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

L10n64 L10n66

Knotscape

This page is passe. Go here instead! The 3-Component Link L10n65 Visit L10n65's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

X6172 X3,11,4,10 X11,16,12,17 X13,19,14,18 X17,20,18,9 X19,13,20,12 X15,8,16,5 X7,14,8,15 X2536 X9,1,10,4

Gauss Code:

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

Jones Polynomial:

3q-5 - 5q-4 + 7q-3 - 7q-2 + 8q-1 - 6 + 5q - 2q2 + q3

A2 (sl(3)) Invariant:

q-20 + q-18 + 4q-16 + 2q-14 + 2q-12 + 3q-10 - q-8 + 2q-6 - q-4 + 2q-2 + 3 + 2q2 + 4q4 + q6 + q8 + q10

HOMFLY-PT Polynomial:

a-2z-2 + 2a-2 + a-2z2 - 3z-2 - 6 - 6z2 - 2z4 + 4a2z-2 + 8a2 + 7a2z2 + 4a2z4 + a2z6 - 3a4z-2 - 5a4 - 3a4z2 - a4z4 + a6z-2 + a6

Kauffman Polynomial:

a-2z-2 - 4a-2 + 6a-2z2 - 4a-2z4 + a-2z6 - a-1z-1 + a-1z + 4a-1z3 - 6a-1z5 + 2a-1z7 + 3z-2 - 14 + 28z2 - 23z4 + 4z6 + z8 - az-1 + az + 8az3 - 18az5 + 7az7 + 4a2z-2 - 21a2 + 40a2z2 - 37a2z4 + 10a2z6 + a2z8 - a3z-1 + a3z + 4a3z3 - 9a3z5 + 5a3z7 + 3a4z-2 - 14a4 + 24a4z2 - 18a4z4 + 7a4z6 - a5z-1 + a5z + 3a5z5 + a6z-2 - 4a6 + 6a6z2

Khovanov Homology:

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 7                 1 j = 5               1   j = 3             4 1   j = 1           2 1     j = -1         6 4       j = -3       4 5         j = -5     3 3           j = -7   2 4             j = -9 1 3               j = -11 3

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]=	3
In[3]:=	Show[DrawMorseLink[Link[10, NonAlternating, 65]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, NonAlternating, 65]]
Out[4]=	PD[X[6, 1, 7, 2], X[3, 11, 4, 10], X[11, 16, 12, 17], X[13, 19, 14, 18], > X[17, 20, 18, 9], X[19, 13, 20, 12], X[15, 8, 16, 5], X[7, 14, 8, 15], > X[2, 5, 3, 6], X[9, 1, 10, 4]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, -2, 10}, {9, -1, -8, 7}, > {-10, 2, -3, 6, -4, 8, -7, 3, -5, 4, -6, 5}]
In[6]:=	Jones[L][q]
Out[6]=	3 5 7 7 8 2 3 -6 + -- - -- + -- - -- + - + 5 q - 2 q + q 5 4 3 2 q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-20 -18 4 2 2 3 -8 2 -4 2 2 4 3 + q + q + --- + --- + --- + --- - q + -- - q + -- + 2 q + 4 q + 16 14 12 10 6 2 q q q q q q 6 8 10 > q + q + q
In[8]:=	HOMFLYPT[Link[10, NonAlternating, 65]][a, z]
Out[8]=	2 4 6 2 2 2 4 6 3 1 4 a 3 a a 2 z -6 + -- + 8 a - 5 a + a - -- + ----- + ---- - ---- + -- - 6 z + -- + 2 2 2 2 2 2 2 2 a z a z z z z a 2 2 4 2 4 2 4 4 4 2 6 > 7 a z - 3 a z - 2 z + 4 a z - a z + a z
In[9]:=	Kauffman[Link[10, NonAlternating, 65]][a, z]
Out[9]=	2 4 6 4 2 4 6 3 1 4 a 3 a a 1 a -14 - -- - 21 a - 14 a - 4 a + -- + ----- + ---- + ---- + -- - --- - - - 2 2 2 2 2 2 2 a z z a z a z z z z 3 5 2 a a z 3 5 2 6 z 2 2 4 2 > -- - -- + - + a z + a z + a z + 28 z + ---- + 40 a z + 24 a z + z z a 2 a 3 4 6 2 4 z 3 3 3 4 4 z 2 4 4 4 > 6 a z + ---- + 8 a z + 4 a z - 23 z - ---- - 37 a z - 18 a z - a 2 a 5 6 6 z 5 3 5 5 5 6 z 2 6 4 6 > ---- - 18 a z - 9 a z + 3 a z + 4 z + -- + 10 a z + 7 a z + a 2 a 7 2 z 7 3 7 8 2 8 > ---- + 7 a z + 5 a z + z + a z a
In[10]:=	Kh[L][q, t]
Out[10]=	5 6 3 1 3 2 4 3 3 4 4 t -- + - + ------ + ----- + ----- + ----- + ----- + ----- + ---- + ---- + --- + 3 q 11 4 9 4 9 3 7 3 7 2 5 2 5 3 q q q t q t q t q t q t q t q t q t 2 3 2 3 3 5 3 7 4 > 2 q t + q t + 4 q t + q t + q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L10n65

L10n64 L10n66