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

L11n93 L11n95

Knotscape

This page is passe. Go here instead! The 2-Component Link L11n94 Visit L11n94's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

X6172 X12,4,13,3 X9,14,10,15 X19,22,20,5 X11,21,12,20 X21,11,22,10 X15,19,16,18 X7,17,8,16 X17,9,18,8 X2536 X4,14,1,13

Gauss Code:

{{1, -10, 2, -11}, {10, -1, -8, 9, -3, 6, -5, -2, 11, 3, -7, 8, -9, 7, -4, 5, -6, 4}}

Jones Polynomial:

- q-5/2 + 2q-3/2 - 3q-1/2 + 3q1/2 - 4q3/2 + 2q5/2 - 2q7/2 + q9/2 - q13/2 + q15/2

A2 (sl(3)) Invariant:

q-8 + q-4 + q-2 + q2 + 2q6 + 2q8 + q10 + 2q12 + q16 - q20 - q22 - q24

HOMFLY-PT Polynomial:

a-7z-1 + a-7z - 3a-5z-1 - 4a-5z - a-5z3 + 3a-3z-1 + 6a-3z + 2a-3z3 - 2a-1z-1 - 5a-1z - 4a-1z3 - a-1z5 + az-1 + 2az + az3

Kauffman Polynomial:

a-8 - 5a-8z2 + 5a-8z4 - a-8z6 - a-7z-1 + 3a-7z - 8a-7z3 + 6a-7z5 - a-7z7 + 2a-6 - 5a-6z2 + 3a-6z4 - 3a-5z-1 + 10a-5z - 14a-5z3 + 8a-5z5 - a-5z7 + 8a-4z2 - 18a-4z4 + 12a-4z6 - 2a-4z8 - 3a-3z-1 + 14a-3z - 18a-3z3 + 6a-3z5 + 3a-3z7 - a-3z9 - 2a-2 + 13a-2z2 - 29a-2z4 + 21a-2z6 - 4a-2z8 - 2a-1z-1 + 11a-1z - 19a-1z3 + 9a-1z5 + 2a-1z7 - a-1z9 + 5z2 - 13z4 + 10z6 - 2z8 - az-1 + 4az - 7az3 + 5az5 - az7

Khovanov Homology:

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 j = 16                       1 j = 14                         j = 12                 1 1 1   j = 10               2 1       j = 8             2 2 1       j = 6           2 3 1         j = 4         3 2 1           j = 2       2 4 1             j = 0     1 2 1               j = -2   1 2                   j = -4   1                     j = -6 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[11, NonAlternating, 94]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 94]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 4, 13, 3], X[9, 14, 10, 15], X[19, 22, 20, 5], > X[11, 21, 12, 20], X[21, 11, 22, 10], X[15, 19, 16, 18], X[7, 17, 8, 16], > X[17, 9, 18, 8], X[2, 5, 3, 6], X[4, 14, 1, 13]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, -8, 9, -3, 6, -5, -2, 11, 3, -7, 8, -9, 7, > -4, 5, -6, 4}]
In[6]:=	Jones[L][q]
Out[6]=	-(5/2) 2 3 3/2 5/2 7/2 9/2 -q + ---- - ------- + 3 Sqrt[q] - 4 q + 2 q - 2 q + q - 3/2 Sqrt[q] q 13/2 15/2 > q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-8 -4 -2 2 6 8 10 12 16 20 22 24 q + q + q + q + 2 q + 2 q + q + 2 q + q - q - q - q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 94]][a, z]
Out[8]=	3 3 1 3 3 2 a z 4 z 6 z 5 z z 2 z ---- - ---- + ---- - --- + - + -- - --- + --- - --- + 2 a z - -- + ---- - 7 5 3 a z z 7 5 3 a 5 3 a z a z a z a a a a a 3 5 4 z 3 z > ---- + a z - -- a a
In[9]:=	Kauffman[Link[11, NonAlternating, 94]][a, z]
Out[9]=	-8 2 2 1 3 3 2 a 3 z 10 z 14 z 11 z a + -- - -- - ---- - ---- - ---- - --- - - + --- + ---- + ---- + ---- + 6 2 7 5 3 a z z 7 5 3 a a a a z a z a z a a a 2 2 2 2 3 3 3 3 2 5 z 5 z 8 z 13 z 8 z 14 z 18 z 19 z > 4 a z + 5 z - ---- - ---- + ---- + ----- - ---- - ----- - ----- - ----- - 8 6 4 2 7 5 3 a a a a a a a a 4 4 4 4 5 5 5 5 3 4 5 z 3 z 18 z 29 z 6 z 8 z 6 z 9 z > 7 a z - 13 z + ---- + ---- - ----- - ----- + ---- + ---- + ---- + ---- + 8 6 4 2 7 5 3 a a a a a a a a 6 6 6 7 7 7 7 5 6 z 12 z 21 z z z 3 z 2 z 7 8 > 5 a z + 10 z - -- + ----- + ----- - -- - -- + ---- + ---- - a z - 2 z - 8 4 2 7 5 3 a a a a a a a 8 8 9 9 2 z 4 z z z > ---- - ---- - -- - -- 4 2 3 a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 2 4 1 1 1 -2 2 2 2 q 2 1 + 4 q + 3 q + ----- + ----- + ----- + t + ----- + - + ---- + q t + 6 4 4 3 2 3 2 2 t t q t q t q t q t 4 6 4 2 6 2 8 2 6 3 8 3 10 3 > 2 q t + 2 q t + q t + 3 q t + 2 q t + q t + 2 q t + 2 q t + 8 4 10 4 12 4 12 5 12 6 16 7 > q t + q t + q t + q t + q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n94

L11n93 L11n95