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L11a422

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Acknowledgement

L11a422 as Morse Link
DrawMorseLink

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

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

Jones Polynomial: - q-11 + 3q-10 - 8q-9 + 14q-8 - 19q-7 + 23q-6 - 22q-5 + 21q-4 - 14q-3 + 10q-2 - 4q-1 + 1

A2 (sl(3)) Invariant: - q-34 - q-32 - 5q-28 + q-26 + q-24 - 2q-22 + 7q-20 + q-18 + 8q-16 + 6q-14 + 3q-12 + 8q-10 - 2q-8 + 3q-6 + q-4 - 2q-2 + 1

HOMFLY-PT Polynomial: a2z2 + a2z4 + 2a4z-2 + 7a4 + 6a4z2 - a4z6 - 5a6z-2 - 14a6 - 13a6z2 - 7a6z4 - 2a6z6 + 4a8z-2 + 9a8 + 8a8z2 + 3a8z4 - a10z-2 - 2a10 - a10z2

Kauffman Polynomial: a2z2 - 2a2z4 + a2z6 + 3a3z3 - 8a3z5 + 4a3z7 - 2a4z-2 + 9a4 - 14a4z2 + 18a4z4 - 20a4z6 + 8a4z8 + 5a5z-1 - 13a5z + 13a5z3 - 5a5z5 - 10a5z7 + 7a5z9 - 5a6z-2 + 21a6 - 47a6z2 + 67a6z4 - 56a6z6 + 16a6z8 + 2a6z10 + 9a7z-1 - 29a7z + 34a7z3 - 7a7z5 - 20a7z7 + 13a7z9 - 4a8z-2 + 18a8 - 44a8z2 + 65a8z4 - 51a8z6 + 16a8z8 + 2a8z10 + 5a9z-1 - 21a9z + 33a9z3 - 20a9z5 + 6a9z9 - a10z-2 + 5a10 - 11a10z2 + 14a10z4 - 13a10z6 + 8a10z8 + a11z-1 - 4a11z + 7a11z3 - 9a11z5 + 6a11z7 + a12z2 - 4a12z4 + 3a12z6 + a13z - 2a13z3 + a13z5

Khovanov Homology:
trqj r = -9r = -8r = -7r = -6r = -5r = -4r = -3r = -2r = -1r = 0r = 1r = 2
j = 1           1
j = -1          3 
j = -3         71 
j = -5        84  
j = -7       136   
j = -9      1110    
j = -11     1211     
j = -13    812      
j = -15   611       
j = -17  28        
j = -19 16         
j = -21 2          
j = -231           


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, Alternating, 422]]]
Out[3]=   
 -Graphics- 
In[4]:=
PD[L = Link[11, Alternating, 422]]
Out[4]=   
PD[X[6, 1, 7, 2], X[12, 3, 13, 4], X[18, 13, 19, 14], X[22, 17, 11, 18], 
 
>   X[16, 7, 17, 8], X[8, 22, 9, 21], X[20, 10, 21, 9], X[10, 15, 5, 16], 
 
>   X[14, 19, 15, 20], X[2, 5, 3, 6], X[4, 11, 1, 12]]
In[5]:=
GaussCode[L]
Out[5]=   
GaussCode[{1, -10, 2, -11}, {10, -1, 5, -6, 7, -8}, 
 
>   {11, -2, 3, -9, 8, -5, 4, -3, 9, -7, 6, -4}]
In[6]:=
Jones[L][q]
Out[6]=   
     -11    3    8    14   19   23   22   21   14   10   4
1 - q    + --- - -- + -- - -- + -- - -- + -- - -- + -- - -
            10    9    8    7    6    5    4    3    2   q
           q     q    q    q    q    q    q    q    q
In[7]:=
A2Invariant[L][q]
Out[7]=   
     -34    -32    5     -26    -24    2     7     -18    8     6     3
1 - q    - q    - --- + q    + q    - --- + --- + q    + --- + --- + --- + 
                   28                  22    20           16    14    12
                  q                   q     q            q     q     q
 
     8    2    3     -4   2
>   --- - -- + -- + q   - --
     10    8    6          2
    q     q    q          q
In[8]:=
HOMFLYPT[Link[11, Alternating, 422]][a, z]
Out[8]=   
                                 4      6      8    10
   4       6      8      10   2 a    5 a    4 a    a      2  2      4  2
7 a  - 14 a  + 9 a  - 2 a   + ---- - ---- + ---- - --- + a  z  + 6 a  z  - 
                                2      2      2     2
                               z      z      z     z
 
        6  2      8  2    10  2    2  4      6  4      8  4    4  6      6  6
>   13 a  z  + 8 a  z  - a   z  + a  z  - 7 a  z  + 3 a  z  - a  z  - 2 a  z
In[9]:=
Kauffman[Link[11, Alternating, 422]][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
9 a  + 21 a  + 18 a  + 5 a   - ---- - ---- - ---- - --- + ---- + ---- + ---- + 
                                 2      2      2     2     z      z      z
                                z      z      z     z
 
     11
    a         5         7         9        11      13      2  2       4  2
>   --- - 13 a  z - 29 a  z - 21 a  z - 4 a   z + a   z + a  z  - 14 a  z  - 
     z
 
        6  2       8  2       10  2    12  2      3  3       5  3       7  3
>   47 a  z  - 44 a  z  - 11 a   z  + a   z  + 3 a  z  + 13 a  z  + 34 a  z  + 
 
        9  3      11  3      13  3      2  4       4  4       6  4       8  4
>   33 a  z  + 7 a   z  - 2 a   z  - 2 a  z  + 18 a  z  + 67 a  z  + 65 a  z  + 
 
        10  4      12  4      3  5      5  5      7  5       9  5      11  5
>   14 a   z  - 4 a   z  - 8 a  z  - 5 a  z  - 7 a  z  - 20 a  z  - 9 a   z  + 
 
     13  5    2  6       4  6       6  6       8  6       10  6      12  6
>   a   z  + a  z  - 20 a  z  - 56 a  z  - 51 a  z  - 13 a   z  + 3 a   z  + 
 
       3  7       5  7       7  7      11  7      4  8       6  8       8  8
>   4 a  z  - 10 a  z  - 20 a  z  + 6 a   z  + 8 a  z  + 16 a  z  + 16 a  z  + 
 
       10  8      5  9       7  9      9  9      6  10      8  10
>   8 a   z  + 7 a  z  + 13 a  z  + 6 a  z  + 2 a  z   + 2 a  z
In[10]:=
Kh[L][q, t]
Out[10]=   
4    7      1        2        1        6        2        8        6
-- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 
 5    3    23  9    21  8    19  8    19  7    17  7    17  6    15  6
q    q    q   t    q   t    q   t    q   t    q   t    q   t    q   t
 
      11       8        12       12       11      11      10      13      6
>   ------ + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ---- + 
     15  5    13  5    13  4    11  4    11  3    9  3    9  2    7  2    7
    q   t    q   t    q   t    q   t    q   t    q  t    q  t    q  t    q  t
 
     8     t    3 t      2
>   ---- + -- + --- + q t
     5      3    q
    q  t   q


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