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

L10n19 L10n21

Knotscape

This page is passe. Go here instead! The 2-Component Link L10n20 Visit L10n20's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

X6172 X18,7,19,8 X4,19,1,20 X11,14,12,15 X8493 X5,13,6,12 X13,5,14,20 X9,16,10,17 X15,10,16,11 X2,18,3,17

Gauss Code:

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

Jones Polynomial:

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

A2 (sl(3)) Invariant:

q-16 + 3q-14 + q-10 - 3q-6 - 2q-2 + 2 + q2 + q4 + 3q6 + q10 + q12

HOMFLY-PT Polynomial:

- a-3z-1 - a-3z + a-1z-1 + 3a-1z + 2a-1z3 + az-1 - az - 2az3 - az5 - 2a3z-1 - a3z + a3z3 + a5z-1

Kauffman Polynomial:

a-3z-1 - 3a-3z + 3a-3z3 - a-3z5 - a-2 - a-2z2 + 4a-2z4 - 2a-2z6 + a-1z-1 - 5a-1z + 6a-1z3 + a-1z5 - 2a-1z7 - 2 + 3z2 + 4z4 - 2z6 - z8 - az-1 + 4az - 3az3 + 5az5 - 4az7 - 3a2 + 6a2z2 - 2a2z4 - a2z6 - a2z8 - 2a3z-1 + 9a3z - 9a3z3 + 3a3z5 - 2a3z7 - a4 + 2a4z2 - 2a4z4 - a4z6 - a5z-1 + 3a5z - 3a5z3

Khovanov Homology:

trqj 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             4 1   j = 2           2 1     j = 0         5 4       j = -2       4 4         j = -4     2 3           j = -6   2 4             j = -8   2               j = -10 2

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

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L10n20

L10n19 L10n21