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

L11a418 L11a420

Knotscape

This page is passe. Go here instead! The 3-Component Link L11a419 Visit L11a419's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

X6172 X12,4,13,3 X14,12,15,11 X22,16,11,15 X20,18,21,17 X16,8,17,7 X10,22,5,21 X8,19,9,20 X18,9,19,10 X2536 X4,14,1,13

Gauss Code:

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

Jones Polynomial:

q-3 - 2q-2 + 6q-1 - 9 + 15q - 16q2 + 18q3 - 16q4 + 12q5 - 8q6 + 4q7 - q8

A2 (sl(3)) Invariant:

q-10 + q-8 + q-6 + 5q-4 + 2q-2 + 3 + 7q2 + q4 + 6q6 + q8 + q12 - 4q14 + 3q16 - q18 - q20 + 2q22 - q24

HOMFLY-PT Polynomial:

- a-6z2 - a-6z4 - a-4 + a-4z2 + 2a-4z4 + a-4z6 + a-2z-2 + 4a-2 + 3a-2z2 + 2a-2z4 + a-2z6 - 2z-2 - 5 - 5z2 - 2z4 + a2z-2 + 2a2 + a2z2

Kauffman Polynomial:

- a-9z3 + a-9z5 + a-8z2 - 6a-8z4 + 4a-8z6 + 5a-7z3 - 12a-7z5 + 7a-7z7 - a-6z2 + 4a-6z4 - 10a-6z6 + 7a-6z8 + a-5z + 5a-5z3 - 9a-5z5 + a-5z7 + 4a-5z9 - 2a-4 + 11a-4z4 - 18a-4z6 + 9a-4z8 + a-4z10 + 3a-3z - 4a-3z3 + 3a-3z5 - 7a-3z7 + 6a-3z9 + a-2z-2 - 8a-2 + 15a-2z2 - 10a-2z4 - 4a-2z6 + 4a-2z8 + a-2z10 - 2a-1z-1 + 5a-1z - a-1z3 - 6a-1z5 + a-1z7 + 2a-1z9 + 2z-2 - 9 + 19z2 - 15z4 + z6 + 2z8 - 2az-1 + 3az + 2az3 - 5az5 + 2az7 + a2z-2 - 4a2 + 6a2z2 - 4a2z4 + a2z6

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 = 17                       1 j = 15                     3   j = 13                   5 1   j = 11                 7 3     j = 9               9 5       j = 7             9 7         j = 5           8 10           j = 3         7 8             j = 1       4 10               j = -1     2 5                 j = -3   1 5                   j = -5   1                     j = -7 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, 419]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 419]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 4, 13, 3], X[14, 12, 15, 11], X[22, 16, 11, 15], > X[20, 18, 21, 17], X[16, 8, 17, 7], X[10, 22, 5, 21], X[8, 19, 9, 20], > X[18, 9, 19, 10], X[2, 5, 3, 6], X[4, 14, 1, 13]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 6, -8, 9, -7}, > {3, -2, 11, -3, 4, -6, 5, -9, 8, -5, 7, -4}]
In[6]:=	Jones[L][q]
Out[6]=	-3 2 6 2 3 4 5 6 7 8 -9 + q - -- + - + 15 q - 16 q + 18 q - 16 q + 12 q - 8 q + 4 q - q 2 q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-10 -8 -6 5 2 2 4 6 8 12 14 16 3 + q + q + q + -- + -- + 7 q + q + 6 q + q + q - 4 q + 3 q - 4 2 q q 18 20 22 24 > q - q + 2 q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 419]][a, z]
Out[8]=	2 2 2 2 -4 4 2 2 1 a 2 z z 3 z 2 2 4 -5 - a + -- + 2 a - -- + ----- + -- - 5 z - -- + -- + ---- + a z - 2 z - 2 2 2 2 2 6 4 2 a z a z z a a a 4 4 4 6 6 z 2 z 2 z z z > -- + ---- + ---- + -- + -- 6 4 2 4 2 a a a a a
In[9]:=	Kauffman[Link[11, Alternating, 419]][a, z]
Out[9]=	2 2 8 2 2 1 a 2 2 a z 3 z 5 z -9 - -- - -- - 4 a + -- + ----- + -- - --- - --- + -- + --- + --- + 3 a z + 4 2 2 2 2 2 a z z 5 3 a a a z a z z a a 2 2 2 3 3 3 3 3 2 z z 15 z 2 2 z 5 z 5 z 4 z z 3 > 19 z + -- - -- + ----- + 6 a z - -- + ---- + ---- - ---- - -- + 2 a z - 8 6 2 9 7 5 3 a a a a a a a a 4 4 4 4 5 5 5 5 4 6 z 4 z 11 z 10 z 2 4 z 12 z 9 z 3 z > 15 z - ---- + ---- + ----- - ----- - 4 a z + -- - ----- - ---- + ---- - 8 6 4 2 9 7 5 3 a a a a a a a a 5 6 6 6 6 7 7 6 z 5 6 4 z 10 z 18 z 4 z 2 6 7 z z > ---- - 5 a z + z + ---- - ----- - ----- - ---- + a z + ---- + -- - a 8 6 4 2 7 5 a a a a a a 7 7 8 8 8 9 9 9 10 7 z z 7 8 7 z 9 z 4 z 4 z 6 z 2 z z > ---- + -- + 2 a z + 2 z + ---- + ---- + ---- + ---- + ---- + ---- + --- + 3 a 6 4 2 5 3 a 4 a a a a a a a 10 z > --- 2 a
In[10]:=	Kh[L][q, t]
Out[10]=	3 1 1 1 5 2 5 4 q 3 10 q + 7 q + ----- + ----- + ----- + ----- + ---- + --- + --- + 8 q t + 7 4 5 3 3 3 3 2 2 q t t q t q t q t q t q t 5 5 2 7 2 7 3 9 3 9 4 11 4 > 8 q t + 10 q t + 9 q t + 7 q t + 9 q t + 5 q t + 7 q t + 11 5 13 5 13 6 15 6 17 7 > 3 q t + 5 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a419

L11a418 L11a420