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

L10a131 L10a133

Knotscape

This page is passe. Go here instead! The 3-Component Link L10a132 Visit L10a132's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

X6172 X12,3,13,4 X20,14,11,13 X10,15,5,16 X8,17,9,18 X16,7,17,8 X18,9,19,10 X14,20,15,19 X2536 X4,11,1,12

Gauss Code:

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

Jones Polynomial:

q-10 - 2q-9 + 5q-8 - 7q-7 + 10q-6 - 9q-5 + 10q-4 - 7q-3 + 5q-2 - 3q-1 + 1

A2 (sl(3)) Invariant:

q-32 + 2q-30 + q-28 + 3q-26 + 4q-24 + 2q-22 + 5q-20 + 2q-18 + 3q-16 + 3q-14 + 3q-10 - 2q-8 - q-2 + 1

HOMFLY-PT Polynomial:

2a2z2 + a2z4 - 2a4z2 - 3a4z4 - a4z6 + a6z-2 + 5a6 + 8a6z2 + 3a6z4 - 2a8z-2 - 6a8 - 3a8z2 + a10z-2 + a10

Kauffman Polynomial:

2a2z2 - 3a2z4 + a2z6 + 8a3z3 - 10a3z5 + 3a3z7 - 2a4z2 + 5a4z4 - 8a4z6 + 3a4z8 + 6a5z3 - 12a5z5 + 3a5z7 + a5z9 + a6z-2 - 5a6 + 6a6z2 - 9a6z6 + 5a6z8 - 2a7z-1 + 6a7z - 3a7z3 - 5a7z5 + 3a7z7 + a7z9 + 2a8z-2 - 8a8 + 15a8z2 - 12a8z4 + 3a8z6 + 2a8z8 - 2a9z-1 + 6a9z - 3a9z3 - a9z5 + 3a9z7 + a10z-2 - 3a10 + 3a10z2 - 3a10z4 + 3a10z6 - 2a11z3 + 2a11z5 + a12 - 2a12z2 + a12z4

Khovanov Homology:

trqj r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 j = 1                     1 j = -1                   2   j = -3                 3 1   j = -5               5 3     j = -7             5 2       j = -9           4 5         j = -11         6 5           j = -13       2 5             j = -15     3 5               j = -17   1 4                 j = -19   1                   j = -21 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, Alternating, 132]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 132]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 3, 13, 4], X[20, 14, 11, 13], X[10, 15, 5, 16], > X[8, 17, 9, 18], X[16, 7, 17, 8], X[18, 9, 19, 10], X[14, 20, 15, 19], > X[2, 5, 3, 6], X[4, 11, 1, 12]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 2, -10}, {9, -1, 6, -5, 7, -4}, > {10, -2, 3, -8, 4, -6, 5, -7, 8, -3}]
In[6]:=	Jones[L][q]
Out[6]=	-10 2 5 7 10 9 10 7 5 3 1 + q - -- + -- - -- + -- - -- + -- - -- + -- - - 9 8 7 6 5 4 3 2 q q q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-32 2 -28 3 4 2 5 2 3 3 3 2 -2 1 + q + --- + q + --- + --- + --- + --- + --- + --- + --- + --- - -- - q 30 26 24 22 20 18 16 14 10 8 q q q q q q q q q q
In[8]:=	HOMFLYPT[Link[10, Alternating, 132]][a, z]
Out[8]=	6 8 10 6 8 10 a 2 a a 2 2 4 2 6 2 8 2 5 a - 6 a + a + -- - ---- + --- + 2 a z - 2 a z + 8 a z - 3 a z + 2 2 2 z z z 2 4 4 4 6 4 4 6 > a z - 3 a z + 3 a z - a z
In[9]:=	Kauffman[Link[10, Alternating, 132]][a, z]
Out[9]=	6 8 10 7 9 6 8 10 12 a 2 a a 2 a 2 a 7 9 -5 a - 8 a - 3 a + a + -- + ---- + --- - ---- - ---- + 6 a z + 6 a z + 2 2 2 z z z z z 2 2 4 2 6 2 8 2 10 2 12 2 3 3 > 2 a z - 2 a z + 6 a z + 15 a z + 3 a z - 2 a z + 8 a z + 5 3 7 3 9 3 11 3 2 4 4 4 8 4 > 6 a z - 3 a z - 3 a z - 2 a z - 3 a z + 5 a z - 12 a z - 10 4 12 4 3 5 5 5 7 5 9 5 11 5 > 3 a z + a z - 10 a z - 12 a z - 5 a z - a z + 2 a z + 2 6 4 6 6 6 8 6 10 6 3 7 5 7 > a z - 8 a z - 9 a z + 3 a z + 3 a z + 3 a z + 3 a z + 7 7 9 7 4 8 6 8 8 8 5 9 7 9 > 3 a z + 3 a z + 3 a z + 5 a z + 2 a z + a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	3 3 1 1 1 4 3 5 2 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 5 3 21 8 19 7 17 7 17 6 15 6 15 5 13 5 q q q t q t q t q t q t q t q t 5 6 5 4 5 5 2 5 t 2 t > ------ + ------ + ------ + ----- + ----- + ----- + ---- + ---- + -- + --- + 13 4 11 4 11 3 9 3 9 2 7 2 7 5 3 q q t q t q t q t q t q t q t q t q 2 > q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L10a132

L10a131 L10a133