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

L11a181 L11a183

Knotscape

This page is passe. Go here instead! The 2-Component Link L11a182 Visit L11a182's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

X8192 X20,9,21,10 X6,21,1,22 X18,8,19,7 X10,4,11,3 X12,16,13,15 X14,6,15,5 X4,14,5,13 X16,12,17,11 X22,18,7,17 X2,20,3,19

Gauss Code:

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

Jones Polynomial:

q-3/2 - 4q-1/2 + 8q1/2 - 15q3/2 + 19q5/2 - 23q7/2 + 23q9/2 - 20q11/2 + 15q13/2 - 9q15/2 + 4q17/2 - q19/2

A2 (sl(3)) Invariant:

- q-4 + 2q-2 - 1 + q2 + 5q4 - 2q6 + 6q8 - q10 + 2q14 - 4q16 + 4q18 - 3q20 + 2q24 - 2q26 + q28

HOMFLY-PT Polynomial:

- a-7z - 2a-7z3 - a-7z5 + a-5z + 3a-5z3 + 3a-5z5 + a-5z7 - a-3z-1 + a-3z + 3a-3z3 + 3a-3z5 + a-3z7 + a-1z-1 - 2a-1z3 - a-1z5

Kauffman Polynomial:

a-11z3 - a-11z5 - a-10z2 + 5a-10z4 - 4a-10z6 + a-9z - 6a-9z3 + 12a-9z5 - 8a-9z7 + 3a-8z2 - 11a-8z4 + 16a-8z6 - 10a-8z8 + a-7z - 7a-7z3 + 8a-7z5 + 3a-7z7 - 7a-7z9 + 6a-6z2 - 26a-6z4 + 36a-6z6 - 15a-6z8 - 2a-6z10 + a-5z - 5a-5z3 + a-5z5 + 16a-5z7 - 12a-5z9 + 2a-4z2 - 15a-4z4 + 28a-4z6 - 11a-4z8 - 2a-4z10 + a-3z-1 + 2a-3z - 13a-3z3 + 16a-3z5 + a-3z7 - 5a-3z9 - a-2 - a-2z2 - 3a-2z4 + 11a-2z6 - 6a-2z8 + a-1z-1 + a-1z - 8a-1z3 + 10a-1z5 - 4a-1z7 - z2 + 2z4 - z6

Khovanov Homology:

trqj r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 r = 8 j = 20                       1 j = 18                     3   j = 16                   6 1   j = 14                 9 3     j = 12               11 6       j = 10             12 9         j = 8           11 11           j = 6         9 13             j = 4       6 10               j = 2     3 10                 j = 0   1 5                   j = -2   3                     j = -4 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, Alternating, 182]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 182]]
Out[4]=	PD[X[8, 1, 9, 2], X[20, 9, 21, 10], X[6, 21, 1, 22], X[18, 8, 19, 7], > X[10, 4, 11, 3], X[12, 16, 13, 15], X[14, 6, 15, 5], X[4, 14, 5, 13], > X[16, 12, 17, 11], X[22, 18, 7, 17], X[2, 20, 3, 19]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -11, 5, -8, 7, -3}, > {4, -1, 2, -5, 9, -6, 8, -7, 6, -9, 10, -4, 11, -2, 3, -10}]
In[6]:=	Jones[L][q]
Out[6]=	-(3/2) 4 3/2 5/2 7/2 9/2 q - ------- + 8 Sqrt[q] - 15 q + 19 q - 23 q + 23 q - Sqrt[q] 11/2 13/2 15/2 17/2 19/2 > 20 q + 15 q - 9 q + 4 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-4 2 2 4 6 8 10 14 16 18 20 -1 - q + -- + q + 5 q - 2 q + 6 q - q + 2 q - 4 q + 4 q - 3 q + 2 q 24 26 28 > 2 q - 2 q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 182]][a, z]
Out[8]=	3 3 3 3 5 5 5 1 1 z z z 2 z 3 z 3 z 2 z z 3 z 3 z -(----) + --- - -- + -- + -- - ---- + ---- + ---- - ---- - -- + ---- + ---- - 3 a z 7 5 3 7 5 3 a 7 5 3 a z a a a a a a a a a 5 7 7 z z z > -- + -- + -- a 5 3 a a
In[9]:=	Kauffman[Link[11, Alternating, 182]][a, z]
Out[9]=	2 2 2 2 -2 1 1 z z z 2 z z 2 z 3 z 6 z 2 z -a + ---- + --- + -- + -- + -- + --- + - - z - --- + ---- + ---- + ---- - 3 a z 9 7 5 3 a 10 8 6 4 a z a a a a a a a a 2 3 3 3 3 3 3 4 4 z z 6 z 7 z 5 z 13 z 8 z 4 5 z 11 z > -- + --- - ---- - ---- - ---- - ----- - ---- + 2 z + ---- - ----- - 2 11 9 7 5 3 a 10 8 a a a a a a a a 4 4 4 5 5 5 5 5 5 26 z 15 z 3 z z 12 z 8 z z 16 z 10 z 6 > ----- - ----- - ---- - --- + ----- + ---- + -- + ----- + ----- - z - 6 4 2 11 9 7 5 3 a a a a a a a a a 6 6 6 6 6 7 7 7 7 7 4 z 16 z 36 z 28 z 11 z 8 z 3 z 16 z z 4 z > ---- + ----- + ----- + ----- + ----- - ---- + ---- + ----- + -- - ---- - 10 8 6 4 2 9 7 5 3 a a a a a a a a a a 8 8 8 8 9 9 9 10 10 10 z 15 z 11 z 6 z 7 z 12 z 5 z 2 z 2 z > ----- - ----- - ----- - ---- - ---- - ----- - ---- - ----- - ----- 8 6 4 2 7 5 3 6 4 a a a a a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 2 4 1 -2 3 5 3 q 4 6 6 2 10 q + 6 q + ----- + t + ----- + - + ---- + 10 q t + 9 q t + 13 q t + 4 3 2 2 t t q t q t 8 2 8 3 10 3 10 4 12 4 12 5 > 11 q t + 11 q t + 12 q t + 9 q t + 11 q t + 6 q t + 14 5 14 6 16 6 16 7 18 7 20 8 > 9 q t + 3 q t + 6 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a182

L11a181 L11a183