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

L11n188 L11n190

Knotscape

This page is passe. Go here instead! The 2-Component Link L11n189 Visit L11n189's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

X8192 X12,3,13,4 X22,10,7,9 X10,14,11,13 X5,16,6,17 X15,21,16,20 X21,19,22,18 X19,15,20,14 X2738 X4,11,5,12 X17,6,18,1

Gauss Code:

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

Jones Polynomial:

- q-11/2 - 3q-5/2 + 4q-3/2 - 5q-1/2 + 6q1/2 - 5q3/2 + 4q5/2 - 3q7/2 + q9/2

A2 (sl(3)) Invariant:

q-18 + 2q-16 + 3q-14 + 2q-12 + 2q-10 + 3q-8 - q-6 - 2q-2 - 2 - q4 + 2q6 + q12 - q14

HOMFLY-PT Polynomial:

a-3z + a-3z3 - 3a-1z - 3a-1z3 - a-1z5 + az-1 + 3az + 2az3 - 3a3z-1 - 5a3z - a3z3 + 2a5z-1 + a5z

Kauffman Polynomial:

- a-4z2 + 3a-4z4 - a-4z6 + a-3z - 8a-3z3 + 11a-3z5 - 3a-3z7 + 3a-2z2 - 9a-2z4 + 11a-2z6 - 3a-2z8 + 2a-1z - 12a-1z3 + 11a-1z5 - a-1z9 + 1 + 11z2 - 25z4 + 18z6 - 4z8 - az-1 + az - az3 - 4az5 + 4az7 - az9 + 3a2 - 10a2z4 + 6a2z6 - a2z8 - 3a3z-1 + 10a3z - 11a3z3 + 3a3z5 + 3a4 - 7a4z2 + 3a4z4 - 2a5z-1 + 10a5z - 14a5z3 + 7a5z5 - a5z7

Khovanov Homology:

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 10                       1 j = 8                     2   j = 6                   2 1   j = 4                 3 2     j = 2               3 2       j = 0           1 3 3         j = -2           3 4           j = -4       1 2 2             j = -6         3               j = -8     1 1                 j = -10 1                       j = -12 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, NonAlternating, 189]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 189]]
Out[4]=	PD[X[8, 1, 9, 2], X[12, 3, 13, 4], X[22, 10, 7, 9], X[10, 14, 11, 13], > X[5, 16, 6, 17], X[15, 21, 16, 20], X[21, 19, 22, 18], X[19, 15, 20, 14], > X[2, 7, 3, 8], X[4, 11, 5, 12], X[17, 6, 18, 1]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 2, -10, -5, 11}, > {9, -1, 3, -4, 10, -2, 4, 8, -6, 5, -11, 7, -8, 6, -7, -3}]
In[6]:=	Jones[L][q]
Out[6]=	-(11/2) 3 4 5 3/2 5/2 7/2 9/2 -q - ---- + ---- - ------- + 6 Sqrt[q] - 5 q + 4 q - 3 q + q 5/2 3/2 Sqrt[q] q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-18 2 3 2 2 3 -6 2 4 6 12 14 -2 + q + --- + --- + --- + --- + -- - q - -- - q + 2 q + q - q 16 14 12 10 8 2 q q q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 189]][a, z]
Out[8]=	3 5 3 3 a 3 a 2 a z 3 z 3 5 z 3 z 3 - - ---- + ---- + -- - --- + 3 a z - 5 a z + a z + -- - ---- + 2 a z - z z z 3 a 3 a a a 5 3 3 z > a z - -- a
In[9]:=	Kauffman[Link[11, NonAlternating, 189]][a, z]
Out[9]=	3 5 2 4 a 3 a 2 a z 2 z 3 5 1 + 3 a + 3 a - - - ---- - ---- + -- + --- + a z + 10 a z + 10 a z + z z z 3 a a 2 2 3 3 2 z 3 z 4 2 8 z 12 z 3 3 3 5 3 > 11 z - -- + ---- - 7 a z - ---- - ----- - a z - 11 a z - 14 a z - 4 2 3 a a a a 4 4 5 5 4 3 z 9 z 2 4 4 4 11 z 11 z 5 > 25 z + ---- - ---- - 10 a z + 3 a z + ----- + ----- - 4 a z + 4 2 3 a a a a 6 6 7 3 5 5 5 6 z 11 z 2 6 3 z 7 5 7 > 3 a z + 7 a z + 18 z - -- + ----- + 6 a z - ---- + 4 a z - a z - 4 2 3 a a a 8 9 8 3 z 2 8 z 9 > 4 z - ---- - a z - -- - a z 2 a a
In[10]:=	Kh[L][q, t]
Out[10]=	4 1 1 1 1 1 3 2 1 2 3 + -- + ------ + ------ + ----- + ----- + ----- + ----- + ----- + - + ---- + 2 12 6 10 6 8 4 8 3 4 3 6 2 4 2 t 4 q q t q t q t q t q t q t q t q t 3 2 2 2 4 2 4 3 6 3 6 4 > ---- + 3 t + 3 q t + 2 q t + 3 q t + 2 q t + 2 q t + q t + 2 q t 8 4 10 5 > 2 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n189

L11n188 L11n190