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

L11n23 L11n25

Knotscape

This page is passe. Go here instead! The 2-Component Link L11n24 Visit L11n24's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

X6172 X16,7,17,8 X4,17,1,18 X9,21,10,20 X8493 X21,18,22,19 X11,14,12,15 X5,13,6,12 X13,5,14,22 X19,11,20,10 X2,16,3,15

Gauss Code:

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

Jones Polynomial:

q-9/2 - 3q-7/2 + 4q-5/2 - 5q-3/2 + 4q-1/2 - 5q1/2 + 3q3/2 - q5/2 + q9/2 - q11/2

A2 (sl(3)) Invariant:

- q-14 + q-12 + 2q-6 + q-4 + 4q-2 + 2 + q2 - 3q6 - q10 + q14 + q16 + q18

HOMFLY-PT Polynomial:

- a-5z-1 - a-5z + 3a-3z-1 + 4a-3z + a-3z3 - 4a-1z-1 - 5a-1z - 2a-1z3 + 2az-1 + 3az + 3az3 + az5 - a3z - a3z3

Kauffman Polynomial:

- a-5z-1 + 4a-5z - 9a-5z3 + 6a-5z5 - a-5z7 - a-4 + 9a-4z2 - 13a-4z4 + 7a-4z6 - a-4z8 - 3a-3z-1 + 14a-3z - 21a-3z3 + 10a-3z5 - a-3z7 - 3a-2 + 20a-2z2 - 31a-2z4 + 15a-2z6 - 2a-2z8 - 4a-1z-1 + 18a-1z - 27a-1z3 + 8a-1z5 + 3a-1z7 - a-1z9 - 2 + 13z2 - 28z4 + 20z6 - 4z8 - 2az-1 + 11az - 23az3 + 15az5 - az9 - a2 + a2z2 - 7a2z4 + 11a2z6 - 3a2z8 + 3a3z - 8a3z3 + 11a3z5 - 3a3z7 - a4z2 + 3a4z4 - a4z6

Khovanov Homology:

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 j = 12                       1 j = 10                         j = 8                 1 1 1   j = 6               2 1       j = 4             2 1 1       j = 2           4 3 1         j = 0         3 4             j = -2       2 3 2             j = -4     2 3                 j = -6   1 2                   j = -8   2                     j = -10 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, 24]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 24]]
Out[4]=	PD[X[6, 1, 7, 2], X[16, 7, 17, 8], X[4, 17, 1, 18], X[9, 21, 10, 20], > X[8, 4, 9, 3], X[21, 18, 22, 19], X[11, 14, 12, 15], X[5, 13, 6, 12], > X[13, 5, 14, 22], X[19, 11, 20, 10], X[2, 16, 3, 15]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -11, 5, -3}, {-8, -1, 2, -5, -4, 10, -7, 8, -9, 7, 11, -2, 3, 6, > -10, 4, -6, 9}]
In[6]:=	Jones[L][q]
Out[6]=	-(9/2) 3 4 5 4 3/2 5/2 9/2 q - ---- + ---- - ---- + ------- - 5 Sqrt[q] + 3 q - q + q - 7/2 5/2 3/2 Sqrt[q] q q q 11/2 > q
In[7]:=	A2Invariant[L][q]
Out[7]=	-14 -12 2 -4 4 2 6 10 14 16 18 2 - q + q + -- + q + -- + q - 3 q - q + q + q + q 6 2 q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 24]][a, z]
Out[8]=	3 3 1 3 4 2 a z 4 z 5 z 3 z 2 z -(----) + ---- - --- + --- - -- + --- - --- + 3 a z - a z + -- - ---- + 5 3 a z z 5 3 a 3 a a z a z a a a 3 3 3 5 > 3 a z - a z + a z
In[9]:=	Kauffman[Link[11, NonAlternating, 24]][a, z]
Out[9]=	-4 3 2 1 3 4 2 a 4 z 14 z 18 z -2 - a - -- - a - ---- - ---- - --- - --- + --- + ---- + ---- + 11 a z + 2 5 3 a z z 5 3 a a a z a z a a 2 2 3 3 3 3 2 9 z 20 z 2 2 4 2 9 z 21 z 27 z > 3 a z + 13 z + ---- + ----- + a z - a z - ---- - ----- - ----- - 4 2 5 3 a a a a a 4 4 5 3 3 3 4 13 z 31 z 2 4 4 4 6 z > 23 a z - 8 a z - 28 z - ----- - ----- - 7 a z + 3 a z + ---- + 4 2 5 a a a 5 5 6 6 10 z 8 z 5 3 5 6 7 z 15 z 2 6 > ----- + ---- + 15 a z + 11 a z + 20 z + ---- + ----- + 11 a z - 3 a 4 2 a a a 7 7 7 8 8 9 4 6 z z 3 z 3 7 8 z 2 z 2 8 z 9 > a z - -- - -- + ---- - 3 a z - 4 z - -- - ---- - 3 a z - -- - a z 5 3 a 4 2 a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 2 1 2 1 2 2 3 2 3 4 + -- + 4 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + - + 2 10 5 8 4 6 4 6 3 4 3 4 2 2 2 t q q t q t q t q t q t q t q t 3 2 4 2 2 4 2 6 2 4 3 6 3 8 3 > ---- + 3 q t + 2 q t + q t + q t + 2 q t + q t + q t + q t + 2 q t 8 4 8 5 12 6 > q t + q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n24

L11n23 L11n25