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

L11a462 L11a464

Knotscape

This page is passe. Go here instead! The 3-Component Link L11a463 Visit L11a463's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-2 + 4q-1 - 7 + 12q - 15q2 + 17q3 - 17q4 + 15q5 - 9q6 + 7q7 - 3q8 + q9

A2 (sl(3)) Invariant:

- q-6 + 2q-4 + 4q2 - 3q4 + 2q6 - q10 + 3q12 + 7q16 + 5q18 + 3q20 + 5q22 + q28

HOMFLY-PT Polynomial:

a-8z-2 + a-8 + a-8z2 - 2a-6z-2 - 3a-6 - 4a-6z2 - 2a-6z4 + a-4z-2 + a-4 + 2a-4z2 + 2a-4z4 + a-4z6 + a-2z2 + 2a-2z4 + a-2z6 + 1 - z2 - z4

Kauffman Polynomial:

a-10z2 - 3a-10z4 + a-10z6 + 2a-9z3 - 8a-9z5 + 3a-9z7 - a-8z-2 + 8a-8 - 20a-8z2 + 28a-8z4 - 22a-8z6 + 6a-8z8 + 2a-7z-1 - 11a-7z + 11a-7z3 + 5a-7z5 - 13a-7z7 + 5a-7z9 - 2a-6z-2 + 13a-6 - 31a-6z2 + 49a-6z4 - 30a-6z6 + 4a-6z8 + 2a-6z10 + 2a-5z-1 - 11a-5z + 32a-5z5 - 30a-5z7 + 10a-5z9 - a-4z-2 + 5a-4 - 12a-4z2 + 24a-4z4 - 18a-4z6 + 4a-4z8 + 2a-4z10 - 7a-3z3 + 9a-3z5 - 8a-3z7 + 5a-3z9 - a-2z4 - 7a-2z6 + 6a-2z8 + a-1z3 - 9a-1z5 + 6a-1z7 + 1 + 2z2 - 7z4 + 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 1 j = 15                   4     j = 13                 5 3     j = 11               10 4       j = 9             8 6         j = 7           9 9           j = 5         6 8             j = 3       6 9               j = 1     3 8                 j = -1   1 4                   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, 463]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 463]]
Out[4]=	PD[X[6, 1, 7, 2], X[14, 4, 15, 3], X[8, 14, 9, 13], X[22, 8, 13, 7], > X[20, 15, 21, 16], X[16, 6, 17, 5], X[18, 12, 19, 11], X[12, 18, 5, 17], > X[10, 20, 11, 19], X[2, 9, 3, 10], X[4, 22, 1, 21]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {6, -1, 4, -3, 10, -9, 7, -8}, > {3, -2, 5, -6, 8, -7, 9, -5, 11, -4}]
In[6]:=	Jones[L][q]
Out[6]=	-2 4 2 3 4 5 6 7 8 9 -7 - q + - + 12 q - 15 q + 17 q - 17 q + 15 q - 9 q + 7 q - 3 q + q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-6 2 2 4 6 10 12 16 18 20 22 -q + -- + 4 q - 3 q + 2 q - q + 3 q + 7 q + 5 q + 3 q + 5 q + 4 q 28 > q
In[8]:=	HOMFLYPT[Link[11, Alternating, 463]][a, z]
Out[8]=	2 2 2 2 -8 3 -4 1 2 1 2 z 4 z 2 z z 4 1 + a - -- + a + ----- - ----- + ----- - z + -- - ---- + ---- + -- - z - 6 8 2 6 2 4 2 8 6 4 2 a a z a z a z a a a a 4 4 4 6 6 2 z 2 z 2 z z z > ---- + ---- + ---- + -- + -- 6 4 2 4 2 a a a a a
In[9]:=	Kauffman[Link[11, Alternating, 463]][a, z]
Out[9]=	8 13 5 1 2 1 2 2 11 z 11 z 2 1 + -- + -- + -- - ----- - ----- - ----- + ---- + ---- - ---- - ---- + 2 z + 8 6 4 8 2 6 2 4 2 7 5 7 5 a a a a z a z a z a z a z a a 2 2 2 2 3 3 3 3 z 20 z 31 z 12 z 2 z 11 z 7 z z 3 4 > --- - ----- - ----- - ----- + ---- + ----- - ---- + -- - a z - 7 z - 10 8 6 4 9 7 3 a a a a a a a a 4 4 4 4 4 5 5 5 5 5 3 z 28 z 49 z 24 z z 8 z 5 z 32 z 9 z 9 z > ---- + ----- + ----- + ----- - -- - ---- + ---- + ----- + ---- - ---- + 10 8 6 4 2 9 7 5 3 a a a a a a a a a a 6 6 6 6 6 7 7 7 5 6 z 22 z 30 z 18 z 7 z 3 z 13 z 30 z > a z + 4 z + --- - ----- - ----- - ----- - ---- + ---- - ----- - ----- - 10 8 6 4 2 9 7 5 a a a a a a a a 7 7 8 8 8 8 9 9 9 10 8 z 6 z 6 z 4 z 4 z 6 z 5 z 10 z 5 z 2 z > ---- + ---- + ---- + ---- + ---- + ---- + ---- + ----- + ---- + ----- + 3 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 4 3 q 3 5 5 2 8 q + 6 q + ----- + ----- + ---- + --- + --- + 9 q t + 6 q t + 8 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 13 5 > 9 q t + 9 q t + 8 q t + 6 q t + 10 q t + 4 q t + 5 q t + 13 6 15 6 17 7 17 8 19 8 > 3 q t + 4 q t + 3 q t + q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a463

L11a462 L11a464