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

L9a4 L9a6

Knotscape

This page is passe. Go here instead! The 2-Component Link L9a5 Visit L9a5's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-7/2 + 2q-5/2 - 6q-3/2 + 7q-1/2 - 9q1/2 + 9q3/2 - 8q5/2 + 6q7/2 - 3q9/2 + q11/2

A2 (sl(3)) Invariant:

q-12 + q-10 + 4q-6 + 2q-4 + 2q-2 + 4 - q2 + q4 - 3q6 - q8 - 2q12 + 2q14 - q18

HOMFLY-PT Polynomial:

a-5z + a-3z-1 - a-3z - 2a-3z3 - 2a-1z-1 - a-1z + a-1z3 + a-1z5 - 2az - 2az3 + a3z-1 + a3z

Kauffman Polynomial:

a-6z2 - a-6z4 - a-5z + 3a-5z3 - 3a-5z5 + 2a-4 - 5a-4z2 + 7a-4z4 - 5a-4z6 - a-3z-1 + 2a-3z3 + 2a-3z5 - 4a-3z7 + 5a-2 - 19a-2z2 + 24a-2z4 - 10a-2z6 - a-2z8 - 2a-1z-1 + 6a-1z - 7a-1z3 + 11a-1z5 - 7a-1z7 + 3 - 13z2 + 19z4 - 7z6 - z8 + 2az - 3az3 + 5az5 - 3az7 - a2 + 3a2z4 - 2a2z6 + a3z-1 - 3a3z + 3a3z3 - a3z5

Khovanov Homology:

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 12                   1 j = 10                 2   j = 8               4 1   j = 6             4 2     j = 4           5 4       j = 2         4 4         j = 0       4 6           j = -2     2 3             j = -4     4               j = -6 1 2                 j = -8 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, Alternating, 5]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[9, Alternating, 5]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[14, 8, 15, 7], X[18, 16, 5, 15], > X[16, 12, 17, 11], X[12, 18, 13, 17], X[8, 14, 9, 13], X[2, 5, 3, 6], > X[4, 9, 1, 10]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -8, 2, -9}, {8, -1, 3, -7, 9, -2, 5, -6, 7, -3, 4, -5, 6, -4}]
In[6]:=	Jones[L][q]
Out[6]=	-(7/2) 2 6 7 3/2 5/2 7/2 -q + ---- - ---- + ------- - 9 Sqrt[q] + 9 q - 8 q + 6 q - 5/2 3/2 Sqrt[q] q q 9/2 11/2 > 3 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-12 -10 4 2 2 2 4 6 8 12 14 18 4 + q + q + -- + -- + -- - q + q - 3 q - q - 2 q + 2 q - q 6 4 2 q q q
In[8]:=	HOMFLYPT[Link[9, Alternating, 5]][a, z]
Out[8]=	3 3 3 5 1 2 a z z z 3 2 z z 3 z ---- - --- + -- + -- - -- - - - 2 a z + a z - ---- + -- - 2 a z + -- 3 a z z 5 3 a 3 a a a z a a a
In[9]:=	Kauffman[Link[9, Alternating, 5]][a, z]
Out[9]=	3 2 2 5 2 1 2 a z 6 z 3 2 z 3 + -- + -- - a - ---- - --- + -- - -- + --- + 2 a z - 3 a z - 13 z + -- - 4 2 3 a z z 5 a 6 a a a z a a 2 2 3 3 3 4 4 5 z 19 z 3 z 2 z 7 z 3 3 3 4 z 7 z > ---- - ----- + ---- + ---- - ---- - 3 a z + 3 a z + 19 z - -- + ---- + 4 2 5 3 a 6 4 a a a a a a 4 5 5 5 6 24 z 2 4 3 z 2 z 11 z 5 3 5 6 5 z > ----- + 3 a z - ---- + ---- + ----- + 5 a z - a z - 7 z - ---- - 2 5 3 a 4 a a a a 6 7 7 8 10 z 2 6 4 z 7 z 7 8 z > ----- - 2 a z - ---- - ---- - 3 a z - z - -- 2 3 a 2 a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 1 2 4 2 4 3 2 4 6 + 4 q + ----- + ----- + ----- + ----- + ----- + - + ---- + 4 q t + 5 q t + 8 4 6 4 6 3 4 2 2 2 t 2 q t q t q t q t q t q t 4 2 6 2 6 3 8 3 8 4 10 4 12 5 > 4 q t + 4 q t + 2 q t + 4 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L9a5

L9a4 L9a6