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

L11n326 L11n328

Knotscape

This page is passe. Go here instead! The 3-Component Link L11n327 Visit L11n327's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

X6172 X5,14,6,15 X3849 X2,16,3,15 X16,7,17,8 X13,18,14,19 X9,21,10,20 X19,5,20,12 X11,13,12,22 X21,11,22,10 X17,1,18,4

Gauss Code:

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

Jones Polynomial:

- q-4 + 2q-3 - 3q-2 + 6q-1 - 6 + 8q - 6q2 + 6q3 - 4q4 + 2q5

A2 (sl(3)) Invariant:

- q-12 - q-8 + 3q-4 + 2q-2 + 5 + 4q2 + 3q4 + 4q6 + q8 + 3q10 + q12 + q14 + 2q16

HOMFLY-PT Polynomial:

a-4z-2 + 3a-4 + 2a-4z2 - 2a-2z-2 - 9a-2 - 10a-2z2 - 3a-2z4 + z-2 + 8 + 10z2 + 5z4 + z6 - 2a2 - 3a2z2 - a2z4

Kauffman Polynomial:

- a-6 + 3a-6z2 - a-5z + 3a-5z3 + a-5z5 - a-4z-2 + 3a-4 - 3a-4z4 + 3a-4z6 + 2a-3z-1 - 8a-3z + 12a-3z3 - 10a-3z5 + 4a-3z7 - 2a-2z-2 + 11a-2 - 19a-2z2 + 12a-2z4 - 8a-2z6 + 3a-2z8 + 2a-1z-1 - 12a-1z + 18a-1z3 - 15a-1z5 + 2a-1z7 + a-1z9 - z-2 + 11 - 26z2 + 31z4 - 21z6 + 5z8 - 7az + 16az3 - 9az5 - az7 + az9 + 3a2 - 10a2z2 + 16a2z4 - 10a2z6 + 2a2z8 - 2a3z + 7a3z3 - 5a3z5 + a3z7

Khovanov Homology:

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 11                   2 j = 9                 3 1 j = 7               3 1   j = 5             3 3     j = 3           5 3       j = 1         4 6         j = -1       2 2           j = -3     1 4             j = -5   1 2               j = -7   1                 j = -9 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]=	3
In[3]:=	Show[DrawMorseLink[Link[11, NonAlternating, 327]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 327]]
Out[4]=	PD[X[6, 1, 7, 2], X[5, 14, 6, 15], X[3, 8, 4, 9], X[2, 16, 3, 15], > X[16, 7, 17, 8], X[13, 18, 14, 19], X[9, 21, 10, 20], X[19, 5, 20, 12], > X[11, 13, 12, 22], X[21, 11, 22, 10], X[17, 1, 18, 4]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -4, -3, 11}, {-2, -1, 5, 3, -7, 10, -9, 8}, > {-6, 2, 4, -5, -11, 6, -8, 7, -10, 9}]
In[6]:=	Jones[L][q]
Out[6]=	-4 2 3 6 2 3 4 5 -6 - q + -- - -- + - + 8 q - 6 q + 6 q - 4 q + 2 q 3 2 q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-12 -8 3 2 2 4 6 8 10 12 14 16 5 - q - q + -- + -- + 4 q + 3 q + 4 q + q + 3 q + q + q + 2 q 4 2 q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 327]][a, z]
Out[8]=	2 2 3 9 2 -2 1 2 2 2 z 10 z 2 2 8 + -- - -- - 2 a + z + ----- - ----- + 10 z + ---- - ----- - 3 a z + 4 2 4 2 2 2 4 2 a a a z a z a a 4 4 3 z 2 4 6 > 5 z - ---- - a z + z 2 a
In[9]:=	Kauffman[Link[11, NonAlternating, 327]][a, z]
Out[9]=	-6 3 11 2 -2 1 2 2 2 z 8 z 11 - a + -- + -- + 3 a - z - ----- - ----- + ---- + --- - -- - --- - 4 2 4 2 2 2 3 a z 5 3 a a a z a z a z a a 2 2 3 3 12 z 3 2 3 z 19 z 2 2 3 z 12 z > ---- - 7 a z - 2 a z - 26 z + ---- - ----- - 10 a z + ---- + ----- + a 6 2 5 3 a a a a 3 4 4 5 5 18 z 3 3 3 4 3 z 12 z 2 4 z 10 z > ----- + 16 a z + 7 a z + 31 z - ---- + ----- + 16 a z + -- - ----- - a 4 2 5 3 a a a a 5 6 6 7 7 15 z 5 3 5 6 3 z 8 z 2 6 4 z 2 z > ----- - 9 a z - 5 a z - 21 z + ---- - ---- - 10 a z + ---- + ---- - a 4 2 3 a a a a 8 9 7 3 7 8 3 z 2 8 z 9 > a z + a z + 5 z + ---- + 2 a z + -- + a z 2 a a
In[10]:=	Kh[L][q, t]
Out[10]=	3 1 1 1 2 1 4 2 2 4 q 6 q + 5 q + ----- + ----- + ----- + ----- + ----- + ----- + ---- + --- + --- + 9 5 7 4 5 4 5 3 3 3 3 2 2 q t t q t q t q t q t q t q t q t 3 5 5 2 7 2 7 3 9 3 9 4 11 4 > 3 q t + 3 q t + 3 q t + 3 q t + q t + 3 q t + q t + 2 q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n327

L11n326 L11n328