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L11a114

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Acknowledgement

L11a114 as Morse Link
DrawMorseLink

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

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

Jones Polynomial: - q-1/2 + 3q1/2 - 7q3/2 + 10q5/2 - 15q7/2 + 15q9/2 - 15q11/2 + 13q13/2 - 9q15/2 + 5q17/2 - 2q19/2 + q21/2

A2 (sl(3)) Invariant: q-2 - 1 + 3q4 - 2q6 + 4q8 + 3q10 + 2q12 + 5q14 - q16 + 2q18 - 2q20 - 2q22 + 2q24 - 3q26 - q28 - q32

HOMFLY-PT Polynomial: a-9z-1 + 2a-9z + a-9z3 - a-7z-1 - 3a-7z - 2a-7z3 - a-7z5 - 2a-5z-1 - 3a-5z - 4a-5z3 - 2a-5z5 + 2a-3z-1 + 3a-3z - a-3z5 + a-1z + a-1z3

Kauffman Polynomial: - 4a-12z2 + 4a-12z4 - a-12z6 - 4a-11z3 + 6a-11z5 - 2a-11z7 - 4a-10 + 12a-10z2 - 12a-10z4 + 9a-10z6 - 3a-10z8 + a-9z-1 - 2a-9z + 11a-9z3 - 12a-9z5 + 8a-9z7 - 3a-9z9 - 9a-8 + 34a-8z2 - 39a-8z4 + 22a-8z6 - 5a-8z8 - a-8z10 + a-7z-1 + a-7z - 3a-7z3 - 8a-7z5 + 12a-7z7 - 6a-7z9 - 4a-6 + 15a-6z2 - 27a-6z4 + 21a-6z6 - 7a-6z8 - a-6z10 - 2a-5z-1 + 11a-5z - 27a-5z3 + 20a-5z5 - 3a-5z7 - 3a-5z9 + 2a-4 - 4a-4z2 + a-4z4 + 6a-4z6 - 5a-4z8 - 2a-3z-1 + 7a-3z - 7a-3z3 + 9a-3z5 - 5a-3z7 - a-2z2 + 5a-2z4 - 3a-2z6 - a-1z + 2a-1z3 - a-1z5

Khovanov Homology:
trqj r = -2r = -1r = 0r = 1r = 2r = 3r = 4r = 5r = 6r = 7r = 8r = 9
j = 22           1
j = 20          1 
j = 18         41 
j = 16        51  
j = 14       84   
j = 12      75    
j = 10     88     
j = 8    77      
j = 6   38       
j = 4  47        
j = 2 15         
j = 0 2          
j = -21           


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, 114]]]
Out[3]=   
 -Graphics- 
In[4]:=
PD[L = Link[11, Alternating, 114]]
Out[4]=   
PD[X[6, 1, 7, 2], X[14, 4, 15, 3], X[16, 8, 17, 7], X[20, 14, 21, 13], 
 
>   X[22, 12, 5, 11], X[12, 22, 13, 21], X[8, 20, 9, 19], X[18, 10, 19, 9], 
 
>   X[10, 18, 11, 17], X[2, 5, 3, 6], X[4, 16, 1, 15]]
In[5]:=
GaussCode[L]
Out[5]=   
GaussCode[{1, -10, 2, -11}, {10, -1, 3, -7, 8, -9, 5, -6, 4, -2, 11, -3, 9, -8, 
 
>    7, -4, 6, -5}]
In[6]:=
Jones[L][q]
Out[6]=   
     1                      3/2       5/2       7/2       9/2       11/2
-(-------) + 3 Sqrt[q] - 7 q    + 10 q    - 15 q    + 15 q    - 15 q     + 
  Sqrt[q]
 
        13/2      15/2      17/2      19/2    21/2
>   13 q     - 9 q     + 5 q     - 2 q     + q
In[7]:=
A2Invariant[L][q]
Out[7]=   
      -2      4      6      8      10      12      14    16      18      20
-1 + q   + 3 q  - 2 q  + 4 q  + 3 q   + 2 q   + 5 q   - q   + 2 q   - 2 q   - 
 
       22      24      26    28    32
>   2 q   + 2 q   - 3 q   - q   - q
In[8]:=
HOMFLYPT[Link[11, Alternating, 114]][a, z]
Out[8]=   
                                                         3      3      3    3
 1      1      2      2     2 z   3 z   3 z   3 z   z   z    2 z    4 z    z
---- - ---- - ---- + ---- + --- - --- - --- + --- + - + -- - ---- - ---- + -- - 
 9      7      5      3      9     7     5     3    a    9     7      5    a
a  z   a  z   a  z   a  z   a     a     a     a         a     a      a
 
     5      5    5
    z    2 z    z
>   -- - ---- - --
     7     5     3
    a     a     a
In[9]:=
Kauffman[Link[11, Alternating, 114]][a, z]
Out[9]=   
-4    9    4    2     1      1      2      2     2 z   z    11 z   7 z   z
--- - -- - -- + -- + ---- + ---- - ---- - ---- - --- + -- + ---- + --- - - - 
 10    8    6    4    9      7      5      3      9     7     5     3    a
a     a    a    a    a  z   a  z   a  z   a  z   a     a     a     a
 
       2       2       2       2      2    2      3       3      3       3
    4 z    12 z    34 z    15 z    4 z    z    4 z    11 z    3 z    27 z
>   ---- + ----- + ----- + ----- - ---- - -- - ---- + ----- - ---- - ----- - 
     12      10      8       6       4     2    11      9       7      5
    a       a       a       a       a     a    a       a       a      a
 
       3      3      4       4       4       4    4      4      5       5
    7 z    2 z    4 z    12 z    39 z    27 z    z    5 z    6 z    12 z
>   ---- + ---- + ---- - ----- - ----- - ----- + -- + ---- + ---- - ----- - 
      3     a      12      10      8       6      4     2     11      9
     a            a       a       a       a      a     a     a       a
 
       5       5      5    5    6       6       6       6      6      6
    8 z    20 z    9 z    z    z     9 z    22 z    21 z    6 z    3 z
>   ---- + ----- + ---- - -- - --- + ---- + ----- + ----- + ---- - ---- - 
      7      5       3    a     12    10      8       6       4      2
     a      a       a          a     a       a       a       a      a
 
       7      7       7      7      7      8      8      8      8      9
    2 z    8 z    12 z    3 z    5 z    3 z    5 z    7 z    5 z    3 z
>   ---- + ---- + ----- - ---- - ---- - ---- - ---- - ---- - ---- - ---- - 
     11      9      7       5      3     10      8      6      4      9
    a       a      a       a      a     a       a      a      a      a
 
       9      9    10    10
    6 z    3 z    z     z
>   ---- - ---- - --- - ---
      7      5     8     6
     a      a     a     a
In[10]:=
Kh[L][q, t]
Out[10]=   
                           2
   2      4     1     2   q       4        6        6  2      8  2      8  3
5 q  + 4 q  + ----- + - + -- + 7 q  t + 3 q  t + 8 q  t  + 7 q  t  + 7 q  t  + 
               2  2   t   t
              q  t
 
       10  3      10  4      12  4      12  5      14  5      14  6
>   8 q   t  + 8 q   t  + 7 q   t  + 5 q   t  + 8 q   t  + 4 q   t  + 
 
       16  6    16  7      18  7    18  8    20  8    22  9
>   5 q   t  + q   t  + 4 q   t  + q   t  + q   t  + q   t


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