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

L11a365 L11a367

Knotscape

This page is passe. Go here instead! The 2-Component Link L11a366 Visit L11a366's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-11/2 + 2q-9/2 - 4q-7/2 + 7q-5/2 - 10q-3/2 + 11q-1/2 - 13q1/2 + 11q3/2 - 9q5/2 + 6q7/2 - 3q9/2 + q11/2

A2 (sl(3)) Invariant:

q-16 + q-12 + q-10 - q-8 + 2q-6 - q-4 + 2q-2 + 3 + 3q4 - 2q6 + q8 - q12 + q14 - q16

HOMFLY-PT Polynomial:

2a-3z + 3a-3z3 + a-3z5 - 4a-1z3 - 4a-1z5 - a-1z7 - az-1 - 7az - 9az3 - 5az5 - az7 + a3z-1 + 4a3z + 4a3z3 + a3z5

Kauffman Polynomial:

a-6z2 - a-6z4 + 3a-5z3 - 3a-5z5 - 2a-4z2 + 6a-4z4 - 5a-4z6 + 3a-3z - 9a-3z3 + 10a-3z5 - 6a-3z7 + a-2z2 - 6a-2z4 + 8a-2z6 - 5a-2z8 - a-1z - a-1z3 + a-1z5 + 3a-1z7 - 3a-1z9 + 5z2 - 11z4 + 12z6 - 3z8 - z10 + az-1 - 10az + 23az3 - 24az5 + 17az7 - 5az9 - a2 + 6a2z2 - 10a2z4 + 8a2z6 - a2z10 + a3z-1 - 4a3z + 5a3z3 - 7a3z5 + 7a3z7 - 2a3z9 + 5a4z2 - 12a4z4 + 9a4z6 - 2a4z8 + 2a5z - 7a5z3 + 5a5z5 - 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 = 12                       1 j = 10                     2   j = 8                   4 1   j = 6                 5 2     j = 4               6 4       j = 2             7 5         j = 0           5 7           j = -2         5 6             j = -4       3 6               j = -6     1 4                 j = -8   1 3                   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, Alternating, 366]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 366]]
Out[4]=	PD[X[12, 1, 13, 2], X[14, 3, 15, 4], X[16, 5, 17, 6], X[6, 11, 7, 12], > X[18, 8, 19, 7], X[22, 18, 11, 17], X[20, 10, 21, 9], X[8, 20, 9, 19], > X[10, 22, 1, 21], X[4, 13, 5, 14], X[2, 15, 3, 16]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -11, 2, -10, 3, -4, 5, -8, 7, -9}, > {4, -1, 10, -2, 11, -3, 6, -5, 8, -7, 9, -6}]
In[6]:=	Jones[L][q]
Out[6]=	-(11/2) 2 4 7 10 11 3/2 -q + ---- - ---- + ---- - ---- + ------- - 13 Sqrt[q] + 11 q - 9/2 7/2 5/2 3/2 Sqrt[q] q q q q 5/2 7/2 9/2 11/2 > 9 q + 6 q - 3 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-16 -12 -10 -8 2 -4 2 4 6 8 12 14 3 + q + q + q - q + -- - q + -- + 3 q - 2 q + q - q + q - 6 2 q q 16 > q
In[8]:=	HOMFLYPT[Link[11, Alternating, 366]][a, z]
Out[8]=	3 3 3 5 5 a a 2 z 3 3 z 4 z 3 3 3 z 4 z -(-) + -- + --- - 7 a z + 4 a z + ---- - ---- - 9 a z + 4 a z + -- - ---- - z z 3 3 a 3 a a a a 7 5 3 5 z 7 > 5 a z + a z - -- - a z a
In[9]:=	Kauffman[Link[11, Alternating, 366]][a, z]
Out[9]=	3 2 2 2 2 a a 3 z z 3 5 2 z 2 z z -a + - + -- + --- - - - 10 a z - 4 a z + 2 a z + 5 z + -- - ---- + -- + z z 3 a 6 4 2 a a a a 3 3 3 2 2 4 2 3 z 9 z z 3 3 3 5 3 > 6 a z + 5 a z + ---- - ---- - -- + 23 a z + 5 a z - 7 a z - 5 3 a a a 4 4 4 5 5 5 4 z 6 z 6 z 2 4 4 4 3 z 10 z z > 11 z - -- + ---- - ---- - 10 a z - 12 a z - ---- + ----- + -- - 6 4 2 5 3 a a a a a a 6 6 5 3 5 5 5 6 5 z 8 z 2 6 4 6 > 24 a z - 7 a z + 5 a z + 12 z - ---- + ---- + 8 a z + 9 a z - 4 2 a a 7 7 8 9 6 z 3 z 7 3 7 5 7 8 5 z 4 8 3 z > ---- + ---- + 17 a z + 7 a z - a z - 3 z - ---- - 2 a z - ---- - 3 a 2 a a a 9 3 9 10 2 10 > 5 a z - 2 a z - z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 1 1 3 1 4 3 6 7 + 7 q + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ----- + 12 6 10 5 8 5 8 4 6 4 6 3 4 3 4 2 q t q t q t q t q t q t q t q t 5 5 6 2 4 4 2 6 2 6 3 > ----- + - + ---- + 5 q t + 6 q t + 4 q t + 5 q t + 2 q t + 2 2 t 2 q t q t 8 3 8 4 10 4 12 5 > 4 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a366

L11a365 L11a367