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L11n45

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Acknowledgement

L11n45 as Morse Link
DrawMorseLink

PD Presentation: X6172 X16,7,17,8 X17,1,18,4 X5,14,6,15 X3849 X9,18,10,19 X11,20,12,21 X13,22,14,5 X19,10,20,11 X21,12,22,13 X2,16,3,15

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

Jones Polynomial: q-23/2 - q-21/2 + 2q-19/2 - 3q-17/2 + 4q-15/2 - 4q-13/2 + 3q-11/2 - 3q-9/2 + q-7/2 - 2q-5/2

A2 (sl(3)) Invariant: - q-34 - q-32 - 2q-30 - 2q-28 - q-26 - q-24 + q-22 + 2q-18 + 3q-16 + 3q-14 + 4q-12 + 2q-10 + 2q-8

HOMFLY-PT Polynomial: - 4a5z-1 - 13a5z - 10a5z3 - 2a5z5 + 7a7z-1 + 20a7z + 18a7z3 + 7a7z5 + a7z7 - 3a9z-1 - 7a9z - 5a9z3 - a9z5

Kauffman Polynomial: 4a5z-1 - 16a5z + 14a5z3 - 3a5z5 - 7a6 + 19a6z2 - 15a6z4 + 6a6z6 - a6z8 + 7a7z-1 - 26a7z + 36a7z3 - 22a7z5 + 7a7z7 - a7z9 - 7a8 + 23a8z2 - 24a8z4 + 11a8z6 - 2a8z8 + 3a9z-1 - 10a9z + 16a9z3 - 15a9z5 + 6a9z7 - a9z9 - 6a10z4 + 4a10z6 - a10z8 - 4a11z3 + 3a11z5 - a11z7 - a12z2 + 2a12z4 - a12z6 + 2a13z3 - a13z5 - a14 + 3a14z2 - a14z4

Khovanov Homology:
trqj r = -9r = -8r = -7r = -6r = -5r = -4r = -3r = -2r = -1r = 0
j = -4         2
j = -6        12
j = -8       2  
j = -10      11  
j = -12     32   
j = -14    11    
j = -16   23     
j = -18   1      
j = -20 12       
j = -22          
j = -241         


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, NonAlternating, 45]]]
Out[3]=   
 -Graphics- 
In[4]:=
PD[L = Link[11, NonAlternating, 45]]
Out[4]=   
PD[X[6, 1, 7, 2], X[16, 7, 17, 8], X[17, 1, 18, 4], X[5, 14, 6, 15], 
 
>   X[3, 8, 4, 9], X[9, 18, 10, 19], X[11, 20, 12, 21], X[13, 22, 14, 5], 
 
>   X[19, 10, 20, 11], X[21, 12, 22, 13], X[2, 16, 3, 15]]
In[5]:=
GaussCode[L]
Out[5]=   
GaussCode[{1, -11, -5, 3}, {-4, -1, 2, 5, -6, 9, -7, 10, -8, 4, 11, -2, -3, 6, 
 
>    -9, 7, -10, 8}]
In[6]:=
Jones[L][q]
Out[6]=   
 -(23/2)    -(21/2)     2       3       4       4       3      3      -(7/2)
q        - q        + ----- - ----- + ----- - ----- + ----- - ---- + q       - 
                       19/2    17/2    15/2    13/2    11/2    9/2
                      q       q       q       q       q       q
 
     2
>   ----
     5/2
    q
In[7]:=
A2Invariant[L][q]
Out[7]=   
  -34    -32    2     2     -26    -24    -22    2     3     3     4     2    2
-q    - q    - --- - --- - q    - q    + q    + --- + --- + --- + --- + --- + --
                30    28                         18    16    14    12    10    8
               q     q                          q     q     q     q     q     q
In[8]:=
HOMFLYPT[Link[11, NonAlternating, 45]][a, z]
Out[8]=   
    5      7      9
-4 a    7 a    3 a        5         7        9         5  3       7  3
----- + ---- - ---- - 13 a  z + 20 a  z - 7 a  z - 10 a  z  + 18 a  z  - 
  z      z      z
 
       9  3      5  5      7  5    9  5    7  7
>   5 a  z  - 2 a  z  + 7 a  z  - a  z  + a  z
In[9]:=
Kauffman[Link[11, NonAlternating, 45]][a, z]
Out[9]=   
                        5      7      9
    6      8    14   4 a    7 a    3 a        5         7         9
-7 a  - 7 a  - a   + ---- + ---- + ---- - 16 a  z - 26 a  z - 10 a  z + 
                      z      z      z
 
        6  2       8  2    12  2      14  2       5  3       7  3       9  3
>   19 a  z  + 23 a  z  - a   z  + 3 a   z  + 14 a  z  + 36 a  z  + 16 a  z  - 
 
       11  3      13  3       6  4       8  4      10  4      12  4    14  4
>   4 a   z  + 2 a   z  - 15 a  z  - 24 a  z  - 6 a   z  + 2 a   z  - a   z  - 
 
       5  5       7  5       9  5      11  5    13  5      6  6       8  6
>   3 a  z  - 22 a  z  - 15 a  z  + 3 a   z  - a   z  + 6 a  z  + 11 a  z  + 
 
       10  6    12  6      7  7      9  7    11  7    6  8      8  8    10  8
>   4 a   z  - a   z  + 7 a  z  + 6 a  z  - a   z  - a  z  - 2 a  z  - a   z  - 
 
     7  9    9  9
>   a  z  - a  z
In[10]:=
Kh[L][q, t]
Out[10]=   
2    2      1        1        2        1        2        3        1
-- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 
 6    4    24  9    20  8    20  7    18  6    16  6    16  5    14  5
q    q    q   t    q   t    q   t    q   t    q   t    q   t    q   t
 
      1        3        2        1        1        2      1
>   ------ + ------ + ------ + ------ + ------ + ----- + ----
     14  4    12  4    12  3    10  3    10  2    8  2    6
    q   t    q   t    q   t    q   t    q   t    q  t    q  t


Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n45
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