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

L11n389 L11n391

Knotscape

This page is passe. Go here instead! The 3-Component Link L11n390 Visit L11n390's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

X6172 X16,7,17,8 X4,17,1,18 X5,12,6,13 X3849 X22,14,19,13 X20,10,21,9 X10,20,11,19 X14,22,15,21 X11,18,12,5 X15,2,16,3

Gauss Code:

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

Jones Polynomial:

q-7 - 3q-6 + 3q-5 - 3q-4 + 4q-3 - 2q-2 + 2q-1 + 1 + q2

A2 (sl(3)) Invariant:

q-22 - 2q-20 - 2q-18 - q-16 - 2q-14 + q-12 + q-10 + 4q-8 + 5q-6 + 5q-4 + 7q-2 + 4 + 3q2 + 2q4 + q6

HOMFLY-PT Polynomial:

2z-2 + 6 + 5z2 + z4 - 5a2z-2 - 13a2 - 14a2z2 - 7a2z4 - a2z6 + 4a4z-2 + 8a4 + 4a4z2 - a6z-2 - a6 + a6z2

Kauffman Polynomial:

- 2z-2 + 11 - 23z2 + 21z4 - 8z6 + z8 + 5az-1 - 18az + 22az3 - 9az5 + az7 - 5a2z-2 + 21a2 - 37a2z2 + 31a2z4 - 10a2z6 + a2z8 + 9a3z-1 - 27a3z + 28a3z3 - 9a3z5 + a3z7 - 4a4z-2 + 12a4 - 13a4z2 + 6a4z4 + 5a5z-1 - 9a5z + 3a5z5 - a6z-2 + a6 - 3a6z4 + 2a6z6 + a7z-1 - 6a7z3 + 3a7z5 - a8z2 + a8z4

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 j = 5                     1 j = 3                     1 j = 1                 1     j = -1             3         j = -3           2 3 1       j = -5         3 1           j = -7         2 1           j = -9     3 3               j = -11   1 1                 j = -13   2                   j = -15 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, 390]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 390]]
Out[4]=	PD[X[6, 1, 7, 2], X[16, 7, 17, 8], X[4, 17, 1, 18], X[5, 12, 6, 13], > X[3, 8, 4, 9], X[22, 14, 19, 13], X[20, 10, 21, 9], X[10, 20, 11, 19], > X[14, 22, 15, 21], X[11, 18, 12, 5], X[15, 2, 16, 3]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, 11, -5, -3}, {8, -7, 9, -6}, > {-4, -1, 2, 5, 7, -8, -10, 4, 6, -9, -11, -2, 3, 10}]
In[6]:=	Jones[L][q]
Out[6]=	-7 3 3 3 4 2 2 2 1 + q - -- + -- - -- + -- - -- + - + q 6 5 4 3 2 q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-22 2 2 -16 2 -12 -10 4 5 5 7 2 4 + q - --- - --- - q - --- + q + q + -- + -- + -- + -- + 3 q + 20 18 14 8 6 4 2 q q q q q q q 4 6 > 2 q + q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 390]][a, z]
Out[8]=	2 4 6 2 4 6 2 5 a 4 a a 2 2 2 4 2 6 - 13 a + 8 a - a + -- - ---- + ---- - -- + 5 z - 14 a z + 4 a z + 2 2 2 2 z z z z 6 2 4 2 4 2 6 > a z + z - 7 a z - a z
In[9]:=	Kauffman[Link[11, NonAlternating, 390]][a, z]
Out[9]=	2 4 6 3 5 7 2 4 6 2 5 a 4 a a 5 a 9 a 5 a a 11 + 21 a + 12 a + a - -- - ---- - ---- - -- + --- + ---- + ---- + -- - 2 2 2 2 z z z z z z z z 3 5 2 2 2 4 2 8 2 3 > 18 a z - 27 a z - 9 a z - 23 z - 37 a z - 13 a z - a z + 22 a z + 3 3 7 3 4 2 4 4 4 6 4 8 4 > 28 a z - 6 a z + 21 z + 31 a z + 6 a z - 3 a z + a z - 5 3 5 5 5 7 5 6 2 6 6 6 7 > 9 a z - 9 a z + 3 a z + 3 a z - 8 z - 10 a z + 2 a z + a z + 3 7 8 2 8 > a z + z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	3 3 1 2 1 1 3 3 2 3 -- + - + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + 3 q 15 6 13 5 11 5 11 4 9 4 9 3 7 2 5 2 q q t q t q t q t q t q t q t q t 1 1 2 t 2 3 4 5 4 > ---- + ---- + ---- + -- + q t + q t + q t 7 5 3 3 q t q t q t q

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n390

L11n389 L11n391