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

L10a44 L10a46

Knotscape

This page is passe. Go here instead! The 2-Component Link L10a45 Visit L10a45's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-7/2 + 2q-5/2 - 6q-3/2 + 8q-1/2 - 11q1/2 + 11q3/2 - 11q5/2 + 9q7/2 - 5q9/2 + 3q11/2 - q13/2

A2 (sl(3)) Invariant:

q-12 + q-10 + 4q-6 + 2q-4 + q-2 + 4 - q2 + 2q4 - q6 - q8 - 4q12 + q14 - q18 + q20

HOMFLY-PT Polynomial:

- a-5z - a-5z3 + a-3z-1 + 3a-3z + 2a-3z3 + a-3z5 - 2a-1z-1 - 3a-1z + a-1z5 - 2az - 2az3 + a3z-1 + a3z

Kauffman Polynomial:

2a-7z3 - a-7z5 - 4a-6z2 + 7a-6z4 - 3a-6z6 + a-5z - 5a-5z3 + 8a-5z5 - 4a-5z7 + 2a-4 - 6a-4z2 + 4a-4z4 + 2a-4z6 - 3a-4z8 - a-3z-1 + 2a-3z - 6a-3z3 + 10a-3z5 - 5a-3z7 - a-3z9 + 5a-2 - 10a-2z2 + 5a-2z4 + 4a-2z6 - 5a-2z8 - 2a-1z-1 + 4a-1z - 3a-1z3 + 6a-1z5 - 4a-1z7 - a-1z9 + 3 - 8z2 + 11z4 - 3z6 - 2z8 - az3 + 4az5 - 3az7 - a2 + 3a2z4 - 2a2z6 + a3z-1 - 3a3z + 3a3z3 - a3z5

Khovanov Homology:

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 j = 14                     1 j = 12                   2   j = 10                 3 1   j = 8               6 2     j = 6             5 3       j = 4           6 6         j = 2         5 5           j = 0       4 7             j = -2     2 4               j = -4     4                 j = -6 1 2                   j = -8 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[10, Alternating, 45]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 45]]
Out[4]=	PD[X[6, 1, 7, 2], X[14, 3, 15, 4], X[16, 8, 17, 7], X[20, 12, 5, 11], > X[8, 20, 9, 19], X[18, 10, 19, 9], X[10, 18, 11, 17], X[12, 16, 13, 15], > X[2, 5, 3, 6], X[4, 13, 1, 14]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 2, -10}, {9, -1, 3, -5, 6, -7, 4, -8, 10, -2, 8, -3, 7, -6, > 5, -4}]
In[6]:=	Jones[L][q]
Out[6]=	-(7/2) 2 6 8 3/2 5/2 7/2 -q + ---- - ---- + ------- - 11 Sqrt[q] + 11 q - 11 q + 9 q - 5/2 3/2 Sqrt[q] q q 9/2 11/2 13/2 > 5 q + 3 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-12 -10 4 2 -2 2 4 6 8 12 14 18 20 4 + q + q + -- + -- + q - q + 2 q - q - q - 4 q + q - q + q 6 4 q q
In[8]:=	HOMFLYPT[Link[10, Alternating, 45]][a, z]
Out[8]=	3 3 3 5 5 1 2 a z 3 z 3 z 3 z 2 z 3 z z ---- - --- + -- - -- + --- - --- - 2 a z + a z - -- + ---- - 2 a z + -- + -- 3 a z z 5 3 a 5 3 3 a a z a a a a a
In[9]:=	Kauffman[Link[10, Alternating, 45]][a, z]
Out[9]=	3 2 2 5 2 1 2 a z 2 z 4 z 3 2 4 z 3 + -- + -- - a - ---- - --- + -- + -- + --- + --- - 3 a z - 8 z - ---- - 4 2 3 a z z 5 3 a 6 a a a z a a a 2 2 3 3 3 3 4 6 z 10 z 2 z 5 z 6 z 3 z 3 3 3 4 7 z > ---- - ----- + ---- - ---- - ---- - ---- - a z + 3 a z + 11 z + ---- + 4 2 7 5 3 a 6 a a a a a a 4 4 5 5 5 5 4 z 5 z 2 4 z 8 z 10 z 6 z 5 3 5 6 > ---- + ---- + 3 a z - -- + ---- + ----- + ---- + 4 a z - a z - 3 z - 4 2 7 5 3 a a a a a a 6 6 6 7 7 7 8 3 z 2 z 4 z 2 6 4 z 5 z 4 z 7 8 3 z > ---- + ---- + ---- - 2 a z - ---- - ---- - ---- - 3 a z - 2 z - ---- - 6 4 2 5 3 a 4 a a a a a a 8 9 9 5 z z z > ---- - -- - -- 2 3 a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 1 2 4 2 4 4 2 4 7 + 5 q + ----- + ----- + ----- + ----- + ----- + - + ---- + 5 q t + 6 q t + 8 4 6 4 6 3 4 2 2 2 t 2 q t q t q t q t q t q t 4 2 6 2 6 3 8 3 8 4 10 4 10 5 > 6 q t + 5 q t + 3 q t + 6 q t + 2 q t + 3 q t + q t + 12 5 14 6 > 2 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L10a45

L10a44 L10a46