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

L11a89 L11a91

Knotscape

This page is passe. Go here instead! The 2-Component Link L11a90 Visit L11a90's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-5/2 + 2q-3/2 - 6q-1/2 + 9q1/2 - 13q3/2 + 16q5/2 - 17q7/2 + 15q9/2 - 12q11/2 + 8q13/2 - 4q15/2 + q17/2

A2 (sl(3)) Invariant:

q-8 + q-6 + q-4 + 4q-2 + 2q2 + q4 - 4q6 + 2q8 - 3q10 + 2q12 + q14 - q16 + 3q18 - 2q20 + q22 + q24 - q26

HOMFLY-PT Polynomial:

a-7z + a-7z3 - a-5z-1 - 3a-5z - 5a-5z3 - 2a-5z5 + 3a-3z-1 + 7a-3z + 7a-3z3 + 4a-3z5 + a-3z7 - 4a-1z-1 - 8a-1z - 7a-1z3 - 2a-1z5 + 2az-1 + 3az + az3

Kauffman Polynomial:

- a-10z4 + 2a-9z3 - 4a-9z5 - 2a-8z2 + 8a-8z4 - 8a-8z6 + a-7z - 6a-7z3 + 13a-7z5 - 10a-7z7 + a-6 - 2a-6z2 + 3a-6z4 + 7a-6z6 - 8a-6z8 - a-5z-1 + 4a-5z - 15a-5z3 + 23a-5z5 - 5a-5z7 - 4a-5z9 + 3a-4 - 6a-4z2 - 9a-4z4 + 25a-4z6 - 10a-4z8 - a-4z10 - 3a-3z-1 + 13a-3z - 24a-3z3 + 14a-3z5 + 8a-3z7 - 6a-3z9 + 3a-2 - 7a-2z2 - 9a-2z4 + 17a-2z6 - 4a-2z8 - a-2z10 - 4a-1z-1 + 17a-1z - 26a-1z3 + 13a-1z5 + 2a-1z7 - 2a-1z9 + 2 - z2 - 6z4 + 7z6 - 2z8 - 2az-1 + 7az - 9az3 + 5az5 - 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               8 5       j = 8             9 7         j = 6           7 8           j = 4         6 9             j = 2       5 9               j = 0     1 4                 j = -2   1 5                   j = -4   1                     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, 90]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 90]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 4, 13, 3], X[18, 8, 19, 7], X[20, 10, 21, 9], > X[22, 15, 5, 16], X[8, 20, 9, 19], X[16, 21, 17, 22], X[14, 12, 15, 11], > X[10, 18, 11, 17], X[2, 5, 3, 6], X[4, 14, 1, 13]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 3, -6, 4, -9, 8, -2, 11, -8, 5, -7, 9, -3, > 6, -4, 7, -5}]
In[6]:=	Jones[L][q]
Out[6]=	-(5/2) 2 6 3/2 5/2 7/2 9/2 -q + ---- - ------- + 9 Sqrt[q] - 13 q + 16 q - 17 q + 15 q - 3/2 Sqrt[q] q 11/2 13/2 15/2 17/2 > 12 q + 8 q - 4 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-8 -6 -4 4 2 4 6 8 10 12 14 16 q + q + q + -- + 2 q + q - 4 q + 2 q - 3 q + 2 q + q - q + 2 q 18 20 22 24 26 > 3 q - 2 q + q + q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 90]][a, z]
Out[8]=	3 3 3 1 3 4 2 a z 3 z 7 z 8 z z 5 z 7 z -(----) + ---- - --- + --- + -- - --- + --- - --- + 3 a z + -- - ---- + ---- - 5 3 a z z 7 5 3 a 7 5 3 a z a z a a a a a a 3 5 5 5 7 7 z 3 2 z 4 z 2 z z > ---- + a z - ---- + ---- - ---- + -- a 5 3 a 3 a a a
In[9]:=	Kauffman[Link[11, Alternating, 90]][a, z]
Out[9]=	-6 3 3 1 3 4 2 a z 4 z 13 z 17 z 2 + a + -- + -- - ---- - ---- - --- - --- + -- + --- + ---- + ---- + 7 a z - 4 2 5 3 a z z 7 5 3 a a a a z a z a a a 2 2 2 2 3 3 3 3 3 2 2 z 2 z 6 z 7 z 2 z 6 z 15 z 24 z 26 z > z - ---- - ---- - ---- - ---- + ---- - ---- - ----- - ----- - ----- - 8 6 4 2 9 7 5 3 a a a a a a a a a 4 4 4 4 4 5 5 5 3 4 z 8 z 3 z 9 z 9 z 4 z 13 z 23 z > 9 a z - 6 z - --- + ---- + ---- - ---- - ---- - ---- + ----- + ----- + 10 8 6 4 2 9 7 5 a a a a a a a a 5 5 6 6 6 6 7 14 z 13 z 5 6 8 z 7 z 25 z 17 z 10 z > ----- + ----- + 5 a z + 7 z - ---- + ---- + ----- + ----- - ----- - 3 a 8 6 4 2 7 a a a a a a 7 7 7 8 8 8 9 9 5 z 8 z 2 z 7 8 8 z 10 z 4 z 4 z 6 z > ---- + ---- + ---- - a z - 2 z - ---- - ----- - ---- - ---- - ---- - 5 3 a 6 4 2 5 3 a a a a a a a 9 10 10 2 z z z > ---- - --- - --- a 4 2 a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 2 4 1 1 1 -2 5 4 5 q 4 9 q + 6 q + ----- + ----- + ----- + t + ----- + - + ---- + 9 q t + 6 4 4 3 2 3 2 2 t t q t q t q t q t 6 6 2 8 2 8 3 10 3 10 4 12 4 > 7 q t + 8 q t + 9 q t + 7 q t + 8 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 L11a90

L11a89 L11a91