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

L11n434 L11n436

Knotscape

This page is passe. Go here instead! The 3-Component Link L11n435 Visit L11n435's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

X8192 X14,4,15,3 X12,14,7,13 X2738 X22,10,13,9 X6,22,1,21 X20,16,21,15 X16,5,17,6 X11,19,12,18 X17,11,18,10 X4,19,5,20

Gauss Code:

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

Jones Polynomial:

3q-1 - 6 + 11q - 13q2 + 16q3 - 14q4 + 12q5 - 8q6 + 4q7 - q8

A2 (sl(3)) Invariant:

3q-4 + q-2 + 1 + 6q2 + 5q6 + 3q8 + 2q10 + 5q12 - q14 + 4q16 - q18 - 2q20 + 2q22 - q24

HOMFLY-PT Polynomial:

- a-6 - a-6z2 - a-6z4 + a-4z-2 + 5a-4 + 6a-4z2 + 3a-4z4 + a-4z6 - 2a-2z-2 - 8a-2 - 10a-2z2 - 4a-2z4 + z-2 + 4 + 3z2

Kauffman Polynomial:

- a-9z3 + a-9z5 + a-8z2 - 6a-8z4 + 4a-8z6 - a-7z + 4a-7z3 - 12a-7z5 + 7a-7z7 + a-6 - 4a-6z2 + 7a-6z4 - 12a-6z6 + 7a-6z8 - 3a-5z + 14a-5z3 - 17a-5z5 + 4a-5z7 + 3a-5z9 - a-4z-2 + 8a-4 - 24a-4z2 + 41a-4z4 - 34a-4z6 + 13a-4z8 + 2a-3z-1 - 8a-3z + 12a-3z3 - 7a-3z5 + 3a-3z9 - 2a-2z-2 + 12a-2 - 31a-2z2 + 34a-2z4 - 18a-2z6 + 6a-2z8 + 2a-1z-1 - 6a-1z + 3a-1z3 - 3a-1z5 + 3a-1z7 - z-2 + 6 - 12z2 + 6z4

Khovanov Homology:

trqj r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 j = 17                   1 j = 15                 3   j = 13               5 1   j = 11             7 3     j = 9           8 6       j = 7         8 6         j = 5       6 9           j = 3     5 7             j = 1   2 7               j = -1 1 4                 j = -3 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[11, NonAlternating, 435]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 435]]
Out[4]=	PD[X[8, 1, 9, 2], X[14, 4, 15, 3], X[12, 14, 7, 13], X[2, 7, 3, 8], > X[22, 10, 13, 9], X[6, 22, 1, 21], X[20, 16, 21, 15], X[16, 5, 17, 6], > X[11, 19, 12, 18], X[17, 11, 18, 10], X[4, 19, 5, 20]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -4, 2, -11, 8, -6}, {4, -1, 5, 10, -9, -3}, > {3, -2, 7, -8, -10, 9, 11, -7, 6, -5}]
In[6]:=	Jones[L][q]
Out[6]=	3 2 3 4 5 6 7 8 -6 + - + 11 q - 13 q + 16 q - 14 q + 12 q - 8 q + 4 q - q q
In[7]:=	A2Invariant[L][q]
Out[7]=	3 -2 2 6 8 10 12 14 16 18 20 1 + -- + q + 6 q + 5 q + 3 q + 2 q + 5 q - q + 4 q - q - 2 q + 4 q 22 24 > 2 q - q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 435]][a, z]
Out[8]=	2 2 2 4 -6 5 8 -2 1 2 2 z 6 z 10 z z 4 - a + -- - -- + z + ----- - ----- + 3 z - -- + ---- - ----- - -- + 4 2 4 2 2 2 6 4 2 6 a a a z a z a a a a 4 4 6 3 z 4 z z > ---- - ---- + -- 4 2 4 a a a
In[9]:=	Kauffman[Link[11, NonAlternating, 435]][a, z]
Out[9]=	-6 8 12 -2 1 2 2 2 z 3 z 8 z 6 z 6 + a + -- + -- - z - ----- - ----- + ---- + --- - -- - --- - --- - --- - 4 2 4 2 2 2 3 a z 7 5 3 a a a a z a z a z a a a 2 2 2 2 3 3 3 3 3 2 z 4 z 24 z 31 z z 4 z 14 z 12 z 3 z > 12 z + -- - ---- - ----- - ----- - -- + ---- + ----- + ----- + ---- + 8 6 4 2 9 7 5 3 a a a a a a a a a 4 4 4 4 5 5 5 5 5 4 6 z 7 z 41 z 34 z z 12 z 17 z 7 z 3 z > 6 z - ---- + ---- + ----- + ----- + -- - ----- - ----- - ---- - ---- + 8 6 4 2 9 7 5 3 a a a a a a a a a 6 6 6 6 7 7 7 8 8 8 4 z 12 z 34 z 18 z 7 z 4 z 3 z 7 z 13 z 6 z > ---- - ----- - ----- - ----- + ---- + ---- + ---- + ---- + ----- + ---- + 8 6 4 2 7 5 a 6 4 2 a a a a a a a a a 9 9 3 z 3 z > ---- + ---- 5 3 a a
In[10]:=	Kh[L][q, t]
Out[10]=	3 3 1 4 2 q 3 5 5 2 7 2 7 q + 5 q + ----- + ---- + --- + --- + 7 q t + 6 q t + 9 q t + 8 q t + 3 2 2 q t t q t q t 7 3 9 3 9 4 11 4 11 5 13 5 13 6 > 6 q t + 8 q t + 6 q t + 7 q t + 3 q t + 5 q t + q t + 15 6 17 7 > 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n435

L11n434 L11n436