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L7n1

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Acknowledgement

L7n1 as Morse Link
DrawMorseLink

PD Presentation: X6172 X12,7,13,8 X4,13,1,14 X5,10,6,11 X3849 X9,14,10,5 X11,2,12,3

Gauss Code: {{1, 7, -5, -3}, {-4, -1, 2, 5, -6, 4, -7, -2, 3, 6}}

Jones Polynomial: q-15/2 - q-13/2 - q-9/2 - q-5/2

A2 (sl(3)) Invariant: - q-28 - q-26 - q-24 + q-20 + 2q-18 + 3q-16 + 2q-14 + 2q-12 + q-10 + q-8

HOMFLY-PT Polynomial: - 2a5z-1 - 6a5z - 5a5z3 - a5z5 + 3a7z-1 + 4a7z + a7z3 - a9z-1

Kauffman Polynomial: 2a5z-1 - 6a5z + 5a5z3 - a5z5 - 3a6 + 4a6z2 - a6z4 + 3a7z-1 - 7a7z + 5a7z3 - a7z5 - 3a8 + 4a8z2 - a8z4 + a9z-1 - a9z - a10

Khovanov Homology:
trqj r = -5r = -4r = -3r = -2r = -1r = 0
j = -4     1
j = -6     1
j = -8   1  
j = -10 1    
j = -12 21   
j = -14      
j = -161     

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

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