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

L11n210 L11n212

Knotscape

This page is passe. Go here instead! The 2-Component Link L11n211 Visit L11n211's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

X10,1,11,2 X15,21,16,20 X5,14,6,15 X3,12,4,13 X13,4,14,5 X2,19,3,20 X16,7,17,8 X8,9,1,10 X18,12,19,11 X22,18,9,17 X21,6,22,7

Gauss Code:

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

Jones Polynomial:

- q-19/2 + 2q-17/2 - 5q-15/2 + 7q-13/2 - 8q-11/2 + 8q-9/2 - 8q-7/2 + 5q-5/2 - 3q-3/2 + q-1/2

A2 (sl(3)) Invariant:

q-28 + 3q-24 + q-22 + 2q-18 - q-16 + 3q-14 + q-10 + q-8 - 2q-6 + q-4 - q-2

HOMFLY-PT Polynomial:

a3z-1 + 3a3z + 3a3z3 + a3z5 - 3a5z-1 - 13a5z - 14a5z3 - 6a5z5 - a5z7 + 2a7z-1 + 6a7z + 4a7z3 + a7z5

Kauffman Polynomial:

- a2 + 2a2z2 - a2z4 + a3z-1 - 3a3z + 5a3z3 - 3a3z5 - 3a4 + 12a4z2 - 11a4z4 + 3a4z6 - a4z8 + 3a5z-1 - 16a5z + 25a5z3 - 17a5z5 + 4a5z7 - a5z9 - 3a6 + 12a6z2 - 14a6z4 + 7a6z6 - 3a6z8 + 2a7z-1 - 8a7z + 11a7z3 - 6a7z5 + a7z7 - a7z9 + a8z2 + 2a8z6 - 2a8z8 + 3a9z - 6a9z3 + 7a9z5 - 3a9z7 - a10z2 + 4a10z4 - 2a10z6 - 2a11z + 3a11z3 - a11z5

Khovanov Homology:

trqj r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 j = 0                   1 j = -2                 2   j = -4               4 2   j = -6             4 1     j = -8           4 4       j = -10         4 4         j = -12       3 4           j = -14     2 4             j = -16     3               j = -18 1 2                 j = -20 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, 211]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 211]]
Out[4]=	PD[X[10, 1, 11, 2], X[15, 21, 16, 20], X[5, 14, 6, 15], X[3, 12, 4, 13], > X[13, 4, 14, 5], X[2, 19, 3, 20], X[16, 7, 17, 8], X[8, 9, 1, 10], > X[18, 12, 19, 11], X[22, 18, 9, 17], X[21, 6, 22, 7]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -6, -4, 5, -3, 11, 7, -8}, > {8, -1, 9, 4, -5, 3, -2, -7, 10, -9, 6, 2, -11, -10}]
In[6]:=	Jones[L][q]
Out[6]=	-(19/2) 2 5 7 8 8 8 5 3 1 -q + ----- - ----- + ----- - ----- + ---- - ---- + ---- - ---- + ------- 17/2 15/2 13/2 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-28 3 -22 2 -16 3 -10 -8 2 -4 -2 q + --- + q + --- - q + --- + q + q - -- + q - q 24 18 14 6 q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 211]][a, z]
Out[8]=	3 5 7 a 3 a 2 a 3 5 7 3 3 5 3 7 3 -- - ---- + ---- + 3 a z - 13 a z + 6 a z + 3 a z - 14 a z + 4 a z + z z z 3 5 5 5 7 5 5 7 > a z - 6 a z + a z - a z
In[9]:=	Kauffman[Link[11, NonAlternating, 211]][a, z]
Out[9]=	3 5 7 2 4 6 a 3 a 2 a 3 5 7 9 -a - 3 a - 3 a + -- + ---- + ---- - 3 a z - 16 a z - 8 a z + 3 a z - z z z 11 2 2 4 2 6 2 8 2 10 2 3 3 > 2 a z + 2 a z + 12 a z + 12 a z + a z - a z + 5 a z + 5 3 7 3 9 3 11 3 2 4 4 4 6 4 > 25 a z + 11 a z - 6 a z + 3 a z - a z - 11 a z - 14 a z + 10 4 3 5 5 5 7 5 9 5 11 5 4 6 > 4 a z - 3 a z - 17 a z - 6 a z + 7 a z - a z + 3 a z + 6 6 8 6 10 6 5 7 7 7 9 7 4 8 > 7 a z + 2 a z - 2 a z + 4 a z + a z - 3 a z - a z - 6 8 8 8 5 9 7 9 > 3 a z - 2 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	2 2 1 1 2 3 2 4 3 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 4 2 20 8 18 8 18 7 16 6 14 6 14 5 12 5 q q q t q t q t q t q t q t q t 4 4 4 4 4 4 1 4 > ------ + ------ + ------ + ----- + ----- + ----- + ---- + ---- + t 12 4 10 4 10 3 8 3 8 2 6 2 6 4 q t q t q t q t q t q t q t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n211

L11n210 L11n212