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The 2-Component Link

L6a3

Visit L6a3's page at Knotilus!

Acknowledgement

L6a3 as Morse Link
DrawMorseLink

Further views:   (Ruberman, Cochran), Melvin, Akbulut, Gompf, Kirby
(Ruberman, Cochran),
Melvin, Akbulut, Gompf, Kirby 		Rich Schwartz' 72
Rich Schwartz' "72"

PD Presentation: X8192 X2,9,3,10 X10,3,11,4 X12,5,7,6 X6718 X4,11,5,12

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

Jones Polynomial: - q-17/2 + q-15/2 - q-13/2 + q-11/2 - q-9/2 - q-5/2

A2 (sl(3)) Invariant: q-26 + q-24 + q-22 + q-16 + q-14 + 2q-12 + q-10 + q-8

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

Kauffman Polynomial: a5z-1 - 6a5z + 5a5z3 - a5z5 - a6 + 3a6z2 - a6z4 + a7z-1 - 4a7z + 4a7z3 - a7z5 + 2a8z2 - a8z4 + a9z - a9z3 - a10z2 - a11z

Khovanov Homology:
trqj r = -6r = -5r = -4r = -3r = -2r = -1r = 0
j = -4      1
j = -6      1
j = -8    1  
j = -10       
j = -12  11   
j = -14       
j = -1611     
j = -181      

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

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