© | Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table:

L11n459 L4a1

Knotscape

This page is passe. Go here instead! The 2-Component Link L2a1 Also known as "The Hopf Link". Visit L2a1's page at Knotilus! Acknowledgement

DrawMorseLink

Further views:

Are they forever linked?

PD Presentation:

X4132 X2314

Gauss Code:

{{1, -2}, {2, -1}}

Jones Polynomial:

- q-5/2 - q-1/2

A2 (sl(3)) Invariant:

q-10 + 2q-8 + 2q-6 + 2q-4 + q-2 + 1

HOMFLY-PT Polynomial:

- az-1 - az + a3z-1

Kauffman Polynomial:

az-1 - az - a2 + a3z-1 - a3z

Khovanov Homology:

trqj r = -2 r = -1 r = 0 j = 0     1 j = -2     1 j = -4 1     j = -6 1

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

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L2a1

L11n459 L4a1