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

L6a4 L6n1

Knotscape

This page is passe. Go here instead! The 3-Component Link L6a5 Visit L6a5's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

X6172 X10,3,11,4 X12,7,9,8 X8,11,5,12 X2536 X4,9,1,10

Gauss Code:

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

Jones Polynomial:

q-7 - q-6 + 3q-5 - q-4 + 3q-3 - 2q-2 + q-1

A2 (sl(3)) Invariant:

q-24 + 3q-22 + 3q-20 + 4q-18 + 5q-16 + 4q-14 + 4q-12 + 2q-10 + q-8 - q-4 + q-2

HOMFLY-PT Polynomial:

a2z2 + a4z-2 + 3a4 + 2a4z2 - 2a6z-2 - 3a6 + a8z-2

Kauffman Polynomial:

a2z2 + 2a3z3 - a4z-2 + 3a4 - 5a4z2 + 3a4z4 + 2a5z-1 - 3a5z + a5z3 + a5z5 - 2a6z-2 + 5a6 - 9a6z2 + 4a6z4 + 2a7z-1 - 3a7z - a7z3 + a7z5 - a8z-2 + 3a8 - 3a8z2 + a8z4

Khovanov Homology:

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 j = -1             1 j = -3           2 1 j = -5         1     j = -7         2     j = -9     3 1       j = -11   1 3         j = -13               j = -15 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, 5]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[6, Alternating, 5]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[12, 7, 9, 8], X[8, 11, 5, 12], > X[2, 5, 3, 6], X[4, 9, 1, 10]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -5, 2, -6}, {5, -1, 3, -4}, {6, -2, 4, -3}]
In[6]:=	Jones[L][q]
Out[6]=	-7 -6 3 -4 3 2 1 q - q + -- - q + -- - -- + - 5 3 2 q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-24 3 3 4 5 4 4 2 -8 -4 -2 q + --- + --- + --- + --- + --- + --- + --- + q - q + q 22 20 18 16 14 12 10 q q q q q q q
In[8]:=	HOMFLYPT[Link[6, Alternating, 5]][a, z]
Out[8]=	4 6 8 4 6 a 2 a a 2 2 4 2 3 a - 3 a + -- - ---- + -- + a z + 2 a z 2 2 2 z z z
In[9]:=	Kauffman[Link[6, Alternating, 5]][a, z]
Out[9]=	4 6 8 5 7 4 6 8 a 2 a a 2 a 2 a 5 7 2 2 3 a + 5 a + 3 a - -- - ---- - -- + ---- + ---- - 3 a z - 3 a z + a z - 2 2 2 z z z z z 4 2 6 2 8 2 3 3 5 3 7 3 4 4 6 4 > 5 a z - 9 a z - 3 a z + 2 a z + a z - a z + 3 a z + 4 a z + 8 4 5 5 7 5 > a z + a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	-3 1 1 1 3 3 1 2 1 2 q + - + ------ + ------ + ------ + ----- + ----- + ----- + ----- + ---- q 15 6 11 5 11 4 9 4 9 3 7 2 5 2 3 q t q t q t q t q t q t q t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L6a5

L6a4 L6n1