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

L11a284 L11a286

Knotscape

This page is passe. Go here instead! The 2-Component Link L11a285 Visit L11a285's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-13/2 + 3q-11/2 - 6q-9/2 + 11q-7/2 - 15q-5/2 + 16q-3/2 - 17q-1/2 + 14q1/2 - 11q3/2 + 6q5/2 - 3q7/2 + q9/2

A2 (sl(3)) Invariant:

q-18 - q-16 + 2q-14 - 3q-12 + q-8 - 2q-6 + 5q-4 - 2q-2 + 5 + q2 + q4 + 3q6 - 2q8 + q10 - q12

HOMFLY-PT Polynomial:

- a-1z-1 + 3a-1z + 8a-1z3 + 5a-1z5 + a-1z7 + az-1 - 6az - 19az3 - 18az5 - 7az7 - az9 + 3a3z + 8a3z3 + 5a3z5 + a3z7

Kauffman Polynomial:

- 2a-4z2 + 3a-4z4 - a-4z6 + a-3z - 7a-3z3 + 9a-3z5 - 3a-3z7 - a-2z2 - 3a-2z4 + 9a-2z6 - 4a-2z8 - a-1z-1 - 2a-1z + 9a-1z3 - 9a-1z5 + 9a-1z7 - 4a-1z9 + 1 + 4z2 - 5z4 + 5z6 - 2z10 - az-1 - 6az + 32az3 - 44az5 + 28az7 - 9az9 + 7a2z2 - 17a2z4 + 11a2z6 - 2a2z8 - 2a2z10 - 2a3z + 8a3z3 - 15a3z5 + 11a3z7 - 5a3z9 + 2a4z2 - 12a4z4 + 13a4z6 - 6a4z8 + a5z - 6a5z3 + 10a5z5 - 5a5z7 - 2a6z2 + 6a6z4 - 3a6z6 + 2a7z3 - a7z5

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                   4 1   j = 4                 7 2     j = 2               7 4       j = 0             10 7         j = -2           8 9           j = -4         7 8             j = -6       4 8               j = -8     2 7                 j = -10   1 4                   j = -12   2                     j = -14 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, 285]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 285]]
Out[4]=	PD[X[10, 1, 11, 2], X[2, 11, 3, 12], X[12, 3, 13, 4], X[20, 14, 21, 13], > X[14, 6, 15, 5], X[4, 21, 5, 22], X[16, 9, 17, 10], X[22, 15, 9, 16], > X[6, 18, 7, 17], X[18, 8, 19, 7], X[8, 20, 1, 19]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 3, -6, 5, -9, 10, -11}, > {7, -1, 2, -3, 4, -5, 8, -7, 9, -10, 11, -4, 6, -8}]
In[6]:=	Jones[L][q]
Out[6]=	-(13/2) 3 6 11 15 16 17 -q + ----- - ---- + ---- - ---- + ---- - ------- + 14 Sqrt[q] - 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q 3/2 5/2 7/2 9/2 > 11 q + 6 q - 3 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-18 -16 2 3 -8 2 5 2 2 4 6 8 5 + q - q + --- - --- + q - -- + -- - -- + q + q + 3 q - 2 q + 14 12 6 4 2 q q q q q 10 12 > q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 285]][a, z]
Out[8]=	3 5 1 a 3 z 3 8 z 3 3 3 5 z 5 -(---) + - + --- - 6 a z + 3 a z + ---- - 19 a z + 8 a z + ---- - 18 a z + a z z a a a 7 3 5 z 7 3 7 9 > 5 a z + -- - 7 a z + a z - a z a
In[9]:=	Kauffman[Link[11, Alternating, 285]][a, z]
Out[9]=	2 2 1 a z 2 z 3 5 2 2 z z 2 2 1 - --- - - + -- - --- - 6 a z - 2 a z + a z + 4 z - ---- - -- + 7 a z + a z z 3 a 4 2 a a a 3 3 4 2 6 2 7 z 9 z 3 3 3 5 3 7 3 > 2 a z - 2 a z - ---- + ---- + 32 a z + 8 a z - 6 a z + 2 a z - 3 a a 4 4 5 5 4 3 z 3 z 2 4 4 4 6 4 9 z 9 z > 5 z + ---- - ---- - 17 a z - 12 a z + 6 a z + ---- - ---- - 4 2 3 a a a a 6 6 5 3 5 5 5 7 5 6 z 9 z 2 6 > 44 a z - 15 a z + 10 a z - a z + 5 z - -- + ---- + 11 a z + 4 2 a a 7 7 8 4 6 6 6 3 z 9 z 7 3 7 5 7 4 z > 13 a z - 3 a z - ---- + ---- + 28 a z + 11 a z - 5 a z - ---- - 3 a 2 a a 9 2 8 4 8 4 z 9 3 9 10 2 10 > 2 a z - 6 a z - ---- - 9 a z - 5 a z - 2 z - 2 a z a
In[10]:=	Kh[L][q, t]
Out[10]=	9 1 2 1 4 2 7 4 8 10 + -- + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + 2 14 6 12 5 10 5 10 4 8 4 8 3 6 3 6 2 q q t q t q t q t q t q t q t q t 7 8 8 2 2 2 4 2 4 3 > ----- + ---- + ---- + 7 t + 7 q t + 4 q t + 7 q t + 2 q t + 4 2 4 2 q t q t q t 6 3 6 4 8 4 10 5 > 4 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a285

L11a284 L11a286