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

L11n231 L11n233

Knotscape

This page is passe. Go here instead! The 2-Component Link L11n232 Visit L11n232's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

X10,1,11,2 X18,11,19,12 X8,9,1,10 X22,19,9,20 X20,6,21,5 X4,22,5,21 X7,15,8,14 X12,4,13,3 X13,16,14,17 X15,7,16,6 X2,18,3,17

Gauss Code:

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

Jones Polynomial:

- q-5/2 + 3q-3/2 - 4q-1/2 + 4q1/2 - 5q3/2 + 4q5/2 - 4q7/2 + 2q9/2 - q11/2

A2 (sl(3)) Invariant:

q-8 - q-6 - 1 + q2 + 2q6 + 2q8 + q10 + 2q12 + q16 + q18

HOMFLY-PT Polynomial:

- a-5z-1 - a-5z + a-3z-1 + 4a-3z + 2a-3z3 - 2a-1z - 3a-1z3 - a-1z5 + az + az3

Kauffman Polynomial:

- a-5z-1 + 5a-5z - 8a-5z3 + 5a-5z5 - a-5z7 + a-4 + 2a-4z2 - 10a-4z4 + 9a-4z6 - 2a-4z8 - a-3z-1 + 8a-3z - 18a-3z3 + 11a-3z5 + a-3z7 - a-3z9 + 4a-2z2 - 17a-2z4 + 17a-2z6 - 4a-2z8 + 5a-1z - 15a-1z3 + 10a-1z5 + a-1z7 - a-1z9 + z2 - 7z4 + 8z6 - 2z8 + 2az - 5az3 + 4az5 - az7 - a2z2

Khovanov Homology:

trqj r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 j = 12                 1 j = 10               1   j = 8             3 1   j = 6           2 2     j = 4         3 2       j = 2     1 2 2         j = 0     3 3           j = -2   1 2             j = -4   2               j = -6 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, 232]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 232]]
Out[4]=	PD[X[10, 1, 11, 2], X[18, 11, 19, 12], X[8, 9, 1, 10], X[22, 19, 9, 20], > X[20, 6, 21, 5], X[4, 22, 5, 21], X[7, 15, 8, 14], X[12, 4, 13, 3], > X[13, 16, 14, 17], X[15, 7, 16, 6], X[2, 18, 3, 17]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -11, 8, -6, 5, 10, -7, -3}, > {3, -1, 2, -8, -9, 7, -10, 9, 11, -2, 4, -5, 6, -4}]
In[6]:=	Jones[L][q]
Out[6]=	-(5/2) 3 4 3/2 5/2 7/2 9/2 -q + ---- - ------- + 4 Sqrt[q] - 5 q + 4 q - 4 q + 2 q - 3/2 Sqrt[q] q 11/2 > q
In[7]:=	A2Invariant[L][q]
Out[7]=	-8 -6 2 6 8 10 12 16 18 -1 + q - q + q + 2 q + 2 q + q + 2 q + q + q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 232]][a, z]
Out[8]=	3 3 5 1 1 z 4 z 2 z 2 z 3 z 3 z -(----) + ---- - -- + --- - --- + a z + ---- - ---- + a z - -- 5 3 5 3 a 3 a a a z a z a a a
In[9]:=	Kauffman[Link[11, NonAlternating, 232]][a, z]
Out[9]=	2 2 3 -4 1 1 5 z 8 z 5 z 2 2 z 4 z 2 2 8 z a - ---- - ---- + --- + --- + --- + 2 a z + z + ---- + ---- - a z - ---- - 5 3 5 3 a 4 2 5 a z a z a a a a a 3 3 4 4 5 5 5 18 z 15 z 3 4 10 z 17 z 5 z 11 z 10 z > ----- - ----- - 5 a z - 7 z - ----- - ----- + ---- + ----- + ----- + 3 a 4 2 5 3 a a a a a a 6 6 7 7 7 8 8 5 6 9 z 17 z z z z 7 8 2 z 4 z > 4 a z + 8 z + ---- + ----- - -- + -- + -- - a z - 2 z - ---- - ---- - 4 2 5 3 a 4 2 a a a a a a 9 9 z z > -- - -- 3 a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 2 1 2 1 2 2 2 4 2 3 + -- + q + ----- + ---- + ---- + 3 t + 2 q t + 2 q t + 3 q t + 2 6 2 4 2 q q t q t q t 4 3 6 3 6 4 8 4 8 5 10 5 12 6 > 2 q t + 2 q t + 2 q t + 3 q t + q t + q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n232

L11n231 L11n233