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

L10a82 L10a84

Knotscape

This page is passe. Go here instead! The 2-Component Link L10a83 Visit L10a83's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-19/2 + 3q-17/2 - 6q-15/2 + 9q-13/2 - 12q-11/2 + 12q-9/2 - 12q-7/2 + 9q-5/2 - 6q-3/2 + 3q-1/2 - q1/2

A2 (sl(3)) Invariant:

q-30 - 2q-26 + 2q-24 + 4q-18 + 3q-14 + 2q-8 - 3q-6 + 2q-4 - 1 + q2

HOMFLY-PT Polynomial:

- az - az3 + a3z3 + a3z5 - a5z-1 - 2a5z + a5z5 + a7z-1 - a7z - 2a7z3 + a9z

Kauffman Polynomial:

- az + 2az3 - az5 - 3a2z2 + 6a2z4 - 3a2z6 + a3z - 2a3z3 + 6a3z5 - 4a3z7 - 3a4z2 + 6a4z4 - 3a4z8 + a5z-1 - a5z - 7a5z3 + 14a5z5 - 7a5z7 - a5z9 - a6 + 6a6z6 - 6a6z8 + a7z-1 - a7z - 7a7z3 + 14a7z5 - 7a7z7 - a7z9 - 3a8z2 + 6a8z4 - 3a8z8 + a9z - 2a9z3 + 6a9z5 - 4a9z7 - 3a10z2 + 6a10z4 - 3a10z6 - a11z + 2a11z3 - a11z5

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 = 2                     1 j = 0                   2   j = -2                 4 1   j = -4               6 3     j = -6             6 3       j = -8           6 6         j = -10         6 6           j = -12       3 6             j = -14     3 6               j = -16   1 4                 j = -18   2                   j = -20 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, 83]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 83]]
Out[4]=	PD[X[8, 1, 9, 2], X[12, 3, 13, 4], X[20, 10, 7, 9], X[14, 17, 15, 18], > X[16, 5, 17, 6], X[4, 15, 5, 16], X[18, 13, 19, 14], X[10, 20, 11, 19], > X[2, 7, 3, 8], X[6, 11, 1, 12]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 2, -6, 5, -10}, > {9, -1, 3, -8, 10, -2, 7, -4, 6, -5, 4, -7, 8, -3}]
In[6]:=	Jones[L][q]
Out[6]=	-(19/2) 3 6 9 12 12 12 9 6 -q + ----- - ----- + ----- - ----- + ---- - ---- + ---- - ---- + 17/2 15/2 13/2 11/2 9/2 7/2 5/2 3/2 q q q q q q q q 3 > ------- - Sqrt[q] Sqrt[q]
In[7]:=	A2Invariant[L][q]
Out[7]=	-30 2 2 4 3 2 3 2 2 -1 + q - --- + --- + --- + --- + -- - -- + -- + q 26 24 18 14 8 6 4 q q q q q q q
In[8]:=	HOMFLYPT[Link[10, Alternating, 83]][a, z]
Out[8]=	5 7 a a 5 7 9 3 3 3 7 3 3 5 5 5 -(--) + -- - a z - 2 a z - a z + a z - a z + a z - 2 a z + a z + a z z z
In[9]:=	Kauffman[Link[10, Alternating, 83]][a, z]
Out[9]=	5 7 6 a a 3 5 7 9 11 2 2 4 2 -a + -- + -- - a z + a z - a z - a z + a z - a z - 3 a z - 3 a z - z z 8 2 10 2 3 3 3 5 3 7 3 9 3 > 3 a z - 3 a z + 2 a z - 2 a z - 7 a z - 7 a z - 2 a z + 11 3 2 4 4 4 8 4 10 4 5 3 5 > 2 a z + 6 a z + 6 a z + 6 a z + 6 a z - a z + 6 a z + 5 5 7 5 9 5 11 5 2 6 6 6 10 6 > 14 a z + 14 a z + 6 a z - a z - 3 a z + 6 a z - 3 a z - 3 7 5 7 7 7 9 7 4 8 6 8 8 8 > 4 a z - 7 a z - 7 a z - 4 a z - 3 a z - 6 a z - 3 a z - 5 9 7 9 > a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	3 4 1 2 1 4 3 6 3 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 4 2 20 8 18 7 16 7 16 6 14 6 14 5 12 5 q q q t q t q t q t q t q t q t 6 6 6 6 6 6 3 6 t > ------ + ------ + ------ + ----- + ----- + ----- + ---- + ---- + 2 t + -- + 12 4 10 4 10 3 8 3 8 2 6 2 6 4 2 q t q t q t q t q t q t q t q t q 2 2 > q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L10a83

L10a82 L10a84