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

L10a61 L10a63

Knotscape

This page is passe. Go here instead! The 2-Component Link L10a62 Visit L10a62's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

X8192 X10,4,11,3 X20,10,7,9 X2738 X16,12,17,11 X14,5,15,6 X4,15,5,16 X18,14,19,13 X12,18,13,17 X6,20,1,19

Gauss Code:

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

Jones Polynomial:

- q-5/2 + 2q-3/2 - 5q-1/2 + 6q1/2 - 9q3/2 + 9q5/2 - 9q7/2 + 8q9/2 - 5q11/2 + 3q13/2 - q15/2

A2 (sl(3)) Invariant:

q-8 + q-6 + q-4 + 3q-2 + 1 + 3q2 + 2q4 + 2q8 - 3q10 - 2q14 - q16 + q18 - q20 + q22

HOMFLY-PT Polynomial:

- 2a-5z - 3a-5z3 - a-5z5 + a-3z-1 + 7a-3z + 9a-3z3 + 5a-3z5 + a-3z7 - 3a-1z-1 - 9a-1z - 8a-1z3 - 2a-1z5 + 2az-1 + 3az + az3

Kauffman Polynomial:

- a-9z3 + a-8z2 - 3a-8z4 + 3a-7z3 - 5a-7z5 - 4a-6z2 + 10a-6z4 - 7a-6z6 + 4a-5z - 9a-5z3 + 14a-5z5 - 7a-5z7 + a-4 - 11a-4z2 + 13a-4z4 + 3a-4z6 - 4a-4z8 - a-3z-1 + 13a-3z - 36a-3z3 + 35a-3z5 - 8a-3z7 - a-3z9 + 3a-2 - 7a-2z2 - 8a-2z4 + 18a-2z6 - 6a-2z8 - 3a-1z-1 + 16a-1z - 32a-1z3 + 21a-1z5 - 2a-1z7 - a-1z9 + 3 - z2 - 8z4 + 8z6 - 2z8 - 2az-1 + 7az - 9az3 + 5az5 - az7

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 = 16                     1 j = 14                   2   j = 12                 3 1   j = 10               5 2     j = 8             4 3       j = 6           5 5         j = 4         4 4           j = 2       3 6             j = 0     2 3               j = -2   1 4                 j = -4   1                   j = -6 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, 62]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 62]]
Out[4]=	PD[X[8, 1, 9, 2], X[10, 4, 11, 3], X[20, 10, 7, 9], X[2, 7, 3, 8], > X[16, 12, 17, 11], X[14, 5, 15, 6], X[4, 15, 5, 16], X[18, 14, 19, 13], > X[12, 18, 13, 17], X[6, 20, 1, 19]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -4, 2, -7, 6, -10}, > {4, -1, 3, -2, 5, -9, 8, -6, 7, -5, 9, -8, 10, -3}]
In[6]:=	Jones[L][q]
Out[6]=	-(5/2) 2 5 3/2 5/2 7/2 9/2 -q + ---- - ------- + 6 Sqrt[q] - 9 q + 9 q - 9 q + 8 q - 3/2 Sqrt[q] q 11/2 13/2 15/2 > 5 q + 3 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-8 -6 -4 3 2 4 8 10 14 16 18 1 + q + q + q + -- + 3 q + 2 q + 2 q - 3 q - 2 q - q + q - 2 q 20 22 > q + q
In[8]:=	HOMFLYPT[Link[10, Alternating, 62]][a, z]
Out[8]=	3 3 3 5 1 3 2 a 2 z 7 z 9 z 3 z 9 z 8 z 3 z ---- - --- + --- - --- + --- - --- + 3 a z - ---- + ---- - ---- + a z - -- + 3 a z z 5 3 a 5 3 a 5 a z a a a a a 5 5 7 5 z 2 z z > ---- - ---- + -- 3 a 3 a a
In[9]:=	Kauffman[Link[10, Alternating, 62]][a, z]
Out[9]=	2 2 -4 3 1 3 2 a 4 z 13 z 16 z 2 z 4 z 3 + a + -- - ---- - --- - --- + --- + ---- + ---- + 7 a z - z + -- - ---- - 2 3 a z z 5 3 a 8 6 a a z a a a a 2 2 3 3 3 3 3 4 11 z 7 z z 3 z 9 z 36 z 32 z 3 4 3 z > ----- - ---- - -- + ---- - ---- - ----- - ----- - 9 a z - 8 z - ---- + 4 2 9 7 5 3 a 8 a a a a a a a 4 4 4 5 5 5 5 10 z 13 z 8 z 5 z 14 z 35 z 21 z 5 6 > ----- + ----- - ---- - ---- + ----- + ----- + ----- + 5 a z + 8 z - 6 4 2 7 5 3 a a a a a a a 6 6 6 7 7 7 8 8 9 7 z 3 z 18 z 7 z 8 z 2 z 7 8 4 z 6 z z > ---- + ---- + ----- - ---- - ---- - ---- - a z - 2 z - ---- - ---- - -- - 6 4 2 5 3 a 4 2 3 a a a a a a a a 9 z > -- a
In[10]:=	Kh[L][q, t]
Out[10]=	2 2 4 1 1 1 2 4 3 3 q 4 6 6 q + 4 q + ----- + ----- + ----- + -- + ----- + - + ---- + 4 q t + 5 q t + 6 4 4 3 2 3 2 2 2 t t q t q t q t t q t 6 2 8 2 8 3 10 3 10 4 12 4 12 5 > 5 q t + 4 q t + 3 q t + 5 q t + 2 q t + 3 q t + q t + 14 5 16 6 > 2 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L10a62

L10a61 L10a63