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

L10a150 L10a152

Knotscape

This page is passe. Go here instead! The 3-Component Link L10a151 Visit L10a151's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

X6172 X10,4,11,3 X14,8,5,7 X18,14,19,13 X16,9,17,10 X8,15,9,16 X20,17,15,18 X12,20,13,19 X2536 X4,12,1,11

Gauss Code:

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

Jones Polynomial:

- q-5 + 3q-4 - 6q-3 + 11q-2 - 12q-1 + 14 - 12q + 11q2 - 6q3 + 3q4 - q5

A2 (sl(3)) Invariant:

- q-16 + q-12 - 2q-10 + 4q-8 + 3q-6 + 2q-4 + 6q-2 + 1 + 6q2 + 2q4 + 3q6 + 4q8 - 2q10 + q12 - q16

HOMFLY-PT Polynomial:

- a-4 - a-4z2 + a-2z-2 + 4a-2 + 4a-2z2 + 2a-2z4 - 2z-2 - 6 - 6z2 - 3z4 - z6 + a2z-2 + 4a2 + 4a2z2 + 2a2z4 - a4 - a4z2

Kauffman Polynomial:

a-5z - 2a-5z3 + a-5z5 - 2a-4 + 5a-4z2 - 6a-4z4 + 3a-4z6 + 3a-3z - 2a-3z3 - 4a-3z5 + 4a-3z7 + a-2z-2 - 8a-2 + 18a-2z2 - 18a-2z4 + 4a-2z6 + 3a-2z8 - 2a-1z-1 + 4a-1z + 2a-1z3 - 15a-1z5 + 9a-1z7 + a-1z9 + 2z-2 - 11 + 26z2 - 24z4 + 2z6 + 6z8 - 2az-1 + 4az + 2az3 - 15az5 + 9az7 + az9 + a2z-2 - 8a2 + 18a2z2 - 18a2z4 + 4a2z6 + 3a2z8 + 3a3z - 2a3z3 - 4a3z5 + 4a3z7 - 2a4 + 5a4z2 - 6a4z4 + 3a4z6 + a5z - 2a5z3 + a5z5

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 j = 11                     1 j = 9                   2   j = 7                 4 1   j = 5               7 2     j = 3             5 4       j = 1           9 7         j = -1         7 9           j = -3       4 5             j = -5     2 7               j = -7   1 4                 j = -9   2                   j = -11 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, 151]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 151]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 4, 11, 3], X[14, 8, 5, 7], X[18, 14, 19, 13], > X[16, 9, 17, 10], X[8, 15, 9, 16], X[20, 17, 15, 18], X[12, 20, 13, 19], > X[2, 5, 3, 6], X[4, 12, 1, 11]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 2, -10}, {6, -5, 7, -4, 8, -7}, > {9, -1, 3, -6, 5, -2, 10, -8, 4, -3}]
In[6]:=	Jones[L][q]
Out[6]=	-5 3 6 11 12 2 3 4 5 14 - q + -- - -- + -- - -- - 12 q + 11 q - 6 q + 3 q - q 4 3 2 q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-16 -12 2 4 3 2 6 2 4 6 8 10 1 - q + q - --- + -- + -- + -- + -- + 6 q + 2 q + 3 q + 4 q - 2 q + 10 8 6 4 2 q q q q q 12 16 > q - q
In[8]:=	HOMFLYPT[Link[10, Alternating, 151]][a, z]
Out[8]=	2 2 2 -4 4 2 4 2 1 a 2 z 4 z 2 2 -6 - a + -- + 4 a - a - -- + ----- + -- - 6 z - -- + ---- + 4 a z - 2 2 2 2 2 4 2 a z a z z a a 4 4 2 4 2 z 2 4 6 > a z - 3 z + ---- + 2 a z - z 2 a
In[9]:=	Kauffman[Link[10, Alternating, 151]][a, z]
Out[9]=	2 2 8 2 4 2 1 a 2 2 a z 3 z 4 z -11 - -- - -- - 8 a - 2 a + -- + ----- + -- - --- - --- + -- + --- + --- + 4 2 2 2 2 2 a z z 5 3 a a a z a z z a a 2 2 3 3 5 2 5 z 18 z 2 2 4 2 2 z > 4 a z + 3 a z + a z + 26 z + ---- + ----- + 18 a z + 5 a z - ---- - 4 2 5 a a a 3 3 4 4 2 z 2 z 3 3 3 5 3 4 6 z 18 z > ---- + ---- + 2 a z - 2 a z - 2 a z - 24 z - ---- - ----- - 3 a 4 2 a a a 5 5 5 2 4 4 4 z 4 z 15 z 5 3 5 5 5 6 > 18 a z - 6 a z + -- - ---- - ----- - 15 a z - 4 a z + a z + 2 z + 5 3 a a a 6 6 7 7 3 z 4 z 2 6 4 6 4 z 9 z 7 3 7 8 > ---- + ---- + 4 a z + 3 a z + ---- + ---- + 9 a z + 4 a z + 6 z + 4 2 3 a a a a 8 9 3 z 2 8 z 9 > ---- + 3 a z + -- + a z 2 a a
In[10]:=	Kh[L][q, t]
Out[10]=	9 1 2 1 4 2 7 4 5 7 - + 9 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + ---- + --- + q 11 5 9 4 7 4 7 3 5 3 5 2 3 2 3 q t q t q t q t q t q t q t q t q t 3 3 2 5 2 5 3 7 3 7 4 9 4 > 7 q t + 5 q t + 4 q t + 7 q t + 2 q t + 4 q t + q t + 2 q t + 11 5 > q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L10a151

L10a150 L10a152