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L11n380

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Acknowledgement

L11n380 as Morse Link
DrawMorseLink

PD Presentation: X6172 X14,7,15,8 X4,15,1,16 X5,10,6,11 X8493 X17,22,18,19 X11,20,12,21 X19,12,20,13 X21,18,22,5 X9,16,10,17 X2,14,3,13

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

Jones Polynomial: - q-10 + 2q-9 - 4q-8 + 5q-7 - 4q-6 + 6q-5 - 4q-4 + 4q-3 - q-2 + q-1

A2 (sl(3)) Invariant: - q-32 - 2q-30 - q-28 - 3q-26 - q-24 + 2q-22 + 3q-20 + 7q-18 + 6q-16 + 6q-14 + 4q-12 + 2q-10 + 3q-8 + q-6 + q-2

HOMFLY-PT Polynomial: a2 + a2z2 + 2a4z-2 + 4a4 + 3a4z2 - 5a6z-2 - 10a6 - 7a6z2 - 2a6z4 + 4a8z-2 + 6a8 + 3a8z2 - a10z-2 - a10

Kauffman Polynomial: - a2 + a2z2 + a3z3 - 2a4z-2 + 6a4 - 7a4z2 + 3a4z4 + 5a5z-1 - 15a5z + 17a5z3 - 9a5z5 + 2a5z7 - 5a6z-2 + 16a6 - 23a6z2 + 23a6z4 - 14a6z6 + 3a6z8 + 9a7z-1 - 30a7z + 39a7z3 - 21a7z5 + a7z7 + a7z9 - 4a8z-2 + 13a8 - 21a8z2 + 31a8z4 - 23a8z6 + 5a8z8 + 5a9z-1 - 20a9z + 31a9z3 - 17a9z5 + a9z9 - a10z-2 + 3a10 - 6a10z2 + 11a10z4 - 9a10z6 + 2a10z8 + a11z-1 - 5a11z + 8a11z3 - 5a11z5 + a11z7

Khovanov Homology:
trqj r = -9r = -8r = -7r = -6r = -5r = -4r = -3r = -2r = -1r = 0
j = -1         1
j = -3        22
j = -5       2 1
j = -7      33  
j = -9     42   
j = -11    251   
j = -13   331    
j = -15  12      
j = -17 13       
j = -19 1        
j = -211         


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, 380]]]
Out[3]=   
 -Graphics- 
In[4]:=
PD[L = Link[11, NonAlternating, 380]]
Out[4]=   
PD[X[6, 1, 7, 2], X[14, 7, 15, 8], X[4, 15, 1, 16], X[5, 10, 6, 11], 
 
>   X[8, 4, 9, 3], X[17, 22, 18, 19], X[11, 20, 12, 21], X[19, 12, 20, 13], 
 
>   X[21, 18, 22, 5], X[9, 16, 10, 17], X[2, 14, 3, 13]]
In[5]:=
GaussCode[L]
Out[5]=   
GaussCode[{1, -11, 5, -3}, {-8, 7, -9, 6}, 
 
>   {-4, -1, 2, -5, -10, 4, -7, 8, 11, -2, 3, 10, -6, 9}]
In[6]:=
Jones[L][q]
Out[6]=   
  -10   2    4    5    4    6    4    4     -2   1
-q    + -- - -- + -- - -- + -- - -- + -- - q   + -
         9    8    7    6    5    4    3         q
        q    q    q    q    q    q    q
In[7]:=
A2Invariant[L][q]
Out[7]=   
  -32    2     -28    3     -24    2     3     7     6     6     4     2
-q    - --- - q    - --- - q    + --- + --- + --- + --- + --- + --- + --- + 
         30           26           22    20    18    16    14    12    10
        q            q            q     q     q     q     q     q     q
 
    3     -6    -2
>   -- + q   + q
     8
    q
In[8]:=
HOMFLYPT[Link[11, NonAlternating, 380]][a, z]
Out[8]=   
                                    4      6      8    10
 2      4       6      8    10   2 a    5 a    4 a    a      2  2      4  2
a  + 4 a  - 10 a  + 6 a  - a   + ---- - ---- + ---- - --- + a  z  + 3 a  z  - 
                                   2      2      2     2
                                  z      z      z     z
 
       6  2      8  2      6  4
>   7 a  z  + 3 a  z  - 2 a  z
In[9]:=
Kauffman[Link[11, NonAlternating, 380]][a, z]
Out[9]=   
                                        4      6      8    10      5      7
  2      4       6       8      10   2 a    5 a    4 a    a     5 a    9 a
-a  + 6 a  + 16 a  + 13 a  + 3 a   - ---- - ---- - ---- - --- + ---- + ---- + 
                                       2      2      2     2     z      z
                                      z      z      z     z
 
       9    11
    5 a    a         5         7         9        11      2  2      4  2
>   ---- + --- - 15 a  z - 30 a  z - 20 a  z - 5 a   z + a  z  - 7 a  z  - 
     z      z
 
        6  2       8  2      10  2    3  3       5  3       7  3       9  3
>   23 a  z  - 21 a  z  - 6 a   z  + a  z  + 17 a  z  + 39 a  z  + 31 a  z  + 
 
       11  3      4  4       6  4       8  4       10  4      5  5       7  5
>   8 a   z  + 3 a  z  + 23 a  z  + 31 a  z  + 11 a   z  - 9 a  z  - 21 a  z  - 
 
        9  5      11  5       6  6       8  6      10  6      5  7    7  7
>   17 a  z  - 5 a   z  - 14 a  z  - 23 a  z  - 9 a   z  + 2 a  z  + a  z  + 
 
     11  7      6  8      8  8      10  8    7  9    9  9
>   a   z  + 3 a  z  + 5 a  z  + 2 a   z  + a  z  + a  z
In[10]:=
Kh[L][q, t]
Out[10]=   
 -5   2    1     1        1        1        3        1        2        3
q   + -- + - + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 
       3   q    21  9    19  8    17  8    17  7    15  7    15  6    13  6
      q        q   t    q   t    q   t    q   t    q   t    q   t    q   t
 
      3        2        1        5        4       1        2       3
>   ------ + ------ + ------ + ------ + ----- + ------ + ----- + ----- + 
     13  5    11  5    13  4    11  4    9  4    11  3    9  3    7  3
    q   t    q   t    q   t    q   t    q  t    q   t    q  t    q  t
 
      3       2      2
>   ----- + ----- + ----
     7  2    5  2    3
    q  t    q  t    q  t


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