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

L10a148 L10a150

Knotscape

This page is passe. Go here instead! The 3-Component Link L10a149 Visit L10a149's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-2 + 3q-1 - 6 + 12q - 14q2 + 17q3 - 15q4 + 13q5 - 9q6 + 5q7 - q8

A2 (sl(3)) Invariant:

- q-6 + q-4 - q-2 - 1 + 5q2 + 6q6 + 4q8 + 2q10 + 5q12 - q14 + 5q16 + q18 + 3q22 - q24

HOMFLY-PT Polynomial:

a-6z-2 + 2a-6 - a-6z4 - 2a-4z-2 - 7a-4 - 4a-4z2 + a-4z4 + a-4z6 + a-2z-2 + 6a-2 + 6a-2z2 + 3a-2z4 + a-2z6 - 1 - 2z2 - z4

Kauffman Polynomial:

a-9z5 + a-8 - 6a-8z4 + 5a-8z6 + 4a-7z3 - 15a-7z5 + 9a-7z7 + a-6z-2 - 3a-6 + 5a-6z2 - 8a-6z4 - 5a-6z6 + 7a-6z8 - 2a-5z-1 + 5a-5z + 9a-5z3 - 28a-5z5 + 13a-5z7 + 2a-5z9 + 2a-4z-2 - 10a-4 + 18a-4z2 - 9a-4z4 - 12a-4z6 + 11a-4z8 - 2a-3z-1 + 7a-3z + 4a-3z3 - 16a-3z5 + 8a-3z7 + 2a-3z9 + a-2z-2 - 9a-2 + 18a-2z2 - 13a-2z4 + a-2z6 + 4a-2z8 + 3a-1z - 3a-1z3 - 3a-1z5 + 4a-1z7 - 2 + 5z2 - 6z4 + 3z6 + az - 2az3 + az5

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 j = 17                     1 j = 15                   4   j = 13                 5 1   j = 11               8 4     j = 9             9 7       j = 7           8 6         j = 5         6 9           j = 3       6 8             j = 1     2 8               j = -1   1 4                 j = -3   2                   j = -5 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[10, Alternating, 149]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 149]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 7, 13, 8], X[4, 13, 1, 14], X[16, 6, 17, 5], > X[8, 4, 9, 3], X[20, 18, 15, 17], X[10, 20, 11, 19], X[18, 10, 19, 9], > X[14, 16, 5, 15], X[2, 12, 3, 11]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 5, -3}, {9, -4, 6, -8, 7, -6}, > {4, -1, 2, -5, 8, -7, 10, -2, 3, -9}]
In[6]:=	Jones[L][q]
Out[6]=	-2 3 2 3 4 5 6 7 8 -6 - q + - + 12 q - 14 q + 17 q - 15 q + 13 q - 9 q + 5 q - q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-6 -4 -2 2 6 8 10 12 14 16 18 -1 - q + q - q + 5 q + 6 q + 4 q + 2 q + 5 q - q + 5 q + q + 22 24 > 3 q - q
In[8]:=	HOMFLYPT[Link[10, Alternating, 149]][a, z]
Out[8]=	2 2 4 4 2 7 6 1 2 1 2 4 z 6 z 4 z z -1 + -- - -- + -- + ----- - ----- + ----- - 2 z - ---- + ---- - z - -- + -- + 6 4 2 6 2 4 2 2 2 4 2 6 4 a a a a z a z a z a a a a 4 6 6 3 z z z > ---- + -- + -- 2 4 2 a a a
In[9]:=	Kauffman[Link[10, Alternating, 149]][a, z]
Out[9]=	-8 3 10 9 1 2 1 2 2 5 z 7 z -2 + a - -- - -- - -- + ----- + ----- + ----- - ---- - ---- + --- + --- + 6 4 2 6 2 4 2 2 2 5 3 5 3 a a a a z a z a z a z a z a a 2 2 2 3 3 3 3 3 z 2 5 z 18 z 18 z 4 z 9 z 4 z 3 z > --- + a z + 5 z + ---- + ----- + ----- + ---- + ---- + ---- - ---- - a 6 4 2 7 5 3 a a a a a a a 4 4 4 4 5 5 5 5 3 4 6 z 8 z 9 z 13 z z 15 z 28 z 16 z > 2 a z - 6 z - ---- - ---- - ---- - ----- + -- - ----- - ----- - ----- - 8 6 4 2 9 7 5 3 a a a a a a a a 5 6 6 6 6 7 7 7 3 z 5 6 5 z 5 z 12 z z 9 z 13 z 8 z > ---- + a z + 3 z + ---- - ---- - ----- + -- + ---- + ----- + ---- + a 8 6 4 2 7 5 3 a a a a a a a 7 8 8 8 9 9 4 z 7 z 11 z 4 z 2 z 2 z > ---- + ---- + ----- + ---- + ---- + ---- a 6 4 2 5 3 a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	3 1 2 1 4 2 q 3 5 5 2 8 q + 6 q + ----- + ----- + ---- + --- + --- + 8 q t + 6 q t + 9 q t + 5 3 3 2 2 q t t q t q t q t 7 2 7 3 9 3 9 4 11 4 11 5 13 5 > 8 q t + 6 q t + 9 q t + 7 q t + 8 q t + 4 q t + 5 q t + 13 6 15 6 17 7 > q t + 4 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L10a149

L10a148 L10a150