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

L10a48 L10a50

Knotscape

This page is passe. Go here instead! The 2-Component Link L10a49 Visit L10a49's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

X6172 X16,7,17,8 X4,17,1,18 X10,5,11,6 X14,3,15,4 X18,11,19,12 X20,13,5,14 X12,19,13,20 X2,9,3,10 X8,15,9,16

Gauss Code:

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

Jones Polynomial:

- q-25/2 + 4q-23/2 - 7q-21/2 + 11q-19/2 - 13q-17/2 + 14q-15/2 - 14q-13/2 + 9q-11/2 - 7q-9/2 + 3q-7/2 - q-5/2

A2 (sl(3)) Invariant:

q-38 - 2q-36 - 2q-34 - 4q-30 + q-28 + q-24 + 5q-22 + q-20 + 6q-18 + q-16 + 2q-12 - 2q-10 + q-8

HOMFLY-PT Polynomial:

- 2a5z3 - a5z5 - 3a7z-1 - 10a7z - 10a7z3 - 3a7z5 + 5a9z-1 + 8a9z + a9z3 - a9z5 - 2a11z-1 + a11z3

Kauffman Polynomial:

2a5z3 - a5z5 + 5a6z4 - 3a6z6 - 3a7z-1 + 10a7z - 18a7z3 + 16a7z5 - 6a7z7 + 5a8 - 7a8z2 - 2a8z4 + 8a8z6 - 5a8z8 - 5a9z-1 + 14a9z - 27a9z3 + 22a9z5 - 6a9z7 - 2a9z9 + 5a10 - 4a10z2 - 13a10z4 + 20a10z6 - 10a10z8 - 2a11z-1 + 5a11z - 11a11z3 + 16a11z5 - 6a11z7 - 2a11z9 + a12z2 + a12z4 + 5a12z6 - 5a12z8 + a13z - 3a13z3 + 10a13z5 - 6a13z7 - a14 - 2a14z2 + 7a14z4 - 4a14z6 + a15z3 - a15z5

Khovanov Homology:

trqj r = -10 r = -9 r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 j = -4                     1 j = -6                   3 1 j = -8                 4     j = -10               5 3     j = -12             9 4       j = -14           6 6         j = -16         7 8           j = -18       4 6             j = -20     3 7               j = -22   1 4                 j = -24   3                   j = -26 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, 49]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 49]]
Out[4]=	PD[X[6, 1, 7, 2], X[16, 7, 17, 8], X[4, 17, 1, 18], X[10, 5, 11, 6], > X[14, 3, 15, 4], X[18, 11, 19, 12], X[20, 13, 5, 14], X[12, 19, 13, 20], > X[2, 9, 3, 10], X[8, 15, 9, 16]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 5, -3}, {4, -1, 2, -10, 9, -4, 6, -8, 7, -5, 10, -2, 3, -6, > 8, -7}]
In[6]:=	Jones[L][q]
Out[6]=	-(25/2) 4 7 11 13 14 14 9 7 -q + ----- - ----- + ----- - ----- + ----- - ----- + ----- - ---- + 23/2 21/2 19/2 17/2 15/2 13/2 11/2 9/2 q q q q q q q q 3 -(5/2) > ---- - q 7/2 q
In[7]:=	A2Invariant[L][q]
Out[7]=	-38 2 2 4 -28 -24 5 -20 6 -16 2 2 -8 q - --- - --- - --- + q + q + --- + q + --- + q + --- - --- + q 36 34 30 22 18 12 10 q q q q q q q
In[8]:=	HOMFLYPT[Link[10, Alternating, 49]][a, z]
Out[8]=	7 9 11 -3 a 5 a 2 a 7 9 5 3 7 3 9 3 11 3 ----- + ---- - ----- - 10 a z + 8 a z - 2 a z - 10 a z + a z + a z - z z z 5 5 7 5 9 5 > a z - 3 a z - a z
In[9]:=	Kauffman[Link[10, Alternating, 49]][a, z]
Out[9]=	7 9 11 8 10 14 3 a 5 a 2 a 7 9 11 5 a + 5 a - a - ---- - ---- - ----- + 10 a z + 14 a z + 5 a z + z z z 13 8 2 10 2 12 2 14 2 5 3 7 3 > a z - 7 a z - 4 a z + a z - 2 a z + 2 a z - 18 a z - 9 3 11 3 13 3 15 3 6 4 8 4 10 4 > 27 a z - 11 a z - 3 a z + a z + 5 a z - 2 a z - 13 a z + 12 4 14 4 5 5 7 5 9 5 11 5 13 5 > a z + 7 a z - a z + 16 a z + 22 a z + 16 a z + 10 a z - 15 5 6 6 8 6 10 6 12 6 14 6 7 7 > a z - 3 a z + 8 a z + 20 a z + 5 a z - 4 a z - 6 a z - 9 7 11 7 13 7 8 8 10 8 12 8 9 9 > 6 a z - 6 a z - 6 a z - 5 a z - 10 a z - 5 a z - 2 a z - 11 9 > 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	-6 -4 1 3 1 4 3 7 4 q + q + ------- + ------ + ------ + ------ + ------ + ------ + ------ + 26 10 24 9 22 9 22 8 20 8 20 7 18 7 q t q t q t q t q t q t q t 6 7 8 6 6 9 4 5 > ------ + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 18 6 16 6 16 5 14 5 14 4 12 4 12 3 10 3 q t q t q t q t q t q t q t q t 3 4 3 > ------ + ----- + ---- 10 2 8 2 6 q t q t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L10a49

L10a48 L10a50