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

L8a11 L8a13

Knotscape

This page is passe. Go here instead! The 2-Component Link L8a12 Visit L8a12's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

X10,1,11,2 X16,7,9,8 X12,3,13,4 X14,5,15,6 X4,13,5,14 X6,15,7,16 X2,9,3,10 X8,11,1,12

Gauss Code:

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

Jones Polynomial:

- q-21/2 + q-19/2 - 2q-17/2 + 3q-15/2 - 3q-13/2 + 2q-11/2 - 2q-9/2 + q-7/2 - q-5/2

A2 (sl(3)) Invariant:

q-32 + q-30 + q-28 + q-26 + q-22 + q-18 + q-16 + q-12 + q-8

HOMFLY-PT Polynomial:

- 3a5z - 4a5z3 - a5z5 - a7z-1 - 4a7z - 4a7z3 - a7z5 + a9z-1 + 3a9z + a9z3

Kauffman Polynomial:

- 3a5z + 4a5z3 - a5z5 - a6z2 + 3a6z4 - a6z6 - a7z-1 + 5a7z - 6a7z3 + 4a7z5 - a7z7 + a8 - 5a8z2 + 6a8z4 - 2a8z6 - a9z-1 + 5a9z - 8a9z3 + 4a9z5 - a9z7 - 3a10z2 + 2a10z4 - a10z6 - a11z + a11z3 - a11z5 + a12z2 - a12z4 + 2a13z - a13z3

Khovanov Homology:

trqj r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 j = -4                 1 j = -6               1 1 j = -8             1     j = -10           1 1     j = -12         2 1       j = -14       1 1         j = -16     1 2           j = -18     1             j = -20 1 1               j = -22 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[8, Alternating, 12]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[8, Alternating, 12]]
Out[4]=	PD[X[10, 1, 11, 2], X[16, 7, 9, 8], X[12, 3, 13, 4], X[14, 5, 15, 6], > X[4, 13, 5, 14], X[6, 15, 7, 16], X[2, 9, 3, 10], X[8, 11, 1, 12]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -7, 3, -5, 4, -6, 2, -8}, {7, -1, 8, -3, 5, -4, 6, -2}]
In[6]:=	Jones[L][q]
Out[6]=	-(21/2) -(19/2) 2 3 3 2 2 -(7/2) -(5/2) -q + q - ----- + ----- - ----- + ----- - ---- + q - q 17/2 15/2 13/2 11/2 9/2 q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-32 -30 -28 -26 -22 -18 -16 -12 -8 q + q + q + q + q + q + q + q + q
In[8]:=	HOMFLYPT[Link[8, Alternating, 12]][a, z]
Out[8]=	7 9 a a 5 7 9 5 3 7 3 9 3 5 5 -(--) + -- - 3 a z - 4 a z + 3 a z - 4 a z - 4 a z + a z - a z - z z 7 5 > a z
In[9]:=	Kauffman[Link[8, Alternating, 12]][a, z]
Out[9]=	7 9 8 a a 5 7 9 11 13 6 2 8 2 a - -- - -- - 3 a z + 5 a z + 5 a z - a z + 2 a z - a z - 5 a z - z z 10 2 12 2 5 3 7 3 9 3 11 3 13 3 > 3 a z + a z + 4 a z - 6 a z - 8 a z + a z - a z + 6 4 8 4 10 4 12 4 5 5 7 5 9 5 > 3 a z + 6 a z + 2 a z - a z - a z + 4 a z + 4 a z - 11 5 6 6 8 6 10 6 7 7 9 7 > a z - a z - 2 a z - a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	-6 -4 1 1 1 1 1 2 1 q + q + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 22 8 20 8 20 7 18 6 16 6 16 5 14 5 q t q t q t q t q t q t q t 1 2 1 1 1 1 1 > ------ + ------ + ------ + ------ + ------ + ----- + ---- 14 4 12 4 12 3 10 3 10 2 8 2 6 q t q t q t q t q t q t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L8a12

L8a11 L8a13