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

L11n145 L11n147

Knotscape

This page is passe. Go here instead! The 2-Component Link L11n146 Visit L11n146's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

X8192 X9,19,10,18 X6718 X19,7,20,22 X5,13,6,12 X3,10,4,11 X15,5,16,4 X11,16,12,17 X13,21,14,20 X21,15,22,14 X17,2,18,3

Gauss Code:

{{1, 11, -6, 7, -5, -3}, {3, -1, -2, 6, -8, 5, -9, 10, -7, 8, -11, 2, -4, 9, -10, 4}}

Jones Polynomial:

q-9/2 - 3q-7/2 + 5q-5/2 - 8q-3/2 + 8q-1/2 - 10q1/2 + 8q3/2 - 6q5/2 + 4q7/2 - q9/2

A2 (sl(3)) Invariant:

- q-14 + q-12 - q-8 + 3q-6 + q-4 + 4q-2 + 4 + q2 + 2q4 - 3q6 - q10 - 2q12 + q14

HOMFLY-PT Polynomial:

a-3z-1 - a-3z3 - 3a-1z-1 - 2a-1z + a-1z3 + a-1z5 + 2az-1 + 2az + 2az3 + az5 - a3z - a3z3

Kauffman Polynomial:

- a-5z3 + a-4 + 2a-4z2 - 4a-4z4 - a-3z-1 + a-3z3 - 2a-3z5 - a-3z7 + 3a-2 - 5a-2z4 + 3a-2z6 - 2a-2z8 - 3a-1z-1 + 5a-1z - 7a-1z3 + 7a-1z5 - 2a-1z7 - a-1z9 + 3 - 3z2 - 4z4 + 11z6 - 5z8 - 2az-1 + 8az - 18az3 + 19az5 - 4az7 - az9 - 3a2z2 + 7a2z6 - 3a2z8 + 3a3z - 9a3z3 + 10a3z5 - 3a3z7 - 2a4z2 + 3a4z4 - a4z6

Khovanov Homology:

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 10                   1 j = 8                 3   j = 6               3 1   j = 4             5 3     j = 2           5 3       j = 0         4 6         j = -2       4 4           j = -4     2 5             j = -6   1 3               j = -8   2                 j = -10 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[11, NonAlternating, 146]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 146]]
Out[4]=	PD[X[8, 1, 9, 2], X[9, 19, 10, 18], X[6, 7, 1, 8], X[19, 7, 20, 22], > X[5, 13, 6, 12], X[3, 10, 4, 11], X[15, 5, 16, 4], X[11, 16, 12, 17], > X[13, 21, 14, 20], X[21, 15, 22, 14], X[17, 2, 18, 3]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, 11, -6, 7, -5, -3}, > {3, -1, -2, 6, -8, 5, -9, 10, -7, 8, -11, 2, -4, 9, -10, 4}]
In[6]:=	Jones[L][q]
Out[6]=	-(9/2) 3 5 8 8 3/2 5/2 q - ---- + ---- - ---- + ------- - 10 Sqrt[q] + 8 q - 6 q + 7/2 5/2 3/2 Sqrt[q] q q q 7/2 9/2 > 4 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-14 -12 -8 3 -4 4 2 4 6 10 12 14 4 - q + q - q + -- + q + -- + q + 2 q - 3 q - q - 2 q + q 6 2 q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 146]][a, z]
Out[8]=	3 3 5 1 3 2 a 2 z 3 z z 3 3 3 z 5 ---- - --- + --- - --- + 2 a z - a z - -- + -- + 2 a z - a z + -- + a z 3 a z z a 3 a a a z a
In[9]:=	Kauffman[Link[11, NonAlternating, 146]][a, z]
Out[9]=	2 -4 3 1 3 2 a 5 z 3 2 2 z 3 + a + -- - ---- - --- - --- + --- + 8 a z + 3 a z - 3 z + ---- - 2 3 a z z a 4 a a z a 3 3 3 4 2 2 4 2 z z 7 z 3 3 3 4 4 z > 3 a z - 2 a z - -- + -- - ---- - 18 a z - 9 a z - 4 z - ---- - 5 3 a 4 a a a 4 5 5 6 5 z 4 4 2 z 7 z 5 3 5 6 3 z > ---- + 3 a z - ---- + ---- + 19 a z + 10 a z + 11 z + ---- + 2 3 a 2 a a a 7 7 8 2 6 4 6 z 2 z 7 3 7 8 2 z 2 8 > 7 a z - a z - -- - ---- - 4 a z - 3 a z - 5 z - ---- - 3 a z - 3 a 2 a a 9 z 9 > -- - a z a
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 2 1 3 2 5 4 4 4 6 + 5 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + - + ---- + 10 5 8 4 6 4 6 3 4 3 4 2 2 2 t 2 q t q t q t q t q t q t q t q t 2 4 4 2 6 2 6 3 8 3 10 4 > 3 q t + 5 q t + 3 q t + 3 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n146

L11n145 L11n147