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

L9n2 L9n4

Knotscape

This page is passe. Go here instead! The 2-Component Link L9n3 Visit L9n3's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

X6172 X14,7,15,8 X4,15,1,16 X5,10,6,11 X8493 X11,18,12,5 X17,12,18,13 X9,16,10,17 X2,14,3,13

Gauss Code:

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

Jones Polynomial:

q-15/2 - q-13/2 + q-11/2 - q-9/2 + q-7/2 - 2q-5/2 - q-1/2

A2 (sl(3)) Invariant:

- q-24 - q-22 - q-20 - q-18 + q-12 + 2q-10 + 3q-8 + 3q-6 + 2q-4 + q-2 + 1

HOMFLY-PT Polynomial:

- az-1 - az - a3z + 2a5z-1 + 3a5z + a5z3 - a7z-1 - a7z

Kauffman Polynomial:

az-1 - az - a2 + 2a3z - a3z3 + 3a4 - 7a4z2 + 5a4z4 - a4z6 - 2a5z-1 + 5a5z - 7a5z3 + 5a5z5 - a5z7 + 5a6 - 13a6z2 + 10a6z4 - 2a6z6 - a7z-1 + 2a7z - 6a7z3 + 5a7z5 - a7z7 + 2a8 - 6a8z2 + 5a8z4 - a8z6

Khovanov Homology:

trqj r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 j = 0               1 j = -2             1 2 j = -4           1   1 j = -6         1 2     j = -8       1 1       j = -10       1 1       j = -12   1 1           j = -14                 j = -16 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[9, NonAlternating, 3]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[9, NonAlternating, 3]]
Out[4]=	PD[X[6, 1, 7, 2], X[14, 7, 15, 8], X[4, 15, 1, 16], X[5, 10, 6, 11], > X[8, 4, 9, 3], X[11, 18, 12, 5], X[17, 12, 18, 13], X[9, 16, 10, 17], > X[2, 14, 3, 13]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 5, -3}, {-4, -1, 2, -5, -8, 4, -6, 7, 9, -2, 3, 8, -7, 6}]
In[6]:=	Jones[L][q]
Out[6]=	-(15/2) -(13/2) -(11/2) -(9/2) -(7/2) 2 1 q - q + q - q + q - ---- - ------- 5/2 Sqrt[q] q
In[7]:=	A2Invariant[L][q]
Out[7]=	-24 -22 -20 -18 -12 2 3 3 2 -2 1 - q - q - q - q + q + --- + -- + -- + -- + q 10 8 6 4 q q q q
In[8]:=	HOMFLYPT[Link[9, NonAlternating, 3]][a, z]
Out[8]=	5 7 a 2 a a 3 5 7 5 3 -(-) + ---- - -- - a z - a z + 3 a z - a z + a z z z z
In[9]:=	Kauffman[Link[9, NonAlternating, 3]][a, z]
Out[9]=	5 7 2 4 6 8 a 2 a a 3 5 7 -a + 3 a + 5 a + 2 a + - - ---- - -- - a z + 2 a z + 5 a z + 2 a z - z z z 4 2 6 2 8 2 3 3 5 3 7 3 4 4 > 7 a z - 13 a z - 6 a z - a z - 7 a z - 6 a z + 5 a z + 6 4 8 4 5 5 7 5 4 6 6 6 8 6 5 7 > 10 a z + 5 a z + 5 a z + 5 a z - a z - 2 a z - a z - a z - 7 7 > a z
In[10]:=	Kh[L][q, t]
Out[10]=	-4 2 1 1 1 1 1 1 1 1 + q + -- + ------ + ------ + ------ + ------ + ----- + ------ + ----- + 2 16 7 12 6 12 5 10 4 8 4 10 3 8 3 q q t q t q t q t q t q t q t 1 2 1 1 > ----- + ----- + ----- + ---- 6 3 6 2 4 2 2 q t q t q t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L9n3

L9n2 L9n4