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

L11a449 L11a451

Knotscape

This page is passe. Go here instead! The 3-Component Link L11a450 Visit L11a450's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-3 - 3q-2 + 7q-1 - 11 + 16q - 16q2 + 18q3 - 15q4 + 11q5 - 6q6 + 3q7 - q8

A2 (sl(3)) Invariant:

q-10 - q-6 + 3q-4 - q-2 + 1 + 5q2 + q4 + 8q6 + 3q8 + 4q10 + 3q12 - 2q14 + 4q16 - q18 - q20 + q22 - q24

HOMFLY-PT Polynomial:

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

Kauffman Polynomial:

- 2a-9z3 + a-9z5 + 2a-8z2 - 6a-8z4 + 3a-8z6 - 2a-7z + 7a-7z3 - 10a-7z5 + 5a-7z7 + 3a-6 - 8a-6z2 + 15a-6z4 - 13a-6z6 + 6a-6z8 - 7a-5z + 19a-5z3 - 9a-5z5 - 2a-5z7 + 4a-5z9 - a-4z-2 + 11a-4 - 36a-4z2 + 52a-4z4 - 36a-4z6 + 11a-4z8 + a-4z10 + 2a-3z-1 - 12a-3z + 19a-3z3 - 6a-3z5 - 9a-3z7 + 7a-3z9 - 2a-2z-2 + 11a-2 - 29a-2z2 + 34a-2z4 - 28a-2z6 + 9a-2z8 + a-2z10 + 2a-1z-1 - 8a-1z + 15a-1z3 - 16a-1z5 + a-1z7 + 3a-1z9 - z-2 + 3 - 7z6 + 4z8 - az + 6az3 - 8az5 + 3az7 - a2 + 3a2z2 - 3a2z4 + 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                     2   j = 13                   4 1   j = 11                 7 2     j = 9               9 5       j = 7             9 6         j = 5           7 9           j = 3         9 9             j = 1       5 10               j = -1     2 6                 j = -3   1 5                   j = -5   2                     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, 450]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 450]]
Out[4]=	PD[X[6, 1, 7, 2], X[14, 4, 15, 3], X[20, 18, 21, 17], X[16, 8, 17, 7], > X[10, 14, 11, 13], X[22, 12, 13, 11], X[12, 22, 5, 21], X[8, 19, 9, 20], > X[18, 9, 19, 10], X[2, 5, 3, 6], X[4, 16, 1, 15]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 4, -8, 9, -5, 6, -7}, > {5, -2, 11, -4, 3, -9, 8, -3, 7, -6}]
In[6]:=	Jones[L][q]
Out[6]=	-3 3 7 2 3 4 5 6 7 8 -11 + q - -- + - + 16 q - 16 q + 18 q - 15 q + 11 q - 6 q + 3 q - q 2 q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-10 -6 3 -2 2 4 6 8 10 12 14 1 + q - q + -- - q + 5 q + q + 8 q + 3 q + 4 q + 3 q - 2 q + 4 q 16 18 20 22 24 > 4 q - q - q + q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 450]][a, z]
Out[8]=	2 2 2 -6 3 3 2 -2 1 2 2 2 z 5 z 2 z 2 2 -a + -- - -- + a + z + ----- - ----- - 3 z - ---- + ---- - ---- + a z - 4 2 4 2 2 2 6 4 2 a a a z a z a a a 4 4 4 6 6 4 z 3 z z z z > 2 z - -- + ---- + -- + -- + -- 6 4 2 4 2 a a a a a
In[9]:=	Kauffman[Link[11, Alternating, 450]][a, z]
Out[9]=	3 11 11 2 -2 1 2 2 2 2 z 7 z 12 z 3 + -- + -- + -- - a - z - ----- - ----- + ---- + --- - --- - --- - ---- - 6 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 8 z 2 z 8 z 36 z 29 z 2 2 2 z 7 z 19 z > --- - a z + ---- - ---- - ----- - ----- + 3 a z - ---- + ---- + ----- + a 8 6 4 2 9 7 5 a a a a a a a 3 3 4 4 4 4 5 19 z 15 z 3 6 z 15 z 52 z 34 z 2 4 z > ----- + ----- + 6 a z - ---- + ----- + ----- + ----- - 3 a z + -- - 3 a 8 6 4 2 9 a a a a a a 5 5 5 5 6 6 6 10 z 9 z 6 z 16 z 5 6 3 z 13 z 36 z > ----- - ---- - ---- - ----- - 8 a z - 7 z + ---- - ----- - ----- - 7 5 3 a 8 6 4 a a a a a a 6 7 7 7 7 8 8 28 z 2 6 5 z 2 z 9 z z 7 8 6 z 11 z > ----- + a z + ---- - ---- - ---- + -- + 3 a z + 4 z + ---- + ----- + 2 7 5 3 a 6 4 a a a a a a 8 9 9 9 10 10 9 z 4 z 7 z 3 z z z > ---- + ---- + ---- + ---- + --- + --- 2 5 3 a 4 2 a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	3 1 2 1 5 2 6 5 q 3 10 q + 9 q + ----- + ----- + ----- + ----- + ---- + --- + --- + 9 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 > 7 q t + 9 q t + 9 q t + 6 q t + 9 q t + 5 q t + 7 q t + 11 5 13 5 13 6 15 6 17 7 > 2 q t + 4 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a450

L11a449 L11a451