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L11a34

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Acknowledgement

L11a34 as Morse Link
DrawMorseLink

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

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

Jones Polynomial: q-27/2 - 2q-25/2 + 6q-23/2 - 10q-21/2 + 14q-19/2 - 18q-17/2 + 17q-15/2 - 16q-13/2 + 12q-11/2 - 8q-9/2 + 3q-7/2 - q-5/2

A2 (sl(3)) Invariant: - q-42 - 2q-40 - q-38 - 4q-36 - q-34 + 3q-32 - q-30 + 5q-28 + 2q-26 + 2q-24 + 4q-22 - 2q-20 + 4q-18 - q-16 + 3q-12 - 2q-10 + q-8

HOMFLY-PT Polynomial: - a5z - 2a5z3 - a5z5 - a7z-1 - 5a7z - 8a7z3 - 3a7z5 - a9z-1 - 2a9z - 4a9z3 - 2a9z5 + 4a11z-1 + 7a11z + 3a11z3 - 2a13z-1 - a13z

Kauffman Polynomial: - a5z + 2a5z3 - a5z5 - a6z2 + 4a6z4 - 3a6z6 - a7z-1 + 6a7z - 10a7z3 + 11a7z5 - 6a7z7 + a8 - 3a8z2 + 2a8z4 + 6a8z6 - 6a8z8 + a9z-1 - a9z - 3a9z3 + 6a9z5 - 4a9z9 - 5a10 + 18a10z2 - 28a10z4 + 25a10z6 - 10a10z8 - a10z10 + 4a11z-1 - 16a11z + 22a11z3 - 17a11z5 + 13a11z7 - 7a11z9 - 6a12 + 24a12z2 - 32a12z4 + 23a12z6 - 7a12z8 - a12z10 + 2a13z-1 - 9a13z + 11a13z3 - 6a13z5 + 5a13z7 - 3a13z9 + a14 - a14z2 - 2a14z4 + 6a14z6 - 3a14z8 - a15z - 2a15z3 + 5a15z5 - 2a15z7 + 2a16 - 5a16z2 + 4a16z4 - a16z6

Khovanov Homology:
trqj r = -11r = -10r = -9r = -8r = -7r = -6r = -5r = -4r = -3r = -2r = -1r = 0
j = -4           1
j = -6          31
j = -8         5  
j = -10        73  
j = -12       95   
j = -14      98    
j = -16     98     
j = -18    59      
j = -20   59       
j = -22  15        
j = -24 15         
j = -26 1          
j = -281           


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[11, Alternating, 34]]]
Out[3]=   
 -Graphics- 
In[4]:=
PD[L = Link[11, Alternating, 34]]
Out[4]=   
PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[18, 11, 19, 12], X[16, 7, 17, 8], 
 
>   X[8, 17, 9, 18], X[22, 15, 5, 16], X[12, 21, 13, 22], X[20, 13, 21, 14], 
 
>   X[14, 19, 15, 20], X[2, 5, 3, 6], X[4, 9, 1, 10]]
In[5]:=
GaussCode[L]
Out[5]=   
GaussCode[{1, -10, 2, -11}, {10, -1, 4, -5, 11, -2, 3, -7, 8, -9, 6, -4, 5, -3, 
 
>    9, -8, 7, -6}]
In[6]:=
Jones[L][q]
Out[6]=   
 -(27/2)     2       6      10      14      18      17      16      12
q        - ----- + ----- - ----- + ----- - ----- + ----- - ----- + ----- - 
            25/2    23/2    21/2    19/2    17/2    15/2    13/2    11/2
           q       q       q       q       q       q       q       q
 
     8      3      -(5/2)
>   ---- + ---- - q
     9/2    7/2
    q      q
In[7]:=
A2Invariant[L][q]
Out[7]=   
  -42    2     -38    4     -34    3     -30    5     2     2     4     2
-q    - --- - q    - --- - q    + --- - q    + --- + --- + --- + --- - --- + 
         40           36           32           28    26    24    22    20
        q            q            q            q     q     q     q     q
 
     4     -16    3     2     -8
>   --- - q    + --- - --- + q
     18           12    10
    q            q     q
In[8]:=
HOMFLYPT[Link[11, Alternating, 34]][a, z]
Out[8]=   
   7     9      11      13
  a     a    4 a     2 a      5        7        9        11      13
-(--) - -- + ----- - ----- - a  z - 5 a  z - 2 a  z + 7 a   z - a   z - 
  z     z      z       z
 
       5  3      7  3      9  3      11  3    5  5      7  5      9  5
>   2 a  z  - 8 a  z  - 4 a  z  + 3 a   z  - a  z  - 3 a  z  - 2 a  z
In[9]:=
Kauffman[Link[11, Alternating, 34]][a, z]
Out[9]=   
                                    7    9      11      13
 8      10      12    14      16   a    a    4 a     2 a      5        7
a  - 5 a   - 6 a   + a   + 2 a   - -- + -- + ----- + ----- - a  z + 6 a  z - 
                                   z    z      z       z
 
     9         11        13      15      6  2      8  2       10  2
>   a  z - 16 a   z - 9 a   z - a   z - a  z  - 3 a  z  + 18 a   z  + 
 
        12  2    14  2      16  2      5  3       7  3      9  3       11  3
>   24 a   z  - a   z  - 5 a   z  + 2 a  z  - 10 a  z  - 3 a  z  + 22 a   z  + 
 
        13  3      15  3      6  4      8  4       10  4       12  4
>   11 a   z  - 2 a   z  + 4 a  z  + 2 a  z  - 28 a   z  - 32 a   z  - 
 
       14  4      16  4    5  5       7  5      9  5       11  5      13  5
>   2 a   z  + 4 a   z  - a  z  + 11 a  z  + 6 a  z  - 17 a   z  - 6 a   z  + 
 
       15  5      6  6      8  6       10  6       12  6      14  6    16  6
>   5 a   z  - 3 a  z  + 6 a  z  + 25 a   z  + 23 a   z  + 6 a   z  - a   z  - 
 
       7  7       11  7      13  7      15  7      8  8       10  8
>   6 a  z  + 13 a   z  + 5 a   z  - 2 a   z  - 6 a  z  - 10 a   z  - 
 
       12  8      14  8      9  9      11  9      13  9    10  10    12  10
>   7 a   z  - 3 a   z  - 4 a  z  - 7 a   z  - 3 a   z  - a   z   - a   z
In[10]:=
Kh[L][q, t]
Out[10]=   
 -6    -4      1         1         1        5        1        5        5
q   + q   + ------- + ------- + ------- + ------ + ------ + ------ + ------ + 
             28  11    26  10    24  10    24  9    22  9    22  8    20  8
            q   t     q   t     q   t     q   t    q   t    q   t    q   t
 
      9        5        9        9        8        9        8        9
>   ------ + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 
     20  7    18  7    18  6    16  6    16  5    14  5    14  4    12  4
    q   t    q   t    q   t    q   t    q   t    q   t    q   t    q   t
 
      5        7        3        5      3
>   ------ + ------ + ------ + ----- + ----
     12  3    10  3    10  2    8  2    6
    q   t    q   t    q   t    q  t    q  t


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