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

L11n344 L11n346

Knotscape

This page is passe. Go here instead! The 3-Component Link L11n345 Visit L11n345's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

X6172 X14,4,15,3 X11,19,12,18 X16,8,17,7 X17,21,18,20 X19,5,20,12 X8,22,9,21 X22,10,13,9 X10,14,11,13 X2536 X4,16,1,15

Gauss Code:

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

Jones Polynomial:

q - q2 + 3q3 - 2q4 + 3q5 - q6 + q7 - q8 + q10

A2 (sl(3)) Invariant:

q4 + q6 + 2q8 + 3q10 + 3q12 + 3q14 + 3q16 + 4q18 + q20 + 2q22 + q26 + q28 + q30 + 2q32 - q34

HOMFLY-PT Polynomial:

- a-10z2 + a-8z-2 + 4a-8 + 9a-8z2 + 6a-8z4 + a-8z6 - 2a-6z-2 - 11a-6 - 20a-6z2 - 17a-6z4 - 7a-6z6 - a-6z8 + a-4z-2 + 7a-4 + 11a-4z2 + 6a-4z4 + a-4z6

Kauffman Polynomial:

a-12 - 4a-12z2 + a-12z4 - a-11z3 - a-10z2 - a-10z4 + 11a-9z3 - 11a-9z5 + 2a-9z7 - a-8z-2 + 5a-8 - 16a-8z2 + 29a-8z4 - 18a-8z6 + 3a-8z8 + 2a-7z-1 - 11a-7z + 20a-7z3 - 7a-7z5 - 3a-7z7 + a-7z9 - 2a-6z-2 + 13a-6 - 37a-6z2 + 48a-6z4 - 25a-6z6 + 4a-6z8 + 2a-5z-1 - 11a-5z + 8a-5z3 + 4a-5z5 - 5a-5z7 + a-5z9 - a-4z-2 + 8a-4 - 18a-4z2 + 17a-4z4 - 7a-4z6 + a-4z8

Khovanov Homology:

trqj r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 r = 8 j = 21                     1 j = 19                     1 j = 17               2 1     j = 15             1 1       j = 13           2 3 1       j = 11         1 1 2         j = 9       1 2             j = 7     2 1               j = 5   1 3                 j = 3                       j = 1 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, 345]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 345]]
Out[4]=	PD[X[6, 1, 7, 2], X[14, 4, 15, 3], X[11, 19, 12, 18], X[16, 8, 17, 7], > X[17, 21, 18, 20], X[19, 5, 20, 12], X[8, 22, 9, 21], X[22, 10, 13, 9], > X[10, 14, 11, 13], X[2, 5, 3, 6], X[4, 16, 1, 15]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 4, -7, 8, -9, -3, 6}, > {9, -2, 11, -4, -5, 3, -6, 5, 7, -8}]
In[6]:=	Jones[L][q]
Out[6]=	2 3 4 5 6 7 8 10 q - q + 3 q - 2 q + 3 q - q + q - q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	4 6 8 10 12 14 16 18 20 22 26 q + q + 2 q + 3 q + 3 q + 3 q + 3 q + 4 q + q + 2 q + q + 28 30 32 34 > q + q + 2 q - q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 345]][a, z]
Out[8]=	2 2 2 2 4 4 11 7 1 2 1 z 9 z 20 z 11 z 6 z -- - -- + -- + ----- - ----- + ----- - --- + ---- - ----- + ----- + ---- - 8 6 4 8 2 6 2 4 2 10 8 6 4 8 a a a a z a z a z a a a a a 4 4 6 6 6 8 17 z 6 z z 7 z z z > ----- + ---- + -- - ---- + -- - -- 6 4 8 6 4 6 a a a a a a
In[9]:=	Kauffman[Link[11, NonAlternating, 345]][a, z]
Out[9]=	-12 5 13 8 1 2 1 2 2 11 z 11 z a + -- + -- + -- - ----- - ----- - ----- + ---- + ---- - ---- - ---- - 8 6 4 8 2 6 2 4 2 7 5 7 5 a a a a z a z a z a z a z a a 2 2 2 2 2 3 3 3 3 4 4 z z 16 z 37 z 18 z z 11 z 20 z 8 z z > ---- - --- - ----- - ----- - ----- - --- + ----- + ----- + ---- + --- - 12 10 8 6 4 11 9 7 5 12 a a a a a a a a a a 4 4 4 4 5 5 5 6 6 6 z 29 z 48 z 17 z 11 z 7 z 4 z 18 z 25 z 7 z > --- + ----- + ----- + ----- - ----- - ---- + ---- - ----- - ----- - ---- + 10 8 6 4 9 7 5 8 6 4 a a a a a a a a a a 7 7 7 8 8 8 9 9 2 z 3 z 5 z 3 z 4 z z z z > ---- - ---- - ---- + ---- + ---- + -- + -- + -- 9 7 5 8 6 4 7 5 a a a a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	5 5 7 q q 7 9 9 2 11 2 11 3 13 3 3 q + 2 q + -- + -- + q t + q t + 2 q t + q t + q t + 2 q t + 2 t t 11 4 13 4 15 4 13 5 15 5 17 5 17 6 > 2 q t + 3 q t + q t + q t + q t + 2 q t + q t + 19 8 21 8 > q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n345

L11n344 L11n346