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

L10a78 L10a80

Knotscape

This page is passe. Go here instead! The 2-Component Link L10a79 Visit L10a79's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

X8192 X12,3,13,4 X20,13,7,14 X14,9,15,10 X10,19,11,20 X16,6,17,5 X18,16,19,15 X2738 X4,11,5,12 X6,18,1,17

Gauss Code:

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

Jones Polynomial:

q-15/2 - 3q-13/2 + 7q-11/2 - 11q-9/2 + 13q-7/2 - 16q-5/2 + 14q-3/2 - 12q-1/2 + 8q1/2 - 4q3/2 + q5/2

A2 (sl(3)) Invariant:

- q-24 - q-22 + q-20 - 3q-18 + q-16 + 3q-14 - q-12 + 5q-10 + q-8 + 3q-6 + 2q-4 - 2q-2 + 3 - 3q2 + 2q6 - q8

HOMFLY-PT Polynomial:

a-1z3 + az - az5 - 2a3z-1 - 6a3z - 5a3z3 - 2a3z5 + 3a5z-1 + 5a5z + 3a5z3 - a7z-1 - a7z

Kauffman Polynomial:

- a-2z4 + 2a-1z3 - 4a-1z5 - 2z2 + 8z4 - 8z6 + 2az - 8az3 + 14az5 - 10az7 - 4a2z2 + 7a2z4 + 4a2z6 - 7a2z8 - 2a3z-1 + 9a3z - 27a3z3 + 36a3z5 - 13a3z7 - 2a3z9 + 3a4 - 6a4z2 - 3a4z4 + 20a4z6 - 11a4z8 - 3a5z-1 + 10a5z - 24a5z3 + 26a5z5 - 6a5z7 - 2a5z9 + 3a6 - 7a6z2 + 2a6z4 + 7a6z6 - 4a6z8 - a7z-1 + 3a7z - 7a7z3 + 8a7z5 - 3a7z7 + a8 - 3a8z2 + 3a8z4 - a8z6

Khovanov Homology:

trqj r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 j = 6                     1 j = 4                   3   j = 2                 5 1   j = 0               7 3     j = -2             8 6       j = -4           8 6         j = -6         6 9           j = -8       5 7             j = -10     2 6               j = -12   1 5                 j = -14   2                   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, 79]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 79]]
Out[4]=	PD[X[8, 1, 9, 2], X[12, 3, 13, 4], X[20, 13, 7, 14], X[14, 9, 15, 10], > X[10, 19, 11, 20], X[16, 6, 17, 5], X[18, 16, 19, 15], X[2, 7, 3, 8], > X[4, 11, 5, 12], X[6, 18, 1, 17]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -8, 2, -9, 6, -10}, > {8, -1, 4, -5, 9, -2, 3, -4, 7, -6, 10, -7, 5, -3}]
In[6]:=	Jones[L][q]
Out[6]=	-(15/2) 3 7 11 13 16 14 12 q - ----- + ----- - ---- + ---- - ---- + ---- - ------- + 8 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 > 4 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-24 -22 -20 3 -16 3 -12 5 -8 3 2 2 3 - q - q + q - --- + q + --- - q + --- + q + -- + -- - -- - 18 14 10 6 4 2 q q q q q q 2 6 8 > 3 q + 2 q - q
In[8]:=	HOMFLYPT[Link[10, Alternating, 79]][a, z]
Out[8]=	3 5 7 3 -2 a 3 a a 3 5 7 z 3 3 5 3 ----- + ---- - -- + a z - 6 a z + 5 a z - a z + -- - 5 a z + 3 a z - z z z a 5 3 5 > a z - 2 a z
In[9]:=	Kauffman[Link[10, Alternating, 79]][a, z]
Out[9]=	3 5 7 4 6 8 2 a 3 a a 3 5 7 3 a + 3 a + a - ---- - ---- - -- + 2 a z + 9 a z + 10 a z + 3 a z - z z z 3 2 2 2 4 2 6 2 8 2 2 z 3 3 3 > 2 z - 4 a z - 6 a z - 7 a z - 3 a z + ---- - 8 a z - 27 a z - a 4 5 3 7 3 4 z 2 4 4 4 6 4 8 4 > 24 a z - 7 a z + 8 z - -- + 7 a z - 3 a z + 2 a z + 3 a z - 2 a 5 4 z 5 3 5 5 5 7 5 6 2 6 > ---- + 14 a z + 36 a z + 26 a z + 8 a z - 8 z + 4 a z + a 4 6 6 6 8 6 7 3 7 5 7 7 7 > 20 a z + 7 a z - a z - 10 a z - 13 a z - 6 a z - 3 a z - 2 8 4 8 6 8 3 9 5 9 > 7 a z - 11 a z - 4 a z - 2 a z - 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	6 1 2 1 5 2 6 5 7 7 + -- + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 2 16 7 14 6 12 6 12 5 10 5 10 4 8 4 8 3 q q t q t q t q t q t q t q t q t 6 9 8 6 8 2 2 2 4 2 6 3 > ----- + ----- + ----- + ---- + ---- + 3 t + 5 q t + q t + 3 q t + q t 6 3 6 2 4 2 4 2 q t q t q t q t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L10a79

L10a78 L10a80