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

L10a57 L10a59

Knotscape

This page is passe. Go here instead! The 2-Component Link L10a58 Visit L10a58's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-19/2 + 2q-17/2 - 5q-15/2 + 8q-13/2 - 10q-11/2 + 11q-9/2 - 11q-7/2 + 8q-5/2 - 6q-3/2 + 3q-1/2 - q1/2

A2 (sl(3)) Invariant:

q-30 + q-28 + 3q-24 - 2q-20 + q-18 - 2q-16 + 2q-14 + q-12 + q-10 + 3q-8 - 2q-6 + 2q-4 - 1 + q2

HOMFLY-PT Polynomial:

- az - az3 - a3z-1 - a3z + a3z3 + a3z5 + 2a5z-1 + a5z + a5z3 + a5z5 - 2a7z-1 - 3a7z - 2a7z3 + a9z-1 + a9z

Kauffman Polynomial:

- az + 2az3 - az5 - 2a2z2 + 6a2z4 - 3a2z6 - a3z-1 + 3a3z - 3a3z3 + 7a3z5 - 4a3z7 - a4z2 + 3a4z4 + 2a4z6 - 3a4z8 - 2a5z-1 + 9a5z - 20a5z3 + 18a5z5 - 6a5z7 - a5z9 - a6 + 7a6z2 - 15a6z4 + 13a6z6 - 6a6z8 - 2a7z-1 + 11a7z - 23a7z3 + 17a7z5 - 5a7z7 - a7z9 + 5a8z2 - 8a8z4 + 6a8z6 - 3a8z8 - a9z-1 + 4a9z - 5a9z3 + 6a9z5 - 3a9z7 - a10z2 + 4a10z4 - 2a10z6 - 2a11z + 3a11z3 - 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               5 3     j = -6             6 3       j = -8           5 5         j = -10         5 6           j = -12       3 5             j = -14     2 5               j = -16   1 4                 j = -18   1                   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, 58]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 58]]
Out[4]=	PD[X[8, 1, 9, 2], X[10, 3, 11, 4], X[20, 16, 7, 15], X[14, 5, 15, 6], > X[4, 13, 5, 14], X[12, 17, 13, 18], X[18, 11, 19, 12], X[16, 20, 17, 19], > X[2, 7, 3, 8], X[6, 9, 1, 10]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 2, -5, 4, -10}, > {9, -1, 10, -2, 7, -6, 5, -4, 3, -8, 6, -7, 8, -3}]
In[6]:=	Jones[L][q]
Out[6]=	-(19/2) 2 5 8 10 11 11 8 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 -28 3 2 -18 2 2 -12 -10 3 2 2 -1 + q + q + --- - --- + q - --- + --- + q + q + -- - -- + -- + 24 20 16 14 8 6 4 q q q q q q q 2 > q
In[8]:=	HOMFLYPT[Link[10, Alternating, 58]][a, z]
Out[8]=	3 5 7 9 a 2 a 2 a a 3 5 7 9 3 3 3 -(--) + ---- - ---- + -- - a z - a z + a z - 3 a z + a z - a z + a z + z z z z 5 3 7 3 3 5 5 5 > a z - 2 a z + a z + a z
In[9]:=	Kauffman[Link[10, Alternating, 58]][a, z]
Out[9]=	3 5 7 9 6 a 2 a 2 a a 3 5 7 9 -a - -- - ---- - ---- - -- - a z + 3 a z + 9 a z + 11 a z + 4 a z - z z z z 11 2 2 4 2 6 2 8 2 10 2 3 3 3 > 2 a z - 2 a z - a z + 7 a z + 5 a z - a z + 2 a z - 3 a z - 5 3 7 3 9 3 11 3 2 4 4 4 6 4 > 20 a z - 23 a z - 5 a z + 3 a z + 6 a z + 3 a z - 15 a z - 8 4 10 4 5 3 5 5 5 7 5 9 5 > 8 a z + 4 a z - a z + 7 a z + 18 a z + 17 a z + 6 a z - 11 5 2 6 4 6 6 6 8 6 10 6 3 7 > a z - 3 a z + 2 a z + 13 a z + 6 a z - 2 a z - 4 a z - 5 7 7 7 9 7 4 8 6 8 8 8 5 9 7 9 > 6 a z - 5 a z - 3 a z - 3 a z - 6 a z - 3 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	3 4 1 1 1 4 2 5 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 5 5 6 5 5 6 3 5 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 L10a58

L10a57 L10a59