      9  7    4  8
>   7 a  z  - 26 a  z  - 19 a  z  - 5 a  z  - 3 a  z  + 2 a  z  + a  z  + 
 
       6  8      8  8    5  9    7  9
>   4 a  z  + 3 a  z  + a  z  + a  z
In[10]:=
Kh[L][q, t]
Out[10]=   
2    3      1        1        1        2        1        2        1
-- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 
 7    5    21  7    19  7    19  6    17  6    15  6    17  5    15  5
q    q    q   t    q   t    q   t    q   t    q   t    q   t    q   t
 
      1        1        4        2        3        2        1        3
>   ------ + ------ + ------ + ------ + ------ + ------ + ------ + ----- + 
     13  5    15  4    13  4    11  4    13  3    11  3    11  2    9  2
    q   t    q   t    q   t    q   t    q   t    q   t    q   t    q  t
 
                                2
      1      1      1     t    t
>   ----- + ---- + ---- + -- + --
     7  2    9      7      5   q
    q  t    q  t   q  t   q


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