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

L10n52 L10n54

Knotscape

This page is passe. Go here instead! The 2-Component Link L10n53 Visit L10n53's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

X8192 X12,3,13,4 X20,13,7,14 X9,15,10,14 X19,11,20,10 X5,16,6,17 X15,18,16,19 X2738 X4,11,5,12 X17,6,18,1

Gauss Code:

{{1, -8, 2, -9, -6, 10}, {8, -1, -4, 5, 9, -2, 3, 4, -7, 6, -10, 7, -5, -3}}

Jones Polynomial:

- 2q-15/2 + 4q-13/2 - 6q-11/2 + 7q-9/2 - 8q-7/2 + 6q-5/2 - 5q-3/2 + 3q-1/2 - q1/2

A2 (sl(3)) Invariant:

2q-24 + q-22 - q-20 + 2q-18 + 2q-14 + 2q-12 + 2q-8 - 2q-6 + q-4 - 1 + q2

HOMFLY-PT Polynomial:

- az - az3 + a3z + 2a3z3 + a3z5 - a5z-1 - 5a5z - 3a5z3 + a7z-1 + 2a7z

Kauffman Polynomial:

- az + 2az3 - az5 - 3a2z2 + 7a2z4 - 3a2z6 + 5a3z5 - 3a3z7 - 4a4z2 + 11a4z4 - 4a4z6 - a4z8 + a5z-1 - 3a5z - 2a5z3 + 8a5z5 - 5a5z7 - a6 + a6z2 + 2a6z4 - 2a6z6 - a6z8 + a7z-1 - a7z - 3a7z3 + 2a7z5 - 2a7z7 + 2a8z2 - 2a8z4 - a8z6 + 3a9z - 3a9z3

Khovanov Homology:

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 j = 2                 1 j = 0               2   j = -2             3 1   j = -4           4 3     j = -6         4 2       j = -8       3 4         j = -10     3 4           j = -12   1 3             j = -14 1 3               j = -16 2

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[10, NonAlternating, 53]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, NonAlternating, 53]]
Out[4]=	PD[X[8, 1, 9, 2], X[12, 3, 13, 4], X[20, 13, 7, 14], X[9, 15, 10, 14], > X[19, 11, 20, 10], X[5, 16, 6, 17], X[15, 18, 16, 19], X[2, 7, 3, 8], > X[4, 11, 5, 12], X[17, 6, 18, 1]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -8, 2, -9, -6, 10}, > {8, -1, -4, 5, 9, -2, 3, 4, -7, 6, -10, 7, -5, -3}]
In[6]:=	Jones[L][q]
Out[6]=	-2 4 6 7 8 6 5 3 ----- + ----- - ----- + ---- - ---- + ---- - ---- + ------- - Sqrt[q] 15/2 13/2 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	2 -22 -20 2 2 2 2 2 -4 2 -1 + --- + q - q + --- + --- + --- + -- - -- + q + q 24 18 14 12 8 6 q q q q q q
In[8]:=	HOMFLYPT[Link[10, NonAlternating, 53]][a, z]
Out[8]=	5 7 a a 3 5 7 3 3 3 5 3 3 5 -(--) + -- - a z + a z - 5 a z + 2 a z - a z + 2 a z - 3 a z + a z z z
In[9]:=	Kauffman[Link[10, NonAlternating, 53]][a, z]
Out[9]=	5 7 6 a a 5 7 9 2 2 4 2 6 2 -a + -- + -- - a z - 3 a z - a z + 3 a z - 3 a z - 4 a z + a z + z z 8 2 3 5 3 7 3 9 3 2 4 4 4 > 2 a z + 2 a z - 2 a z - 3 a z - 3 a z + 7 a z + 11 a z + 6 4 8 4 5 3 5 5 5 7 5 2 6 > 2 a z - 2 a z - a z + 5 a z + 8 a z + 2 a z - 3 a z - 4 6 6 6 8 6 3 7 5 7 7 7 4 8 6 8 > 4 a z - 2 a z - a z - 3 a z - 5 a z - 2 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	3 3 2 1 3 1 3 3 4 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 4 2 16 6 14 6 14 5 12 5 12 4 10 4 10 3 q q q t q t q t q t q t q t q t 3 4 4 2 4 t 2 2 > ----- + ----- + ----- + ---- + ---- + 2 t + -- + q t 8 3 8 2 6 2 6 4 2 q t q t q t q t q t q

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L10n53

L10n52 L10n54