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

L10a39 L10a41

Knotscape

This page is passe. Go here instead! The 2-Component Link L10a40 Visit L10a40's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-13/2 + 2q-11/2 - 5q-9/2 + 6q-7/2 - 8q-5/2 + 9q-3/2 - 8q-1/2 + 6q1/2 - 4q3/2 + 2q5/2 - q7/2

A2 (sl(3)) Invariant:

q-22 + 2q-20 + 2q-16 + 3q-14 - q-12 + q-10 - q-8 - 2q-6 - q-2 + 3 + 2q6 - q8 + q12

HOMFLY-PT Polynomial:

- a-3z - a-1z-1 - a-1z + a-1z3 + 2az-1 + 3az + 3az3 - a3z-1 + 2a3z3 - a5z-1 - 3a5z + a7z-1

Kauffman Polynomial:

- a-3z + 3a-3z3 - a-3z5 - a-2z2 + 5a-2z4 - 2a-2z6 - a-1z-1 + 5a-1z - 9a-1z3 + 9a-1z5 - 3a-1z7 - 1 + 7z2 - 14z4 + 10z6 - 3z8 - 2az-1 + 14az - 28az3 + 15az5 - 2az7 - az9 - 3a2 + 13a2z2 - 26a2z4 + 17a2z6 - 5a2z8 - a3z-1 + 8a3z - 13a3z3 + 7a3z5 - a3z7 - a3z9 - 2a4 + 5a4z2 - 3a4z4 + 3a4z6 - 2a4z8 + a5z-1 - 3a5z + 6a5z3 + a5z5 - 2a5z7 - a6 + 4a6z4 - 2a6z6 + a7z-1 - 3a7z + 3a7z3 - a7z5

Khovanov Homology:

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 8                     1 j = 6                   1   j = 4                 3 1   j = 2               3 1     j = 0             5 3       j = -2           5 4         j = -4         3 4           j = -6       3 5             j = -8     2 3               j = -10   1 4                 j = -12   1                   j = -14 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, 40]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 40]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 3, 13, 4], X[18, 8, 19, 7], X[20, 13, 5, 14], > X[14, 19, 15, 20], X[16, 10, 17, 9], X[10, 16, 11, 15], X[8, 18, 9, 17], > X[2, 5, 3, 6], X[4, 11, 1, 12]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 2, -10}, {9, -1, 3, -8, 6, -7, 10, -2, 4, -5, 7, -6, 8, -3, > 5, -4}]
In[6]:=	Jones[L][q]
Out[6]=	-(13/2) 2 5 6 8 9 8 3/2 -q + ----- - ---- + ---- - ---- + ---- - ------- + 6 Sqrt[q] - 4 q + 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q 5/2 7/2 > 2 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-22 2 2 3 -12 -10 -8 2 -2 6 8 12 3 + q + --- + --- + --- - q + q - q - -- - q + 2 q - q + q 20 16 14 6 q q q q
In[8]:=	HOMFLYPT[Link[10, Alternating, 40]][a, z]
Out[8]=	3 5 7 3 1 2 a a a a z z 5 z 3 3 3 -(---) + --- - -- - -- + -- - -- - - + 3 a z - 3 a z + -- + 3 a z + 2 a z a z z z z z 3 a a a
In[9]:=	Kauffman[Link[10, Alternating, 40]][a, z]
Out[9]=	3 5 7 2 4 6 1 2 a a a a z 5 z 3 -1 - 3 a - 2 a - a - --- - --- - -- + -- + -- - -- + --- + 14 a z + 8 a z - a z z z z z 3 a a 2 3 3 5 7 2 z 2 2 4 2 3 z 9 z 3 > 3 a z - 3 a z + 7 z - -- + 13 a z + 5 a z + ---- - ---- - 28 a z - 2 3 a a a 4 3 3 5 3 7 3 4 5 z 2 4 4 4 > 13 a z + 6 a z + 3 a z - 14 z + ---- - 26 a z - 3 a z + 2 a 5 5 6 6 4 z 9 z 5 3 5 5 5 7 5 6 2 z > 4 a z - -- + ---- + 15 a z + 7 a z + a z - a z + 10 z - ---- + 3 a 2 a a 7 2 6 4 6 6 6 3 z 7 3 7 5 7 8 > 17 a z + 3 a z - 2 a z - ---- - 2 a z - a z - 2 a z - 3 z - a 2 8 4 8 9 3 9 > 5 a z - 2 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	4 1 1 1 4 2 3 3 5 5 + -- + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + 2 14 6 12 5 10 5 10 4 8 4 8 3 6 3 6 2 q q t q t q t q t q t q t q t q t 3 4 5 2 2 2 4 2 4 3 6 3 8 4 > ----- + ---- + ---- + 3 t + 3 q t + q t + 3 q t + q t + q t + q t 4 2 4 2 q t q t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L10a40

L10a39 L10a41