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The 2-Component Link

L5a1

Also known as "The Whitehead Link".

Visit L5a1's page at Knotilus!

Acknowledgement

L5a1 as Morse Link
DrawMorseLink

Further views:   Wolfgang Staubach's Medallion
Wolfgang Staubach's Medallion

PD Presentation: X6172 X10,7,5,8 X4516 X2,10,3,9 X8493

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

Jones Polynomial: q-7/2 - 2q-5/2 + q-3/2 - 2q-1/2 + q1/2 - q3/2

A2 (sl(3)) Invariant: - q-12 + q-8 + q-6 + 2q-4 + q-2 + 2 + q2 + q4 + q6

HOMFLY-PT Polynomial: - a-1z-1 - a-1z + az-1 + 2az + az3 - a3z

Kauffman Polynomial: - a-1z-1 + 2a-1z - a-1z3 + 1 + z2 - z4 - az-1 + 4az - 3az3 - a2z4 + 2a3z - 2a3z3 - a4z2

Khovanov Homology:
trqj r = -3r = -2r = -1r = 0r = 1r = 2
j = 4     1
j = 2      
j = 0   21 
j = -2  12  
j = -4 1    
j = -6 1    
j = -81     

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

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L5a1
L4a1
L4a1 		L6a1
L6a1