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

L11a428 L11a430

Knotscape

This page is passe. Go here instead! The 3-Component Link L11a429 Visit L11a429's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

X6172 X12,4,13,3 X18,12,19,11 X16,13,17,14 X14,6,15,5 X10,16,5,15 X22,20,11,19 X8,22,9,21 X20,8,21,7 X2,9,3,10 X4,18,1,17

Gauss Code:

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

Jones Polynomial:

- q-2 + 4q-1 - 8 + 15q - 19q2 + 23q3 - 22q4 + 20q5 - 14q6 + 9q7 - 4q8 + q9

A2 (sl(3)) Invariant:

- q-6 + 2q-4 - q-2 - 1 + 6q2 - 2q4 + 5q6 + 4q8 + q10 + 6q12 - q14 + 6q16 + 2q18 - q20 + 4q22 - 2q24 - q26 + q28

HOMFLY-PT Polynomial:

a-8z2 + a-6z-2 + 2a-6 - a-6z2 - 2a-6z4 - 2a-4z-2 - 6a-4 - 4a-4z2 + a-4z6 + a-2z-2 + 4a-2 + 4a-2z2 + 2a-2z4 + a-2z6 - z2 - z4

Kauffman Polynomial:

a-10z2 - 2a-10z4 + a-10z6 + 5a-9z3 - 9a-9z5 + 4a-9z7 + a-8 - 4a-8z2 + 11a-8z4 - 16a-8z6 + 7a-8z8 + 3a-7z3 - 4a-7z5 - 7a-7z7 + 6a-7z9 + a-6z-2 - 3a-6 - 2a-6z2 + 23a-6z4 - 34a-6z6 + 12a-6z8 + 2a-6z10 - 2a-5z-1 + 6a-5z - 5a-5z3 + 6a-5z5 - 18a-5z7 + 12a-5z9 + 2a-4z-2 - 8a-4 + 8a-4z2 + 12a-4z4 - 28a-4z6 + 13a-4z8 + 2a-4z10 - 2a-3z-1 + 6a-3z + a-3z3 - 9a-3z5 + 6a-3z9 + a-2z-2 - 5a-2 + 8a-2z2 - 4a-2z4 - 7a-2z6 + 8a-2z8 + 3a-1z3 - 9a-1z5 + 7a-1z7 + 3z2 - 6z4 + 4z6 - az3 + az5

Khovanov Homology:

trqj r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 r = 8 j = 19                       1 j = 17                     3   j = 15                   6 1   j = 13                 9 4     j = 11               11 5       j = 9             11 9         j = 7           12 11           j = 5         8 12             j = 3       7 11               j = 1     3 10                 j = -1   1 5                   j = -3   3                     j = -5 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, Alternating, 429]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 429]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 4, 13, 3], X[18, 12, 19, 11], X[16, 13, 17, 14], > X[14, 6, 15, 5], X[10, 16, 5, 15], X[22, 20, 11, 19], X[8, 22, 9, 21], > X[20, 8, 21, 7], X[2, 9, 3, 10], X[4, 18, 1, 17]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {5, -1, 9, -8, 10, -6}, > {3, -2, 4, -5, 6, -4, 11, -3, 7, -9, 8, -7}]
In[6]:=	Jones[L][q]
Out[6]=	-2 4 2 3 4 5 6 7 8 9 -8 - q + - + 15 q - 19 q + 23 q - 22 q + 20 q - 14 q + 9 q - 4 q + q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-6 2 -2 2 4 6 8 10 12 14 16 -1 - q + -- - q + 6 q - 2 q + 5 q + 4 q + q + 6 q - q + 6 q + 4 q 18 20 22 24 26 28 > 2 q - q + 4 q - 2 q - q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 429]][a, z]
Out[8]=	2 2 2 2 4 2 6 4 1 2 1 2 z z 4 z 4 z 4 2 z -- - -- + -- + ----- - ----- + ----- - z + -- - -- - ---- + ---- - z - ---- + 6 4 2 6 2 4 2 2 2 8 6 4 2 6 a a a a z a z a z a a a a a 4 6 6 2 z z z > ---- + -- + -- 2 4 2 a a a
In[9]:=	Kauffman[Link[11, Alternating, 429]][a, z]
Out[9]=	-8 3 8 5 1 2 1 2 2 6 z 6 z 2 a - -- - -- - -- + ----- + ----- + ----- - ---- - ---- + --- + --- + 3 z + 6 4 2 6 2 4 2 2 2 5 3 5 3 a a a a z a z a z a z a z a a 2 2 2 2 2 3 3 3 3 3 z 4 z 2 z 8 z 8 z 5 z 3 z 5 z z 3 z 3 > --- - ---- - ---- + ---- + ---- + ---- + ---- - ---- + -- + ---- - a z - 10 8 6 4 2 9 7 5 3 a a a a a a a a a a 4 4 4 4 4 5 5 5 5 4 2 z 11 z 23 z 12 z 4 z 9 z 4 z 6 z 9 z > 6 z - ---- + ----- + ----- + ----- - ---- - ---- - ---- + ---- - ---- - 10 8 6 4 2 9 7 5 3 a a a a a a a a a 5 6 6 6 6 6 7 7 9 z 5 6 z 16 z 34 z 28 z 7 z 4 z 7 z > ---- + a z + 4 z + --- - ----- - ----- - ----- - ---- + ---- - ---- - a 10 8 6 4 2 9 7 a a a a a a a 7 7 8 8 8 8 9 9 9 10 18 z 7 z 7 z 12 z 13 z 8 z 6 z 12 z 6 z 2 z > ----- + ---- + ---- + ----- + ----- + ---- + ---- + ----- + ---- + ----- + 5 a 8 6 4 2 7 5 3 6 a a a a a a a a a 10 2 z > ----- 4 a
In[10]:=	Kh[L][q, t]
Out[10]=	3 1 3 1 5 3 q 3 5 5 2 10 q + 7 q + ----- + ----- + ---- + --- + --- + 11 q t + 8 q t + 12 q t + 5 3 3 2 2 q t t q t q t q t 7 2 7 3 9 3 9 4 11 4 11 5 > 12 q t + 11 q t + 11 q t + 9 q t + 11 q t + 5 q t + 13 5 13 6 15 6 15 7 17 7 19 8 > 9 q t + 4 q t + 6 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a429

L11a428 L11a430