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

L6a3 L6a5

Knotscape

This page is passe. Go here instead! The 3-Component Link L6a4 Also known as "The Borromean Link" or "The Borromean Rings". Visit Peter Cromwell's page on the Borromean Rings. Visit L6a4's page at Knotilus! Acknowledgement

DrawMorseLink

Further views:	A Borromean link by Dylan Thurston	A Borromean bathroom tile	A Borromean rattle by Sassy	A Borromean link at the Fields Institute
	Borromean paper clips	The Colombo Mall in Lisboa

PD Presentation:

X6172 X12,8,9,7 X4,12,1,11 X10,5,11,6 X8453 X2,9,3,10

Gauss Code:

{{1, -6, 5, -3}, {4, -1, 2, -5}, {6, -4, 3, -2}}

Jones Polynomial:

- q-3 + 3q-2 - 2q-1 + 4 - 2q + 3q2 - q3

A2 (sl(3)) Invariant:

- q-10 + q-8 + 2q-6 + 3q-4 + 6q-2 + 5 + 6q2 + 3q4 + 2q6 + q8 - q10

HOMFLY-PT Polynomial:

a-2z-2 - a-2z2 - 2z-2 + 2z2 + z4 + a2z-2 - a2z2

Kauffman Polynomial:

a-3z3 + a-2z-2 - 4a-2z2 + 3a-2z4 - 2a-1z-1 - a-1z3 + 2a-1z5 + 2z-2 + 1 - 8z2 + 6z4 - 2az-1 - az3 + 2az5 + a2z-2 - 4a2z2 + 3a2z4 + a3z3

Khovanov Homology:

trqj r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 j = 7             1 j = 5           2   j = 3           1   j = 1       4 2     j = -1     2 4       j = -3   1           j = -5   2           j = -7 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]=	3
In[3]:=	Show[DrawMorseLink[Link[6, Alternating, 4]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[6, Alternating, 4]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 8, 9, 7], X[4, 12, 1, 11], X[10, 5, 11, 6], > X[8, 4, 5, 3], X[2, 9, 3, 10]]
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]=	-3 3 2 2 3 4 - q + -- - - - 2 q + 3 q - q 2 q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-10 -8 2 3 6 2 4 6 8 10 5 - q + q + -- + -- + -- + 6 q + 3 q + 2 q + q - q 6 4 2 q q q
In[8]:=	HOMFLYPT[Link[6, Alternating, 4]][a, z]
Out[8]=	2 2 -2 1 a 2 z 2 2 4 -- + ----- + -- + 2 z - -- - a z + z 2 2 2 2 2 z a z z a
In[9]:=	Kauffman[Link[6, Alternating, 4]][a, z]
Out[9]=	2 2 3 3 2 1 a 2 2 a 2 4 z 2 2 z z 3 1 + -- + ----- + -- - --- - --- - 8 z - ---- - 4 a z + -- - -- - a z + 2 2 2 2 a z z 2 3 a z a z z a a 4 5 3 3 4 3 z 2 4 2 z 5 > a z + 6 z + ---- + 3 a z + ---- + 2 a z 2 a a
In[10]:=	Kh[L][q, t]
Out[10]=	4 1 2 1 2 3 2 5 2 7 3 - + 4 q + ----- + ----- + ----- + --- + 2 q t + q t + 2 q t + q t q 7 3 5 2 3 2 q t q t q t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L6a4

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