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

L11a483 L11a485

Knotscape

This page is passe. Go here instead! The 3-Component Link L11a484 Visit L11a484's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

X6172 X14,7,15,8 X4,15,1,16 X12,6,13,5 X8493 X16,10,17,9 X22,17,19,18 X20,12,21,11 X10,20,11,19 X18,21,5,22 X2,14,3,13

Gauss Code:

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

Jones Polynomial:

- q-4 + 4q-3 - 8q-2 + 14q-1 - 16 + 21q - 20q2 + 18q3 - 13q4 + 8q5 - 4q6 + q7

A2 (sl(3)) Invariant:

- q-12 + q-10 + q-8 + 6q-4 + 2q-2 + 7 + 6q2 + q4 + 6q6 - 3q8 + 3q10 - q12 - 2q14 + 2q16 - 2q18 + q20

HOMFLY-PT Polynomial:

2a-4z2 + 3a-4z4 + a-4z6 + a-2z-2 - 6a-2z2 - 9a-2z4 - 5a-2z6 - a-2z8 - 2z-2 + 6z2 + 7z4 + 2z6 + a2z-2 - 2a2z2 - a2z4

Kauffman Polynomial:

a-8z4 - 2a-7z3 + 4a-7z5 + a-6z2 - 7a-6z4 + 8a-6z6 - a-5z + 4a-5z3 - 13a-5z5 + 11a-5z7 - 5a-4z2 + 12a-4z4 - 20a-4z6 + 12a-4z8 - 4a-3z + 20a-3z3 - 18a-3z5 - 6a-3z7 + 8a-3z9 + a-2z-2 - 21a-2z2 + 66a-2z4 - 67a-2z6 + 18a-2z8 + 2a-2z10 - 2a-1z-1 - 6a-1z + 23a-1z3 + a-1z5 - 30a-1z7 + 13a-1z9 + 2z-2 + 1 - 23z2 + 62z4 - 53z6 + 10z8 + 2z10 - 2az-1 - 4az + 12az3 - az5 - 12az7 + 5az9 + a2z-2 - 8a2z2 + 16a2z4 - 14a2z6 + 4a2z8 - a3z + 3a3z3 - 3a3z5 + a3z7

Khovanov Homology:

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 j = 15                       1 j = 13                     3   j = 11                   5 1   j = 9                 8 3     j = 7               10 5       j = 5             10 8         j = 3           11 10           j = 1         9 14             j = -1       5 7               j = -3     3 9                 j = -5   1 5                   j = -7   3                     j = -9 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[11, Alternating, 484]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 484]]
Out[4]=	PD[X[6, 1, 7, 2], X[14, 7, 15, 8], X[4, 15, 1, 16], X[12, 6, 13, 5], > X[8, 4, 9, 3], X[16, 10, 17, 9], X[22, 17, 19, 18], X[20, 12, 21, 11], > X[10, 20, 11, 19], X[18, 21, 5, 22], X[2, 14, 3, 13]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -11, 5, -3}, {9, -8, 10, -7}, > {4, -1, 2, -5, 6, -9, 8, -4, 11, -2, 3, -6, 7, -10}]
In[6]:=	Jones[L][q]
Out[6]=	-4 4 8 14 2 3 4 5 6 7 -16 - q + -- - -- + -- + 21 q - 20 q + 18 q - 13 q + 8 q - 4 q + q 3 2 q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-12 -10 -8 6 2 2 4 6 8 10 12 7 - q + q + q + -- + -- + 6 q + q + 6 q - 3 q + 3 q - q - 4 2 q q 14 16 18 20 > 2 q + 2 q - 2 q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 484]][a, z]
Out[8]=	2 2 2 4 4 -2 1 a 2 2 z 6 z 2 2 4 3 z 9 z 2 4 -- + ----- + -- + 6 z + ---- - ---- - 2 a z + 7 z + ---- - ---- - a z + 2 2 2 2 4 2 4 2 z a z z a a a a 6 6 8 6 z 5 z z > 2 z + -- - ---- - -- 4 2 2 a a a
In[9]:=	Kauffman[Link[11, Alternating, 484]][a, z]
Out[9]=	2 2 2 1 a 2 2 a z 4 z 6 z 3 2 z 1 + -- + ----- + -- - --- - --- - -- - --- - --- - 4 a z - a z - 23 z + -- - 2 2 2 2 a z z 5 3 a 6 z a z z a a a 2 2 3 3 3 3 5 z 21 z 2 2 2 z 4 z 20 z 23 z 3 3 3 > ---- - ----- - 8 a z - ---- + ---- + ----- + ----- + 12 a z + 3 a z + 4 2 7 5 3 a a a a a a 4 4 4 4 5 5 5 5 4 z 7 z 12 z 66 z 2 4 4 z 13 z 18 z z > 62 z + -- - ---- + ----- + ----- + 16 a z + ---- - ----- - ----- + -- - 8 6 4 2 7 5 3 a a a a a a a a 6 6 6 7 7 5 3 5 6 8 z 20 z 67 z 2 6 11 z 6 z > a z - 3 a z - 53 z + ---- - ----- - ----- - 14 a z + ----- - ---- - 6 4 2 5 3 a a a a a 7 8 8 9 9 30 z 7 3 7 8 12 z 18 z 2 8 8 z 13 z > ----- - 12 a z + a z + 10 z + ----- + ----- + 4 a z + ---- + ----- + a 4 2 3 a a a a 10 9 10 2 z > 5 a z + 2 z + ----- 2 a
In[10]:=	Kh[L][q, t]
Out[10]=	3 1 3 1 5 3 9 5 7 14 q + 11 q + ----- + ----- + ----- + ----- + ----- + ----- + ---- + --- + 9 5 7 4 5 4 5 3 3 3 3 2 2 q t q t q t q t q t q t q t q t 9 q 3 5 5 2 7 2 7 3 9 3 > --- + 10 q t + 10 q t + 8 q t + 10 q t + 5 q t + 8 q t + t 9 4 11 4 11 5 13 5 15 6 > 3 q t + 5 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a484

L11a483 L11a485