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

L10a77 L10a79

Knotscape

This page is passe. Go here instead! The 2-Component Link L10a78 Visit L10a78's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

X8192 X2,9,3,10 X10,3,11,4 X14,8,15,7 X6,13,1,14 X18,12,19,11 X16,5,17,6 X12,18,13,17 X20,16,7,15 X4,19,5,20

Gauss Code:

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

Jones Polynomial:

- q-15/2 + 3q-13/2 - 6q-11/2 + 8q-9/2 - 11q-7/2 + 10q-5/2 - 10q-3/2 + 8q-1/2 - 5q1/2 + 3q3/2 - q5/2

A2 (sl(3)) Invariant:

q-22 - q-20 + 2q-18 + q-16 + q-14 + 4q-12 + 4q-8 - 3 - q4 + q8

HOMFLY-PT Polynomial:

- 2a-1z - a-1z3 + az-1 + 6az + 7az3 + 2az5 - 3a3z-1 - 10a3z - 10a3z3 - 5a3z5 - a3z7 + 2a5z-1 + 3a5z + 3a5z3 + a5z5

Kauffman Polynomial:

2a-1z - 5a-1z3 + 4a-1z5 - a-1z7 + 1 + 6z2 - 17z4 + 13z6 - 3z8 - az-1 + 5az - 17az3 + 10az5 + 3az7 - 2az9 + 3a2 + a2z2 - 26a2z4 + 31a2z6 - 9a2z8 - 3a3z-1 + 12a3z - 31a3z3 + 31a3z5 - 5a3z7 - 2a3z9 + 3a4 - 7a4z2 + 4a4z4 + 10a4z6 - 6a4z8 - 2a5z-1 + 7a5z - 13a5z3 + 19a5z5 - 9a5z7 - 2a6z2 + 10a6z4 - 8a6z6 - 2a7z + 5a7z3 - 6a7z5 - 3a8z4 - a9z3

Khovanov Homology:

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 6                     1 j = 4                   2   j = 2                 3 1   j = 0               5 2     j = -2             5 3       j = -4           6 6         j = -6         5 4           j = -8       3 6             j = -10     3 5               j = -12     3                 j = -14 1 3                   j = -16 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]=	2
In[3]:=	Show[DrawMorseLink[Link[10, Alternating, 78]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 78]]
Out[4]=	PD[X[8, 1, 9, 2], X[2, 9, 3, 10], X[10, 3, 11, 4], X[14, 8, 15, 7], > X[6, 13, 1, 14], X[18, 12, 19, 11], X[16, 5, 17, 6], X[12, 18, 13, 17], > X[20, 16, 7, 15], X[4, 19, 5, 20]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 3, -10, 7, -5}, > {4, -1, 2, -3, 6, -8, 5, -4, 9, -7, 8, -6, 10, -9}]
In[6]:=	Jones[L][q]
Out[6]=	-(15/2) 3 6 8 11 10 10 8 -q + ----- - ----- + ---- - ---- + ---- - ---- + ------- - 5 Sqrt[q] + 13/2 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q q 3/2 5/2 > 3 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-22 -20 2 -16 -14 4 4 4 8 -3 + q - q + --- + q + q + --- + -- - q + q 18 12 8 q q q
In[8]:=	HOMFLYPT[Link[10, Alternating, 78]][a, z]
Out[8]=	3 5 3 a 3 a 2 a 2 z 3 5 z 3 3 3 - - ---- + ---- - --- + 6 a z - 10 a z + 3 a z - -- + 7 a z - 10 a z + z z z a a 5 3 5 3 5 5 5 3 7 > 3 a z + 2 a z - 5 a z + a z - a z
In[9]:=	Kauffman[Link[10, Alternating, 78]][a, z]
Out[9]=	3 5 2 4 a 3 a 2 a 2 z 3 5 7 1 + 3 a + 3 a - - - ---- - ---- + --- + 5 a z + 12 a z + 7 a z - 2 a z + z z z a 3 2 2 2 4 2 6 2 5 z 3 3 3 5 3 > 6 z + a z - 7 a z - 2 a z - ---- - 17 a z - 31 a z - 13 a z + a 5 7 3 9 3 4 2 4 4 4 6 4 8 4 4 z > 5 a z - a z - 17 z - 26 a z + 4 a z + 10 a z - 3 a z + ---- + a 5 3 5 5 5 7 5 6 2 6 4 6 > 10 a z + 31 a z + 19 a z - 6 a z + 13 z + 31 a z + 10 a z - 7 6 6 z 7 3 7 5 7 8 2 8 4 8 > 8 a z - -- + 3 a z - 5 a z - 9 a z - 3 z - 9 a z - 6 a z - a 9 3 9 > 2 a z - 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	6 5 1 1 3 3 3 5 3 6 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 4 2 16 6 14 6 14 5 12 4 10 4 10 3 8 3 8 2 q q q t q t q t q t q t q t q t q t 5 4 6 3 t 2 2 2 2 3 4 3 6 4 > ----- + ---- + ---- + 5 t + --- + 2 t + 3 q t + q t + 2 q t + q t 6 2 6 4 2 q t q t q t q

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L10a78

L10a77 L10a79