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

L10n68 L10n70

Knotscape

This page is passe. Go here instead! The 3-Component Link L10n69 Visit L10n69's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

X6172 X5,12,6,13 X3849 X2,14,3,13 X14,7,15,8 X9,18,10,19 X11,16,12,17 X17,20,18,11 X15,1,16,4 X19,10,20,5

Gauss Code:

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

Jones Polynomial:

q-9 - 2q-8 + 3q-7 - 4q-6 + 5q-5 - 4q-4 + 5q-3 - 2q-2 + 2q-1

A2 (sl(3)) Invariant:

q-28 + q-22 - q-20 + q-18 + 2q-16 + 3q-14 + 5q-12 + 4q-10 + 5q-8 + 3q-6 + q-4 + 2q-2

HOMFLY-PT Polynomial:

a2z-2 + 3a2 + 2a2z2 - 2a4z-2 - 3a4 - 2a4z2 - a4z4 + a6z-2 - a6 - 2a6z2 - a6z4 + a8 + a8z2

Kauffman Polynomial:

a2z-2 - 4a2 + 3a2z2 - 2a3z-1 + 4a3z - a3z3 + a3z5 + 2a4z-2 - 6a4 + 9a4z2 - 5a4z4 + 2a4z6 - 2a5z-1 + 5a5z3 - 5a5z5 + 2a5z7 + a6z-2 - 3a6 + 8a6z2 - 7a6z4 + a6z8 - 6a7z + 14a7z3 - 14a7z5 + 4a7z7 - a8 + 6a8z2 - 6a8z4 - a8z6 + a8z8 - 2a9z + 8a9z3 - 8a9z5 + 2a9z7 - a10 + 4a10z2 - 4a10z4 + a10z6

Khovanov Homology:

trqj r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 j = -1                 2 j = -3               2 2 j = -5             3     j = -7           2 3     j = -9         3 2       j = -11       1 2         j = -13     2 3           j = -15   1 2             j = -17   1               j = -19 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[10, NonAlternating, 69]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, NonAlternating, 69]]
Out[4]=	PD[X[6, 1, 7, 2], X[5, 12, 6, 13], X[3, 8, 4, 9], X[2, 14, 3, 13], > X[14, 7, 15, 8], X[9, 18, 10, 19], X[11, 16, 12, 17], X[17, 20, 18, 11], > X[15, 1, 16, 4], X[19, 10, 20, 5]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -4, -3, 9}, {-2, -1, 5, 3, -6, 10}, > {-7, 2, 4, -5, -9, 7, -8, 6, -10, 8}]
In[6]:=	Jones[L][q]
Out[6]=	-9 2 3 4 5 4 5 2 2 q - -- + -- - -- + -- - -- + -- - -- + - 8 7 6 5 4 3 2 q q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-28 -22 -20 -18 2 3 5 4 5 3 -4 2 q + q - q + q + --- + --- + --- + --- + -- + -- + q + -- 16 14 12 10 8 6 2 q q q q q q q
In[8]:=	HOMFLYPT[Link[10, NonAlternating, 69]][a, z]
Out[8]=	2 4 6 2 4 6 8 a 2 a a 2 2 4 2 6 2 8 2 3 a - 3 a - a + a + -- - ---- + -- + 2 a z - 2 a z - 2 a z + a z - 2 2 2 z z z 4 4 6 4 > a z - a z
In[9]:=	Kauffman[Link[10, NonAlternating, 69]][a, z]
Out[9]=	2 4 6 3 5 2 4 6 8 10 a 2 a a 2 a 2 a 3 -4 a - 6 a - 3 a - a - a + -- + ---- + -- - ---- - ---- + 4 a z - 2 2 2 z z z z z 7 9 2 2 4 2 6 2 8 2 10 2 > 6 a z - 2 a z + 3 a z + 9 a z + 8 a z + 6 a z + 4 a z - 3 3 5 3 7 3 9 3 4 4 6 4 8 4 > a z + 5 a z + 14 a z + 8 a z - 5 a z - 7 a z - 6 a z - 10 4 3 5 5 5 7 5 9 5 4 6 8 6 > 4 a z + a z - 5 a z - 14 a z - 8 a z + 2 a z - a z + 10 6 5 7 7 7 9 7 6 8 8 8 > a z + 2 a z + 4 a z + 2 a z + a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	2 2 1 1 1 2 2 3 1 -- + - + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 3 q 19 8 17 7 15 7 15 6 13 6 13 5 11 5 q q t q t q t q t q t q t q t 2 3 2 2 3 3 2 > ------ + ----- + ----- + ----- + ----- + ----- + ---- 11 4 9 4 9 3 7 3 7 2 5 2 3 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 L10n69

L10n68 L10n70