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L11n109

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Acknowledgement

L11n109 as Morse Link
DrawMorseLink

PD Presentation: X6172 X12,3,13,4 X7,18,8,19 X19,22,20,5 X13,20,14,21 X21,14,22,15 X9,16,10,17 X15,10,16,11 X17,8,18,9 X2536 X4,11,1,12

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

Jones Polynomial: q-25/2 - q-23/2 + q-21/2 - q-19/2 + q-15/2 - 2q-13/2 + q-11/2 - 2q-9/2 + q-7/2 - q-5/2

A2 (sl(3)) Invariant: - q-40 - 2q-38 - q-36 - q-34 + q-32 + 2q-30 + q-28 + q-26 - q-24 + q-22 + q-20 + 2q-18 + 3q-16 + q-14 + q-12 + q-8

HOMFLY-PT Polynomial: - a5z-1 - 3a5z - 4a5z3 - a5z5 + a7z-1 - 2a7z - 4a7z3 - a7z5 - a9z-1 + a9z + 2a11z-1 + 2a11z - a13z-1

Kauffman Polynomial: a5z-1 - 3a5z + 4a5z3 - a5z5 - a6 + 3a6z4 - a6z6 + a7z-1 - a7z - 3a7z3 + 4a7z5 - a7z7 - 3a8z2 + 4a8z4 - a8z6 + a9z-1 - 4a9z + 2a9z3 - 4a10 + 14a10z2 - 18a10z4 + 8a10z6 - a10z8 + 2a11z-1 - 10a11z + 21a11z3 - 20a11z5 + 8a11z7 - a11z9 - 7a12 + 28a12z2 - 34a12z4 + 15a12z6 - 2a12z8 + a13z-1 - 4a13z + 12a13z3 - 15a13z5 + 7a13z7 - a13z9 - 3a14 + 11a14z2 - 15a14z4 + 7a14z6 - a14z8

Khovanov Homology:
trqj r = -11r = -10r = -9r = -8r = -7r = -6r = -5r = -4r = -3r = -2r = -1r = 0
j = -4           1
j = -6          11
j = -8         1  
j = -10       111  
j = -12      131   
j = -14       1    
j = -16    122     
j = -18   1        
j = -20   11       
j = -22 11         
j = -24            
j = -261           


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, 109]]]
Out[3]=   
 -Graphics- 
In[4]:=
PD[L = Link[11, NonAlternating, 109]]
Out[4]=   
PD[X[6, 1, 7, 2], X[12, 3, 13, 4], X[7, 18, 8, 19], X[19, 22, 20, 5], 
 
>   X[13, 20, 14, 21], X[21, 14, 22, 15], X[9, 16, 10, 17], X[15, 10, 16, 11], 
 
>   X[17, 8, 18, 9], X[2, 5, 3, 6], X[4, 11, 1, 12]]
In[5]:=
GaussCode[L]
Out[5]=   
GaussCode[{1, -10, 2, -11}, {10, -1, -3, 9, -7, 8, 11, -2, -5, 6, -8, 7, -9, 3, 
 
>    -4, 5, -6, 4}]
In[6]:=
Jones[L][q]
Out[6]=   
 -(25/2)    -(23/2)    -(21/2)    -(19/2)    -(15/2)     2      -(11/2)
q        - q        + q        - q        + q        - ----- + q        - 
                                                        13/2
                                                       q
 
     2      -(7/2)    -(5/2)
>   ---- + q       - q
     9/2
    q
In[7]:=
A2Invariant[L][q]
Out[7]=   
  -40    2     -36    -34    -32    2     -28    -26    -24    -22    -20
-q    - --- - q    - q    + q    + --- + q    + q    - q    + q    + q    + 
         38                         30
        q                          q
 
     2     3     -14    -12    -8
>   --- + --- + q    + q    + q
     18    16
    q     q
In[8]:=
HOMFLYPT[Link[11, NonAlternating, 109]][a, z]
Out[8]=   
   5     7    9      11    13
  a     a    a    2 a     a        5        7      9        11        5  3
-(--) + -- - -- + ----- - --- - 3 a  z - 2 a  z + a  z + 2 a   z - 4 a  z  - 
  z     z    z      z      z
 
       7  3    5  5    7  5
>   4 a  z  - a  z  - a  z
In[9]:=
Kauffman[Link[11, NonAlternating, 109]][a, z]
Out[9]=   
                               5    7    9      11    13
  6      10      12      14   a    a    a    2 a     a        5      7
-a  - 4 a   - 7 a   - 3 a   + -- + -- + -- + ----- + --- - 3 a  z - a  z - 
                              z    z    z      z      z
 
       9         11        13        8  2       10  2       12  2       14  2
>   4 a  z - 10 a   z - 4 a   z - 3 a  z  + 14 a   z  + 28 a   z  + 11 a   z  + 
 
       5  3      7  3      9  3       11  3       13  3      6  4      8  4
>   4 a  z  - 3 a  z  + 2 a  z  + 21 a   z  + 12 a   z  + 3 a  z  + 4 a  z  - 
 
        10  4       12  4       14  4    5  5      7  5       11  5
>   18 a   z  - 34 a   z  - 15 a   z  - a  z  + 4 a  z  - 20 a   z  - 
 
        13  5    6  6    8  6      10  6       12  6      14  6    7  7
>   15 a   z  - a  z  - a  z  + 8 a   z  + 15 a   z  + 7 a   z  - a  z  + 
 
       11  7      13  7    10  8      12  8    14  8    11  9    13  9
>   8 a   z  + 7 a   z  - a   z  - 2 a   z  - a   z  - a   z  - a   z
In[10]:=
Kh[L][q, t]
Out[10]=   
 -6    -4      1         1        1        1        1        1        1
q   + q   + ------- + ------- + ------ + ------ + ------ + ------ + ------ + 
             26  11    22  10    22  9    20  8    18  8    20  7    16  7
            q   t     q   t     q   t    q   t    q   t    q   t    q   t
 
      2        2        1        1        3        1        1        1
>   ------ + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 
     16  6    16  5    12  5    14  4    12  4    10  4    12  3    10  3
    q   t    q   t    q   t    q   t    q   t    q   t    q   t    q   t
 
      1        1      1
>   ------ + ----- + ----
     10  2    8  2    6
    q   t    q  t    q  t


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