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

L11a288 L11a290

Knotscape

This page is passe. Go here instead! The 2-Component Link L11a289 Visit L11a289's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

X10,1,11,2 X12,4,13,3 X22,12,9,11 X2,9,3,10 X20,14,21,13 X14,7,15,8 X18,16,19,15 X16,6,17,5 X6,18,7,17 X4,19,5,20 X8,22,1,21

Gauss Code:

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

Jones Polynomial:

- q-5/2 + 3q-3/2 - 7q-1/2 + 10q1/2 - 15q3/2 + 17q5/2 - 18q7/2 + 16q9/2 - 12q11/2 + 8q13/2 - 4q15/2 + q17/2

A2 (sl(3)) Invariant:

q-8 + 3q-2 - 1 + 3q2 + 3q4 - q6 + 4q8 - 3q10 + q12 - q14 - 2q16 + 3q18 - 2q20 + q22 + q24 - q26

HOMFLY-PT Polynomial:

a-7z + a-7z3 - 3a-5z - 5a-5z3 - 2a-5z5 + 5a-3z + 7a-3z3 + 4a-3z5 + a-3z7 - a-1z-1 - 5a-1z - 6a-1z3 - 2a-1z5 + az-1 + 2az + az3

Kauffman Polynomial:

- a-10z4 + 2a-9z3 - 4a-9z5 - 3a-8z2 + 8a-8z4 - 8a-8z6 + 2a-7z - 6a-7z3 + 12a-7z5 - 10a-7z7 - 4a-6z2 + 9a-6z4 + 4a-6z6 - 8a-6z8 + 6a-5z - 24a-5z3 + 34a-5z5 - 9a-5z7 - 4a-5z9 - 16a-4z4 + 33a-4z6 - 13a-4z8 - a-4z10 + 10a-3z - 32a-3z3 + 24a-3z5 + 7a-3z7 - 7a-3z9 + 4a-2z2 - 28a-2z4 + 32a-2z6 - 8a-2z8 - a-2z10 - a-1z-1 + 10a-1z - 22a-1z3 + 10a-1z5 + 5a-1z7 - 3a-1z9 + 1 + 3z2 - 12z4 + 11z6 - 3z8 - az-1 + 4az - 6az3 + 4az5 - az7

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 = 18                       1 j = 16                     3   j = 14                   5 1   j = 12                 7 3     j = 10               9 5       j = 8             9 7         j = 6           8 9           j = 4         7 9             j = 2       5 10               j = 0     2 5                 j = -2   1 5                   j = -4   2                     j = -6 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]=	2
In[3]:=	Show[DrawMorseLink[Link[11, Alternating, 289]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 289]]
Out[4]=	PD[X[10, 1, 11, 2], X[12, 4, 13, 3], X[22, 12, 9, 11], X[2, 9, 3, 10], > X[20, 14, 21, 13], X[14, 7, 15, 8], X[18, 16, 19, 15], X[16, 6, 17, 5], > X[6, 18, 7, 17], X[4, 19, 5, 20], X[8, 22, 1, 21]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -4, 2, -10, 8, -9, 6, -11}, > {4, -1, 3, -2, 5, -6, 7, -8, 9, -7, 10, -5, 11, -3}]
In[6]:=	Jones[L][q]
Out[6]=	-(5/2) 3 7 3/2 5/2 7/2 -q + ---- - ------- + 10 Sqrt[q] - 15 q + 17 q - 18 q + 3/2 Sqrt[q] q 9/2 11/2 13/2 15/2 17/2 > 16 q - 12 q + 8 q - 4 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-8 3 2 4 6 8 10 12 14 16 18 -1 + q + -- + 3 q + 3 q - q + 4 q - 3 q + q - q - 2 q + 3 q - 2 q 20 22 24 26 > 2 q + q + q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 289]][a, z]
Out[8]=	3 3 3 3 1 a z 3 z 5 z 5 z z 5 z 7 z 6 z 3 -(---) + - + -- - --- + --- - --- + 2 a z + -- - ---- + ---- - ---- + a z - a z z 7 5 3 a 7 5 3 a a a a a a a 5 5 5 7 2 z 4 z 2 z z > ---- + ---- - ---- + -- 5 3 a 3 a a a
In[9]:=	Kauffman[Link[11, Alternating, 289]][a, z]
Out[9]=	2 2 2 1 a 2 z 6 z 10 z 10 z 2 3 z 4 z 4 z 1 - --- - - + --- + --- + ---- + ---- + 4 a z + 3 z - ---- - ---- + ---- + a z z 7 5 3 a 8 6 2 a a a a a a 3 3 3 3 3 4 4 4 2 z 6 z 24 z 32 z 22 z 3 4 z 8 z 9 z > ---- - ---- - ----- - ----- - ----- - 6 a z - 12 z - --- + ---- + ---- - 9 7 5 3 a 10 8 6 a a a a a a a 4 4 5 5 5 5 5 16 z 28 z 4 z 12 z 34 z 24 z 10 z 5 6 > ----- - ----- - ---- + ----- + ----- + ----- + ----- + 4 a z + 11 z - 4 2 9 7 5 3 a a a a a a a 6 6 6 6 7 7 7 7 8 z 4 z 33 z 32 z 10 z 9 z 7 z 5 z 7 8 > ---- + ---- + ----- + ----- - ----- - ---- + ---- + ---- - a z - 3 z - 8 6 4 2 7 5 3 a a a a a a a a 8 8 8 9 9 9 10 10 8 z 13 z 8 z 4 z 7 z 3 z z z > ---- - ----- - ---- - ---- - ---- - ---- - --- - --- 6 4 2 5 3 a 4 2 a a a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 2 4 1 2 1 2 5 5 5 q 4 10 q + 7 q + ----- + ----- + ----- + -- + ----- + - + ---- + 9 q t + 6 4 4 3 2 3 2 2 2 t t q t q t q t t q t 6 6 2 8 2 8 3 10 3 10 4 12 4 > 8 q t + 9 q t + 9 q t + 7 q t + 9 q t + 5 q t + 7 q t + 12 5 14 5 14 6 16 6 18 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 L11a289

L11a288 L11a290