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L11a341

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Acknowledgement

L11a341 as Morse Link
DrawMorseLink

PD Presentation: X12,1,13,2 X16,8,17,7 X10,5,1,6 X6374 X4,9,5,10 X18,14,19,13 X22,20,11,19 X20,15,21,16 X14,21,15,22 X2,11,3,12 X8,18,9,17

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

Jones Polynomial: q-15/2 - 3q-13/2 + 7q-11/2 - 13q-9/2 + 18q-7/2 - 22q-5/2 + 22q-3/2 - 20q-1/2 + 15q1/2 - 10q3/2 + 4q5/2 - q7/2

A2 (sl(3)) Invariant: - q-24 - q-22 + q-20 - 2q-18 + q-16 + 5q-14 - 2q-12 + 4q-10 - q-8 - 2q-6 + 2q-4 - 4q-2 + 6 - q2 + q4 + 4q6 - 2q8 + q10

HOMFLY-PT Polynomial: - 2a-1z-1 - 3a-1z - 2a-1z3 - a-1z5 + 5az-1 + 10az + 9az3 + 4az5 + az7 - 6a3z-1 - 12a3z - 9a3z3 - 3a3z5 + 4a5z-1 + 6a5z + 3a5z3 - a7z-1 - a7z

Kauffman Polynomial: a-3z3 - a-3z5 - a-2z2 + 4a-2z4 - 4a-2z6 - 2a-1z-1 + 7a-1z - 12a-1z3 + 15a-1z5 - 9a-1z7 + 1 - 4z4 + 13z6 - 10z8 - 5az-1 + 24az - 47az3 + 45az5 - 12az7 - 5az9 + a2 - 19a2z4 + 38a2z6 - 19a2z8 - a2z10 - 6a3z-1 + 31a3z - 60a3z3 + 50a3z5 - 6a3z7 - 8a3z9 + 3a4 - 6a4z2 - 10a4z4 + 27a4z6 - 13a4z8 - a4z10 - 4a5z-1 + 18a5z - 34a5z3 + 29a5z5 - 6a5z7 - 3a5z9 + 3a6 - 8a6z2 + 4a6z4 + 5a6z6 - 4a6z8 - a7z-1 + 4a7z - 8a7z3 + 8a7z5 - 3a7z7 + a8 - 3a8z2 + 3a8z4 - a8z6

Khovanov Homology:
trqj r = -7r = -6r = -5r = -4r = -3r = -2r = -1r = 0r = 1r = 2r = 3r = 4
j = 8           1
j = 6          3 
j = 4         71 
j = 2        83  
j = 0       127   
j = -2      1210    
j = -4     1010     
j = -6    812      
j = -8   510       
j = -10  28        
j = -12 15         
j = -14 2          
j = -161           


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, 341]]]
Out[3]=   
 -Graphics- 
In[4]:=
PD[L = Link[11, Alternating, 341]]
Out[4]=   
PD[X[12, 1, 13, 2], X[16, 8, 17, 7], X[10, 5, 1, 6], X[6, 3, 7, 4], 
 
>   X[4, 9, 5, 10], X[18, 14, 19, 13], X[22, 20, 11, 19], X[20, 15, 21, 16], 
 
>   X[14, 21, 15, 22], X[2, 11, 3, 12], X[8, 18, 9, 17]]
In[5]:=
GaussCode[L]
Out[5]=   
GaussCode[{1, -10, 4, -5, 3, -4, 2, -11, 5, -3}, 
 
>   {10, -1, 6, -9, 8, -2, 11, -6, 7, -8, 9, -7}]
In[6]:=
Jones[L][q]
Out[6]=   
 -(15/2)     3       7      13     18     22     22      20
q        - ----- + ----- - ---- + ---- - ---- + ---- - ------- + 15 Sqrt[q] - 
            13/2    11/2    9/2    7/2    5/2    3/2   Sqrt[q]
           q       q       q      q      q      q
 
        3/2      5/2    7/2
>   10 q    + 4 q    - q
In[7]:=
A2Invariant[L][q]
Out[7]=   
     -24    -22    -20    2     -16    5     2     4     -8   2    2    4
6 - q    - q    + q    - --- + q    + --- - --- + --- - q   - -- + -- - -- - 
                          18           14    12    10          6    4    2
                         q            q     q     q           q    q    q
 
     2    4      6      8    10
>   q  + q  + 4 q  - 2 q  + q
In[8]:=
HOMFLYPT[Link[11, Alternating, 341]][a, z]
Out[8]=   
               3      5    7                                               3
-2    5 a   6 a    4 a    a    3 z                3        5      7     2 z
--- + --- - ---- + ---- - -- - --- + 10 a z - 12 a  z + 6 a  z - a  z - ---- + 
a z    z     z      z     z     a                                        a
 
                                  5
         3      3  3      5  3   z         5      3  5      7
>   9 a z  - 9 a  z  + 3 a  z  - -- + 4 a z  - 3 a  z  + a z
                                 a
In[9]:=
Kauffman[Link[11, Alternating, 341]][a, z]
Out[9]=   
                                           3      5    7
     2      4      6    8    2    5 a   6 a    4 a    a    7 z
1 + a  + 3 a  + 3 a  + a  - --- - --- - ---- - ---- - -- + --- + 24 a z + 
                            a z    z     z      z     z     a
 
                                  2                                  3
        3         5        7     z       4  2      6  2      8  2   z
>   31 a  z + 18 a  z + 4 a  z - -- - 6 a  z  - 8 a  z  - 3 a  z  + -- - 
                                  2                                  3
                                 a                                  a
 
        3                                                       4
    12 z          3       3  3       5  3      7  3      4   4 z        2  4
>   ----- - 47 a z  - 60 a  z  - 34 a  z  - 8 a  z  - 4 z  + ---- - 19 a  z  - 
      a                                                        2
                                                              a
 
                                    5       5
        4  4      6  4      8  4   z    15 z          5       3  5       5  5
>   10 a  z  + 4 a  z  + 3 a  z  - -- + ----- + 45 a z  + 50 a  z  + 29 a  z  + 
                                    3     a
                                   a
 
                         6                                              7
       7  5       6   4 z        2  6       4  6      6  6    8  6   9 z
>   8 a  z  + 13 z  - ---- + 38 a  z  + 27 a  z  + 5 a  z  - a  z  - ---- - 
                        2                                             a
                       a
 
          7      3  7      5  7      7  7       8       2  8       4  8
>   12 a z  - 6 a  z  - 6 a  z  - 3 a  z  - 10 z  - 19 a  z  - 13 a  z  - 
 
       6  8        9      3  9      5  9    2  10    4  10
>   4 a  z  - 5 a z  - 8 a  z  - 3 a  z  - a  z   - a  z
In[10]:=
Kh[L][q, t]
Out[10]=   
     10     1        2        1        5        2        8        5      10
12 + -- + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 
      2    16  7    14  6    12  6    12  5    10  5    10  4    8  4    8  3
     q    q   t    q   t    q   t    q   t    q   t    q   t    q  t    q  t
 
      8      12      10      10     12             2        2  2      4  2
>   ----- + ----- + ----- + ---- + ---- + 7 t + 8 q  t + 3 q  t  + 7 q  t  + 
     6  3    6  2    4  2    4      2
    q  t    q  t    q  t    q  t   q  t
 
     4  3      6  3    8  4
>   q  t  + 3 q  t  + q  t


Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a341
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