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

L11n289 L11n291

Knotscape

This page is passe. Go here instead! The 3-Component Link L11n290 Visit L11n290's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

X6172 X5,12,6,13 X8493 X2,14,3,13 X14,7,15,8 X11,18,12,19 X9,21,10,20 X19,5,20,10 X4,15,1,16 X17,22,18,11 X21,16,22,17

Gauss Code:

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

Jones Polynomial:

- q-8 + 3q-7 - 6q-6 + 7q-5 - 8q-4 + 9q-3 - 6q-2 + 6q-1 - 1 + q

A2 (sl(3)) Invariant:

- q-24 + q-22 - 2q-20 - 2q-18 - q-16 - 4q-14 + q-12 + q-10 + 6q-8 + 8q-6 + 6q-4 + 8q-2 + 3 + 2q2 + q4

HOMFLY-PT Polynomial:

2z-2 + 3 + z2 - 5a2z-2 - 8a2 - 6a2z2 - 2a2z4 + 4a4z-2 + 7a4 + 7a4z2 + 4a4z4 + a4z6 - a6z-2 - 2a6 - 2a6z2 - a6z4

Kauffman Polynomial:

- 2z-2 + 5 - 4z2 + z4 + 5az-1 - 7az + az3 + az5 - 5a2z-2 + 11a2 - 18a2z2 + 16a2z4 - 5a2z6 + a2z8 + 9a3z-1 - 17a3z + 10a3z3 + 4a3z5 - 3a3z7 + a3z9 - 4a4z-2 + 10a4 - 20a4z2 + 26a4z4 - 14a4z6 + 4a4z8 + 5a5z-1 - 13a5z + 13a5z3 - 6a5z5 + a5z7 + a5z9 - a6z-2 + 3a6 - 5a6z2 + 5a6z4 - 6a6z6 + 3a6z8 + a7z-1 - 2a7z + 2a7z3 - 8a7z5 + 4a7z7 + a8z2 - 6a8z4 + 3a8z6 + a9z - 2a9z3 + a9z5

Khovanov Homology:

trqj r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 j = 3                   1 j = 1                     j = -1               6 1   j = -3             4 4     j = -5           5 2       j = -7         3 4         j = -9       4 5           j = -11     2 3             j = -13   1 4               j = -15   2                 j = -17 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, 290]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 290]]
Out[4]=	PD[X[6, 1, 7, 2], X[5, 12, 6, 13], X[8, 4, 9, 3], X[2, 14, 3, 13], > X[14, 7, 15, 8], X[11, 18, 12, 19], X[9, 21, 10, 20], X[19, 5, 20, 10], > X[4, 15, 1, 16], X[17, 22, 18, 11], X[21, 16, 22, 17]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -4, 3, -9}, {-2, -1, 5, -3, -7, 8}, > {-6, 2, 4, -5, 9, 11, -10, 6, -8, 7, -11, 10}]
In[6]:=	Jones[L][q]
Out[6]=	-8 3 6 7 8 9 6 6 -1 - q + -- - -- + -- - -- + -- - -- + - + q 7 6 5 4 3 2 q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-24 -22 2 2 -16 4 -12 -10 6 8 6 8 3 - q + q - --- - --- - q - --- + q + q + -- + -- + -- + -- + 20 18 14 8 6 4 2 q q q q q q q 2 4 > 2 q + q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 290]][a, z]
Out[8]=	2 4 6 2 4 6 2 5 a 4 a a 2 2 2 4 2 3 - 8 a + 7 a - 2 a + -- - ---- + ---- - -- + z - 6 a z + 7 a z - 2 2 2 2 z z z z 6 2 2 4 4 4 6 4 4 6 > 2 a z - 2 a z + 4 a z - a z + a z
In[9]:=	Kauffman[Link[11, NonAlternating, 290]][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 5 + 11 a + 10 a + 3 a - -- - ---- - ---- - -- + --- + ---- + ---- + -- - 2 2 2 2 z z z z z z z z 3 5 7 9 2 2 2 4 2 > 7 a z - 17 a z - 13 a z - 2 a z + a z - 4 z - 18 a z - 20 a z - 6 2 8 2 3 3 3 5 3 7 3 9 3 4 > 5 a z + a z + a z + 10 a z + 13 a z + 2 a z - 2 a z + z + 2 4 4 4 6 4 8 4 5 3 5 5 5 > 16 a z + 26 a z + 5 a z - 6 a z + a z + 4 a z - 6 a z - 7 5 9 5 2 6 4 6 6 6 8 6 3 7 > 8 a z + a z - 5 a z - 14 a z - 6 a z + 3 a z - 3 a z + 5 7 7 7 2 8 4 8 6 8 3 9 5 9 > a z + 4 a z + a z + 4 a z + 3 a z + a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	4 6 1 2 1 4 2 3 4 5 -- + - + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 3 q 17 7 15 6 13 6 13 5 11 5 11 4 9 4 9 3 q q t q t q t q t q t q t q t q t 3 4 5 2 4 t 3 2 > ----- + ----- + ----- + ---- + ---- + - + q t 7 3 7 2 5 2 5 3 q q t q t q t q t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n290

L11n289 L11n291