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

L11n345 L11n347

Knotscape

This page is passe. Go here instead! The 3-Component Link L11n346 Visit L11n346's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

X6172 X3,15,4,14 X11,20,12,21 X7,18,8,19 X17,22,18,13 X9,17,10,16 X15,11,16,10 X19,12,20,5 X21,8,22,9 X2536 X13,1,14,4

Gauss Code:

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

Jones Polynomial:

- 2q-6 + 6q-5 - 9q-4 + 13q-3 - 12q-2 + 13q-1 - 10 + 7q - 3q2 + q3

A2 (sl(3)) Invariant:

- q-20 - 2q-18 + 3q-16 + 3q-12 + 6q-10 + 2q-8 + 7q-6 + q-4 + 4q-2 + 2 - q2 + 3q4 - q6 + q10

HOMFLY-PT Polynomial:

a-2 + a-2z2 + z-2 - 3z2 - 2z4 - 2a2z-2 - 3a2 + 2a2z4 + a2z6 + a4z-2 + 3a4 + a4z2 - a4z4 - a6

Kauffman Polynomial:

- a-2 + 3a-2z2 - 3a-2z4 + a-2z6 - a-1z + 6a-1z3 - 8a-1z5 + 3a-1z7 - z-2 + 3 + z4 - 8z6 + 4z8 + 2az-1 - 8az + 17az3 - 21az5 + 4az7 + 2az9 - 2a2z-2 + 11a2 - 23a2z2 + 25a2z4 - 25a2z6 + 10a2z8 + 2a3z-1 - 12a3z + 23a3z3 - 22a3z5 + 6a3z7 + 2a3z9 - a4z-2 + 11a4 - 26a4z2 + 27a4z4 - 15a4z6 + 6a4z8 - 7a5z + 15a5z3 - 9a5z5 + 5a5z7 + 3a6 - 6a6z2 + 6a6z4 + a6z6 - 2a7z + 3a7z3

Khovanov Homology:

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 7                   1 j = 5                 2   j = 3               5 1   j = 1             5 2     j = -1           8 5       j = -3         7 8         j = -5       6 5           j = -7     3 7             j = -9   3 6               j = -11   4                 j = -13 2

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, 346]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 346]]
Out[4]=	PD[X[6, 1, 7, 2], X[3, 15, 4, 14], X[11, 20, 12, 21], X[7, 18, 8, 19], > X[17, 22, 18, 13], X[9, 17, 10, 16], X[15, 11, 16, 10], X[19, 12, 20, 5], > X[21, 8, 22, 9], X[2, 5, 3, 6], X[13, 1, 14, 4]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, -2, 11}, {10, -1, -4, 9, -6, 7, -3, 8}, > {-11, 2, -7, 6, -5, 4, -8, 3, -9, 5}]
In[6]:=	Jones[L][q]
Out[6]=	2 6 9 13 12 13 2 3 -10 - -- + -- - -- + -- - -- + -- + 7 q - 3 q + q 6 5 4 3 2 q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-20 2 3 3 6 2 7 -4 4 2 4 6 10 2 - q - --- + --- + --- + --- + -- + -- + q + -- - q + 3 q - q + q 18 16 12 10 8 6 2 q q q q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 346]][a, z]
Out[8]=	2 4 2 -2 2 4 6 -2 2 a a 2 z 4 2 4 2 4 a - 3 a + 3 a - a + z - ---- + -- - 3 z + -- + a z - 2 z + 2 a z - 2 2 2 z z a 4 4 2 6 > a z + a z
In[9]:=	Kauffman[Link[11, NonAlternating, 346]][a, z]
Out[9]=	2 4 3 -2 2 4 6 -2 2 a a 2 a 2 a z 3 - a + 11 a + 11 a + 3 a - z - ---- - -- + --- + ---- - - - 8 a z - 2 2 z z a z z 2 3 3 5 7 3 z 2 2 4 2 6 2 6 z > 12 a z - 7 a z - 2 a z + ---- - 23 a z - 26 a z - 6 a z + ---- + 2 a a 4 3 3 3 5 3 7 3 4 3 z 2 4 4 4 > 17 a z + 23 a z + 15 a z + 3 a z + z - ---- + 25 a z + 27 a z + 2 a 5 6 6 4 8 z 5 3 5 5 5 6 z 2 6 > 6 a z - ---- - 21 a z - 22 a z - 9 a z - 8 z + -- - 25 a z - a 2 a 7 4 6 6 6 3 z 7 3 7 5 7 8 2 8 > 15 a z + a z + ---- + 4 a z + 6 a z + 5 a z + 4 z + 10 a z + a 4 8 9 3 9 > 6 a z + 2 a z + 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	8 8 2 4 3 6 3 7 6 5 -- + - + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ---- + 3 q 13 5 11 4 9 4 9 3 7 3 7 2 5 2 5 q q t q t q t q t q t q t q t q t 7 5 t 2 3 2 3 3 5 3 7 4 > ---- + --- + 5 q t + 2 q t + 5 q t + q t + 2 q t + q t 3 q q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n346

L11n345 L11n347