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L11a23

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Acknowledgement

L11a23 as Morse Link
DrawMorseLink

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

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

Jones Polynomial: q-27/2 - 3q-25/2 + 8q-23/2 - 13q-21/2 + 17q-19/2 - 20q-17/2 + 20q-15/2 - 18q-13/2 + 12q-11/2 - 8q-9/2 + 3q-7/2 - q-5/2

A2 (sl(3)) Invariant: - q-42 - q-40 + q-38 - 4q-36 - q-34 + 2q-32 - 3q-30 + 4q-28 + 2q-24 + 4q-22 - q-20 + 6q-18 - q-16 + 3q-12 - 2q-10 + q-8

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

Kauffman Polynomial: - a5z + 2a5z3 - a5z5 - a6z2 + 4a6z4 - 3a6z6 - 2a7z-1 + 8a7z - 12a7z3 + 11a7z5 - 6a7z7 + 3a8 - 8a8z2 + 5a8z4 + 4a8z6 - 6a8z8 - 2a9z-1 + 8a9z - 18a9z3 + 19a9z5 - 5a9z7 - 4a9z9 + 3a10z2 - 20a10z4 + 30a10z6 - 14a10z8 - a10z10 + a11z-1 - 3a11z - 5a11z3 + 11a11z5 + 5a11z7 - 8a11z9 - 3a12 + 14a12z2 - 29a12z4 + 34a12z6 - 13a12z8 - a12z10 + a13z-1 - a13z - 6a13z3 + 11a13z5 + a13z7 - 4a13z9 + a14z2 - 5a14z4 + 10a14z6 - 5a14z8 + a15z - 5a15z3 + 7a15z5 - 3a15z7 + a16 - 3a16z2 + 3a16z4 - 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       115   
j = -14      108    
j = -16     1010     
j = -18    710      
j = -20   610       
j = -22  27        
j = -24 16         
j = -26 2          
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, 23]]]
Out[3]=   
 -Graphics- 
In[4]:=
PD[L = Link[11, Alternating, 23]]
Out[4]=   
PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[16, 11, 17, 12], X[14, 7, 15, 8], 
 
>   X[8, 15, 9, 16], X[20, 13, 21, 14], X[22, 17, 5, 18], X[18, 21, 19, 22], 
 
>   X[12, 19, 13, 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, -9, 6, -4, 5, -3, 7, -8, 
 
>    9, -6, 8, -7}]
In[6]:=
Jones[L][q]
Out[6]=   
 -(27/2)     3       8      13      17      20      20      18      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    -40    -38    4     -34    2     3     4     2     4     -20    6
-q    - q    + q    - --- - q    + --- - --- + --- + --- + --- - q    + --- - 
                       36           32    30    28    24    22           18
                      q            q     q     q     q     q            q
 
     -16    3     2     -8
>   q    + --- - --- + q
            12    10
           q     q
In[8]:=
HOMFLYPT[Link[11, Alternating, 23]][a, z]
Out[8]=   
    7      9    11    13
-2 a    2 a    a     a      5        7        9        11      13        5  3
----- + ---- + --- - --- - a  z - 7 a  z + 3 a  z + 4 a   z - a   z - 2 a  z  - 
  z      z      z     z
 
       7  3      9  3      11  3    5  5      7  5      9  5
>   8 a  z  - 2 a  z  + 3 a   z  - a  z  - 3 a  z  - 2 a  z
In[9]:=
Kauffman[Link[11, Alternating, 23]][a, z]
Out[9]=   
                        7      9    11    13
   8      12    16   2 a    2 a    a     a      5        7        9
3 a  - 3 a   + a   - ---- - ---- + --- + --- - a  z + 8 a  z + 8 a  z - 
                      z      z      z     z
 
       11      13      15      6  2      8  2      10  2       12  2    14  2
>   3 a   z - a   z + a   z - a  z  - 8 a  z  + 3 a   z  + 14 a   z  + a   z  - 
 
       16  2      5  3       7  3       9  3      11  3      13  3      15  3
>   3 a   z  + 2 a  z  - 12 a  z  - 18 a  z  - 5 a   z  - 6 a   z  - 5 a   z  + 
 
       6  4      8  4       10  4       12  4      14  4      16  4    5  5
>   4 a  z  + 5 a  z  - 20 a   z  - 29 a   z  - 5 a   z  + 3 a   z  - a  z  + 
 
        7  5       9  5       11  5       13  5      15  5      6  6
>   11 a  z  + 19 a  z  + 11 a   z  + 11 a   z  + 7 a   z  - 3 a  z  + 
 
       8  6       10  6       12  6       14  6    16  6      7  7      9  7
>   4 a  z  + 30 a   z  + 34 a   z  + 10 a   z  - a   z  - 6 a  z  - 5 a  z  + 
 
       11  7    13  7      15  7      8  8       10  8       12  8      14  8
>   5 a   z  + a   z  - 3 a   z  - 6 a  z  - 14 a   z  - 13 a   z  - 5 a   z  - 
 
       9  9      11  9      13  9    10  10    12  10
>   4 a  z  - 8 a   z  - 4 a   z  - a   z   - a   z
In[10]:=
Kh[L][q, t]
Out[10]=   
 -6    -4      1         2         1        6        2        7        6
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
 
      10       7        10       10       10       10       8        11
>   ------ + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 
     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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