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The 3-Component Link

L6n1

"Three fibers in the Hopf fibration"

Visit L6n1's page at Knotilus!

Acknowledgement

L6n1 as Morse Link
DrawMorseLink

Further views:   Rich Schwartz' 98
Rich Schwartz' "98"

PD Presentation: X6172 X12,8,9,7 X4,12,1,11 X5,11,6,10 X3845 X9,3,10,2

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

Jones Polynomial: 2 + q2 + q4

A2 (sl(3)) Invariant: 2q-2 + 3 + 4q2 + 4q4 + 4q6 + 4q8 + 3q10 + 2q12 + q14

HOMFLY-PT Polynomial: a-4z-2 + a-4 - 2a-2z-2 - 3a-2 - a-2z2 + z-2 + 2

Kauffman Polynomial: - a-4z-2 + 3a-4 - 4a-4z2 + a-4z4 + 2a-3z-1 - 3a-3z + a-3z3 - 2a-2z-2 + 5a-2 - 4a-2z2 + a-2z4 + 2a-1z-1 - 3a-1z + a-1z3 - z-2 + 3

Khovanov Homology:
trqj r = 0r = 1r = 2r = 3r = 4
j = 9    1
j = 7    1
j = 5  1  
j = 31    
j = 131   
j = -12    

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[6, NonAlternating, 1]]]
Out[3]=   
 -Graphics- 
In[4]:=
PD[L = Link[6, NonAlternating, 1]]
Out[4]=   
PD[X[6, 1, 7, 2], X[12, 8, 9, 7], X[4, 12, 1, 11], X[5, 11, 6, 10], 
 
>   X[3, 8, 4, 5], X[9, 3, 10, 2]]
In[5]:=
GaussCode[L]
Out[5]=   
GaussCode[{1, 6, -5, -3}, {-4, -1, 2, 5}, {-6, 4, 3, -2}]
In[6]:=
Jones[L][q]
Out[6]=   
     2    4
2 + q  + q
In[7]:=
A2Invariant[L][q]
Out[7]=   
    2       2      4      6      8      10      12    14
3 + -- + 4 q  + 4 q  + 4 q  + 4 q  + 3 q   + 2 q   + q
     2
    q
In[8]:=
HOMFLYPT[Link[6, NonAlternating, 1]][a, z]
Out[8]=   
                                      2
     -4   3     -2     1       2     z
2 + a   - -- + z   + ----- - ----- - --
           2          4  2    2  2    2
          a          a  z    a  z    a
In[9]:=
Kauffman[Link[6, NonAlternating, 1]][a, z]
Out[9]=   
                                                                2      2    3
    3    5     -2     1       2      2      2    3 z   3 z   4 z    4 z    z
3 + -- + -- - z   - ----- - ----- + ---- + --- - --- - --- - ---- - ---- + -- + 
     4    2          4  2    2  2    3     a z    3     a      4      2     3
    a    a          a  z    a  z    a  z         a            a      a     a
 
     3    4    4
    z    z    z
>   -- + -- + --
    a     4    2
         a    a
In[10]:=
Kh[L][q, t]
Out[10]=   
2          3          5  2    7  4    9  4
- + 3 q + q  + q t + q  t  + q  t  + q  t
q

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L6n1
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