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

L11n429 L11n431

Knotscape

This page is passe. Go here instead! The 3-Component Link L11n430 Visit L11n430's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

X8192 X9,20,10,21 X14,5,15,6 X12,14,7,13 X16,8,17,7 X22,18,13,17 X3,10,4,11 X18,11,19,12 X6,15,1,16 X4,20,5,19 X21,3,22,2

Gauss Code:

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

Jones Polynomial:

q-6 - 3q-5 + 5q-4 - 6q-3 + 9q-2 - 7q-1 + 8 - 5q + 3q2 - q3

A2 (sl(3)) Invariant:

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

HOMFLY-PT Polynomial:

- a-2 - a-2z2 + z-2 + 4 + 5z2 + 2z4 - 2a2z-2 - 4a2 - 6a2z2 - 4a2z4 - a2z6 + a4z-2 + a4 + 2a4z2 + a4z4

Kauffman Polynomial:

- a-3z + a-3z3 + 2a-2 - 4a-2z2 + 3a-2z4 - 3a-1z + 4a-1z3 - a-1z5 + a-1z7 - z-2 + 7 - 21z2 + 27z4 - 13z6 + 3z8 + 2az-1 - 4az + 2az3 + 6az5 - 6az7 + 2az9 - 2a2z-2 + 7a2 - 27a2z2 + 43a2z4 - 29a2z6 + 7a2z8 + 2a3z-1 - 2a3z + 5a3z3 - 3a3z5 - 4a3z7 + 2a3z9 - a4z-2 + 3a4 - 9a4z2 + 16a4z4 - 15a4z6 + 4a4z8 + 6a5z3 - 10a5z5 + 3a5z7 + a6z2 - 3a6z4 + a6z6

Khovanov Homology:

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 j = 7                   1 j = 5                 2   j = 3               3 1   j = 1             5 2     j = -1           4 5       j = -3         5 3         j = -5       3 6           j = -7     2 3             j = -9   1 3               j = -11   2                 j = -13 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]=	3
In[3]:=	Show[DrawMorseLink[Link[11, NonAlternating, 430]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 430]]
Out[4]=	PD[X[8, 1, 9, 2], X[9, 20, 10, 21], X[14, 5, 15, 6], X[12, 14, 7, 13], > X[16, 8, 17, 7], X[22, 18, 13, 17], X[3, 10, 4, 11], X[18, 11, 19, 12], > X[6, 15, 1, 16], X[4, 20, 5, 19], X[21, 3, 22, 2]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, 11, -7, -10, 3, -9}, {5, -1, -2, 7, 8, -4}, > {4, -3, 9, -5, 6, -8, 10, 2, -11, -6}]
In[6]:=	Jones[L][q]
Out[6]=	-6 3 5 6 9 7 2 3 8 + q - -- + -- - -- + -- - - - 5 q + 3 q - q 5 4 3 2 q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-18 -16 -14 -12 -10 6 4 6 4 2 4 6 10 2 + q - q + q + q + q + -- + -- + -- + -- + 3 q - q + q - q 8 6 4 2 q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 430]][a, z]
Out[8]=	2 4 2 -2 2 4 -2 2 a a 2 z 2 2 4 2 4 4 - a - 4 a + a + z - ---- + -- + 5 z - -- - 6 a z + 2 a z + 2 z - 2 2 2 z z a 2 4 4 4 2 6 > 4 a z + a z - a z
In[9]:=	Kauffman[Link[11, NonAlternating, 430]][a, z]
Out[9]=	2 4 3 2 2 4 -2 2 a a 2 a 2 a z 3 z 7 + -- + 7 a + 3 a - z - ---- - -- + --- + ---- - -- - --- - 4 a z - 2 2 2 z z 3 a a z z a 2 3 3 3 2 4 z 2 2 4 2 6 2 z 4 z 3 > 2 a z - 21 z - ---- - 27 a z - 9 a z + a z + -- + ---- + 2 a z + 2 3 a a a 4 5 3 3 5 3 4 3 z 2 4 4 4 6 4 z > 5 a z + 6 a z + 27 z + ---- + 43 a z + 16 a z - 3 a z - -- + 2 a a 7 5 3 5 5 5 6 2 6 4 6 6 6 z > 6 a z - 3 a z - 10 a z - 13 z - 29 a z - 15 a z + a z + -- - a 7 3 7 5 7 8 2 8 4 8 9 3 9 > 6 a z - 4 a z + 3 a z + 3 z + 7 a z + 4 a z + 2 a z + 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	5 1 2 1 3 2 3 3 6 - + 5 q + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ----- + q 13 6 11 5 9 5 9 4 7 4 7 3 5 3 5 2 q t q t q t q t q t q t q t q t 5 3 4 3 3 2 5 2 7 3 > ----- + ---- + --- + 2 q t + 3 q t + q t + 2 q t + q t 3 2 3 q t q t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n430

L11n429 L11n431