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

L9a14 L9a16

Knotscape

This page is passe. Go here instead! The 2-Component Link L9a15 Visit L9a15's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

X6172 X12,4,13,3 X14,8,15,7 X18,12,5,11 X8,18,9,17 X16,10,17,9 X10,16,11,15 X2536 X4,14,1,13

Gauss Code:

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

Jones Polynomial:

- q-1/2 + 2q1/2 - 5q3/2 + 5q5/2 - 7q7/2 + 7q9/2 - 6q11/2 + 4q13/2 - 2q15/2 + q17/2

A2 (sl(3)) Invariant:

q-2 + q2 + 3q4 + q6 + 4q8 + q10 + q12 + q14 - q16 + q18 - 2q20 - q22 - q26

HOMFLY-PT Polynomial:

a-7z-1 + 2a-7z + a-7z3 - 2a-5z-1 - 4a-5z - 3a-5z3 - a-5z5 - 2a-3z3 - a-3z5 + a-1z-1 + 2a-1z + a-1z3

Kauffman Polynomial:

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

Khovanov Homology:

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

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L9a15

L9a14 L9a16