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

L11n156 L11n158

Knotscape

This page is passe. Go here instead! The 2-Component Link L11n157 Visit L11n157's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

X8192 X2,9,3,10 X10,3,11,4 X17,15,18,14 X7,17,8,16 X15,7,16,22 X13,19,14,18 X6,20,1,19 X20,12,21,11 X12,6,13,5 X4,21,5,22

Gauss Code:

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

Jones Polynomial:

- q-7/2 + 3q-5/2 - 6q-3/2 + 8q-1/2 - 11q1/2 + 10q3/2 - 10q5/2 + 7q7/2 - 4q9/2 + 2q11/2

A2 (sl(3)) Invariant:

q-10 - q-8 + 2q-6 + 4 + 5q4 + q6 + 2q8 + q10 - 3q12 - 2q16 - q18

HOMFLY-PT Polynomial:

a-5z-1 + a-5z - 2a-3z-1 + 2a-3z3 + a-3z5 - 5a-1z - 9a-1z3 - 5a-1z5 - a-1z7 + az-1 + 3az + 3az3 + az5

Kauffman Polynomial:

- 2a-6 + 6a-6z2 - 3a-6z4 + a-5z-1 - a-5z + 3a-5z3 - a-5z5 - a-5z7 - 5a-4 + 12a-4z2 - 10a-4z4 + 4a-4z6 - 2a-4z8 + 2a-3z-1 - 6a-3z3 + 5a-3z5 - 2a-3z7 - a-3z9 - 3a-2 + 5a-2z2 - 9a-2z4 + 9a-2z6 - 5a-2z8 + 5a-1z - 14a-1z3 + 14a-1z5 - 5a-1z7 - a-1z9 + 1 - 3z2 + 4z4 + 2z6 - 3z8 - az-1 + 3az - 3az3 + 7az5 - 4az7 - 2a2z2 + 6a2z4 - 3a2z6 - a3z + 2a3z3 - a3z5

Khovanov Homology:

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 12                   2 j = 10                 2   j = 8               5 2   j = 6             5 2     j = 4           5 5       j = 2         6 5         j = 0       3 6           j = -2     3 5             j = -4   1 4               j = -6   2                 j = -8 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, 157]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 157]]
Out[4]=	PD[X[8, 1, 9, 2], X[2, 9, 3, 10], X[10, 3, 11, 4], X[17, 15, 18, 14], > X[7, 17, 8, 16], X[15, 7, 16, 22], X[13, 19, 14, 18], X[6, 20, 1, 19], > X[20, 12, 21, 11], X[12, 6, 13, 5], X[4, 21, 5, 22]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 3, -11, 10, -8}, > {-5, -1, 2, -3, 9, -10, -7, 4, -6, 5, -4, 7, 8, -9, 11, 6}]
In[6]:=	Jones[L][q]
Out[6]=	-(7/2) 3 6 8 3/2 5/2 7/2 -q + ---- - ---- + ------- - 11 Sqrt[q] + 10 q - 10 q + 7 q - 5/2 3/2 Sqrt[q] q q 9/2 11/2 > 4 q + 2 q
In[7]:=	A2Invariant[L][q]
Out[7]=	-10 -8 2 4 6 8 10 12 16 18 4 + q - q + -- + 5 q + q + 2 q + q - 3 q - 2 q - q 6 q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 157]][a, z]
Out[8]=	3 3 5 5 1 2 a z 5 z 2 z 9 z 3 z 5 z 5 ---- - ---- + - + -- - --- + 3 a z + ---- - ---- + 3 a z + -- - ---- + a z - 5 3 z 5 a 3 a 3 a a z a z a a a 7 z > -- a
In[9]:=	Kauffman[Link[11, NonAlternating, 157]][a, z]
Out[9]=	2 2 5 3 1 2 a z 5 z 3 2 6 z 1 - -- - -- - -- + ---- + ---- - - - -- + --- + 3 a z - a z - 3 z + ---- + 6 4 2 5 3 z 5 a 6 a a a a z a z a a 2 2 3 3 3 12 z 5 z 2 2 3 z 6 z 14 z 3 3 3 4 > ----- + ---- - 2 a z + ---- - ---- - ----- - 3 a z + 2 a z + 4 z - 4 2 5 3 a a a a a 4 4 4 5 5 5 3 z 10 z 9 z 2 4 z 5 z 14 z 5 3 5 6 > ---- - ----- - ---- + 6 a z - -- + ---- + ----- + 7 a z - a z + 2 z + 6 4 2 5 3 a a a a a a 6 6 7 7 7 8 8 4 z 9 z 2 6 z 2 z 5 z 7 8 2 z 5 z > ---- + ---- - 3 a z - -- - ---- - ---- - 4 a z - 3 z - ---- - ---- - 4 2 5 3 a 4 2 a a a a a a 9 9 z z > -- - -- 3 a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 2 1 4 3 3 5 2 4 6 + 6 q + ----- + ----- + ----- + ----- + ----- + - + ---- + 5 q t + 5 q t + 8 4 6 3 4 3 4 2 2 2 t 2 q t q t q t q t q t q t 4 2 6 2 6 3 8 3 8 4 10 4 12 5 > 5 q t + 5 q t + 2 q t + 5 q t + 2 q t + 2 q t + 2 q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n157

L11n156 L11n158