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L11n411

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Acknowledgement

L11n411 as Morse Link
DrawMorseLink

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

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

Jones Polynomial: - q-13 + 2q-12 - 2q-11 + q-10 + q-8 + 2q-7 - q-6 + 2q-5 - q-4 + q-3

A2 (sl(3)) Invariant: - q-42 - 3q-40 - q-38 - q-36 + 3q-32 + 3q-30 + 6q-28 + 5q-26 + 6q-24 + 4q-22 + q-20 + 2q-18 + q-16 + q-14 + q-10

HOMFLY-PT Polynomial: 2a6 + 6a6z2 + 5a6z4 + a6z6 + 2a8z-2 + 5a8 + 6a8z2 + 5a8z4 + a8z6 - 5a10z-2 - 12a10 - 8a10z2 - a10z4 + 4a12z-2 + 5a12 - a14z-2

Kauffman Polynomial: - 2a6 + 6a6z2 - 5a6z4 + a6z6 + 2a7z3 - 4a7z5 + a7z7 - 2a8z-2 + 7a8 - 10a8z2 + 7a8z4 - 5a8z6 + a8z8 + 5a9z-1 - 22a9z + 32a9z3 - 19a9z5 + 3a9z7 - 5a10z-2 + 21a10 - 43a10z2 + 44a10z4 - 21a10z6 + 3a10z8 + 9a11z-1 - 39a11z + 54a11z3 - 20a11z5 - 2a11z7 + a11z9 - 4a12z-2 + 16a12 - 35a12z2 + 47a12z4 - 26a12z6 + 4a12z8 + 5a13z-1 - 21a13z + 30a13z3 - 10a13z5 - 3a13z7 + a13z9 - a14z-2 + 3a14 - 8a14z2 + 15a14z4 - 11a14z6 + 2a14z8 + a15z-1 - 4a15z + 6a15z3 - 5a15z5 + a15z7

Khovanov Homology:
trqj r = -11r = -10r = -9r = -8r = -7r = -6r = -5r = -4r = -3r = -2r = -1r = 0
j = -5           1
j = -7          11
j = -9        12  
j = -11       111  
j = -13      252   
j = -15     223    
j = -17    252     
j = -19   221      
j = -21  133       
j = -23 11         
j = -25 1          
j = -271           


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, NonAlternating, 411]]]
Out[3]=   
 -Graphics- 
In[4]:=
PD[L = Link[11, NonAlternating, 411]]
Out[4]=   
PD[X[6, 1, 7, 2], X[12, 3, 13, 4], X[13, 22, 14, 19], X[7, 20, 8, 21], 
 
>   X[19, 10, 20, 11], X[9, 16, 10, 17], X[17, 14, 18, 15], X[15, 8, 16, 9], 
 
>   X[21, 18, 22, 5], X[2, 5, 3, 6], X[4, 11, 1, 12]]
In[5]:=
GaussCode[L]
Out[5]=   
GaussCode[{1, -10, 2, -11}, {-5, 4, -9, 3}, 
 
>   {10, -1, -4, 8, -6, 5, 11, -2, -3, 7, -8, 6, -7, 9}]
In[6]:=
Jones[L][q]
Out[6]=   
  -13    2     2     -10    -8   2     -6   2     -4    -3
-q    + --- - --- + q    + q   + -- - q   + -- - q   + q
         12    11                 7          5
        q     q                  q          q
In[7]:=
A2Invariant[L][q]
Out[7]=   
  -42    3     -38    -36    3     3     6     5     6     4     -20    2
-q    - --- - q    - q    + --- + --- + --- + --- + --- + --- + q    + --- + 
         40                  32    30    28    26    24    22           18
        q                   q     q     q     q     q     q            q
 
     -16    -14    -10
>   q    + q    + q
In[8]:=
HOMFLYPT[Link[11, NonAlternating, 411]][a, z]
Out[8]=   
                                  8      10      12    14
   6      8       10      12   2 a    5 a     4 a     a        6  2      8  2
2 a  + 5 a  - 12 a   + 5 a   + ---- - ----- + ----- - --- + 6 a  z  + 6 a  z  - 
                                 2      2       2      2
                                z      z       z      z
 
       10  2      6  4      8  4    10  4    6  6    8  6
>   8 a   z  + 5 a  z  + 5 a  z  - a   z  + a  z  + a  z
In[9]:=
Kauffman[Link[11, NonAlternating, 411]][a, z]
Out[9]=   
                                            8      10      12    14      9
    6      8       10       12      14   2 a    5 a     4 a     a     5 a
-2 a  + 7 a  + 21 a   + 16 a   + 3 a   - ---- - ----- - ----- - --- + ---- + 
                                           2      2       2      2     z
                                          z      z       z      z
 
       11      13    15
    9 a     5 a     a         9         11         13        15        6  2
>   ----- + ----- + --- - 22 a  z - 39 a   z - 21 a   z - 4 a   z + 6 a  z  - 
      z       z      z
 
        8  2       10  2       12  2      14  2      7  3       9  3
>   10 a  z  - 43 a   z  - 35 a   z  - 8 a   z  + 2 a  z  + 32 a  z  + 
 
        11  3       13  3      15  3      6  4      8  4       10  4
>   54 a   z  + 30 a   z  + 6 a   z  - 5 a  z  + 7 a  z  + 44 a   z  + 
 
        12  4       14  4      7  5       9  5       11  5       13  5
>   47 a   z  + 15 a   z  - 4 a  z  - 19 a  z  - 20 a   z  - 10 a   z  - 
 
       15  5    6  6      8  6       10  6       12  6       14  6    7  7
>   5 a   z  + a  z  - 5 a  z  - 21 a   z  - 26 a   z  - 11 a   z  + a  z  + 
 
       9  7      11  7      13  7    15  7    8  8      10  8      12  8
>   3 a  z  - 2 a   z  - 3 a   z  + a   z  + a  z  + 3 a   z  + 4 a   z  + 
 
       14  8    11  9    13  9
>   2 a   z  + a   z  + a   z
In[10]:=
Kh[L][q, t]
Out[10]=   
 -7    -5      1         1         1        1        1        3        2
q   + q   + ------- + ------- + ------- + ------ + ------ + ------ + ------ + 
             27  11    25  10    23  10    23  9    21  9    21  8    19  8
            q   t     q   t     q   t     q   t    q   t    q   t    q   t
 
      3        2        2        1        5        2        2        2
>   ------ + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 
     21  7    19  7    17  7    19  6    17  6    15  6    17  5    15  5
    q   t    q   t    q   t    q   t    q   t    q   t    q   t    q   t
 
      2        3        5        1        2        1        1       1
>   ------ + ------ + ------ + ------ + ------ + ------ + ----- + ------ + 
     13  5    15  4    13  4    11  4    13  3    11  3    9  3    11  2
    q   t    q   t    q   t    q   t    q   t    q   t    q  t    q   t
 
      2      1
>   ----- + ----
     9  2    7
    q  t    q  t


Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n411
L11n410
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