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

L10n76 L10n78

Knotscape

This page is passe. Go here instead! The 3-Component Link L10n77 Visit L10n77's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

X6172 X5,14,6,15 X3849 X15,2,16,3 X16,7,17,8 X9,18,10,19 X11,20,12,13 X13,12,14,5 X4,17,1,18 X19,10,20,11

Gauss Code:

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

Jones Polynomial:

q-12 + q-8 + q-6 + q-4

A2 (sl(3)) Invariant:

q-40 + q-38 + 2q-36 + 2q-34 + 2q-32 + 2q-30 + 2q-28 + 3q-26 + 3q-24 + 3q-22 + 2q-20 + 2q-18 + q-16 + q-14

HOMFLY-PT Polynomial:

a8z-2 + 8a8 + 21a8z2 + 21a8z4 + 8a8z6 + a8z8 - 2a10z-2 - 11a10 - 15a10z2 - 7a10z4 - a10z6 + a12z-2 + 3a12 + a12z2

Kauffman Polynomial:

- a8z-2 + 8a8 - 21a8z2 + 21a8z4 - 8a8z6 + a8z8 + 2a9z-1 - 11a9z + 15a9z3 - 7a9z5 + a9z7 - 2a10z-2 + 13a10 - 26a10z2 + 22a10z4 - 8a10z6 + a10z8 + 2a11z-1 - 11a11z + 15a11z3 - 7a11z5 + a11z7 - a12z-2 + 5a12 - 5a12z2 + a12z4 + a16

Khovanov Homology:

trqj r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 j = -7                 1 j = -9                 1 j = -11             1     j = -13         1         j = -15         2 1       j = -17     1             j = -19     1 1           j = -21 1 1               j = -23 2 1               j = -25 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[10, NonAlternating, 77]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, NonAlternating, 77]]
Out[4]=	PD[X[6, 1, 7, 2], X[5, 14, 6, 15], X[3, 8, 4, 9], X[15, 2, 16, 3], > X[16, 7, 17, 8], X[9, 18, 10, 19], X[11, 20, 12, 13], X[13, 12, 14, 5], > X[4, 17, 1, 18], X[19, 10, 20, 11]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, 4, -3, -9}, {-2, -1, 5, 3, -6, 10, -7, 8}, > {-8, 2, -4, -5, 9, 6, -10, 7}]
In[6]:=	Jones[L][q]
Out[6]=	-12 -8 -6 -4 q + q + q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-40 -38 2 2 2 2 2 3 3 3 2 2 q + q + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + 36 34 32 30 28 26 24 22 20 18 q q q q q q q q q q -16 -14 > q + q
In[8]:=	HOMFLYPT[Link[10, NonAlternating, 77]][a, z]
Out[8]=	8 10 12 8 10 12 a 2 a a 8 2 10 2 12 2 8 a - 11 a + 3 a + -- - ----- + --- + 21 a z - 15 a z + a z + 2 2 2 z z z 8 4 10 4 8 6 10 6 8 8 > 21 a z - 7 a z + 8 a z - a z + a z
In[9]:=	Kauffman[Link[10, NonAlternating, 77]][a, z]
Out[9]=	8 10 12 9 11 8 10 12 16 a 2 a a 2 a 2 a 9 8 a + 13 a + 5 a + a - -- - ----- - --- + ---- + ----- - 11 a z - 2 2 2 z z z z z 11 8 2 10 2 12 2 9 3 11 3 > 11 a z - 21 a z - 26 a z - 5 a z + 15 a z + 15 a z + 8 4 10 4 12 4 9 5 11 5 8 6 10 6 > 21 a z + 22 a z + a z - 7 a z - 7 a z - 8 a z - 8 a z + 9 7 11 7 8 8 10 8 > a z + a z + a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	-9 -7 1 2 1 1 1 1 1 q + q + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 25 8 23 8 21 8 23 7 21 7 19 6 17 6 q t q t q t q t q t q t q t 1 2 1 1 1 > ------ + ------ + ------ + ------ + ------ 19 5 15 4 13 4 15 3 11 2 q t q t q t q t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L10n77

L10n76 L10n78