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

L10a119 L10a121

Knotscape

This page is passe. Go here instead! The 2-Component Link L10a120 Visit L10a120's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

X12,1,13,2 X2,13,3,14 X14,3,15,4 X16,5,17,6 X6,11,7,12 X20,7,11,8 X18,9,19,10 X10,17,1,18 X8,19,9,20 X4,15,5,16

Gauss Code:

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

Jones Polynomial:

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

A2 (sl(3)) Invariant:

q-38 + q-36 + q-34 + q-32 + q-28 + q-26 + q-24 + q-22 - q-20 + q-12 + q-8

HOMFLY-PT Polynomial:

- 3a5z - 4a5z3 - a5z5 - a7z - 3a7z3 - a7z5 - a9z-1 - 4a9z - 4a9z3 - a9z5 + a11z-1 + 3a11z + a11z3

Kauffman Polynomial:

- 3a5z + 4a5z3 - a5z5 - a6z2 + 3a6z4 - a6z6 + a7z - 2a7z3 + 3a7z5 - a7z7 + 3a8z2 - 6a8z4 + 4a8z6 - a8z8 + a9z-1 - 7a9z + 15a9z3 - 15a9z5 + 6a9z7 - a9z9 - a10 + 8a10z2 - 18a10z4 + 10a10z6 - 2a10z8 + a11z-1 - 7a11z + 15a11z3 - 15a11z5 + 6a11z7 - a11z9 + 3a12z2 - 6a12z4 + 4a12z6 - a12z8 + a13z - 2a13z3 + 3a13z5 - a13z7 - a14z2 + 3a14z4 - a14z6 - 3a15z + 4a15z3 - 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                   1 1 j = -8                 1     j = -10               2 1     j = -12             2 1       j = -14           2 2         j = -16         2 2           j = -18       1 2             j = -20     1 2               j = -22     1                 j = -24 1 1                   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, 120]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 120]]
Out[4]=	PD[X[12, 1, 13, 2], X[2, 13, 3, 14], X[14, 3, 15, 4], X[16, 5, 17, 6], > X[6, 11, 7, 12], X[20, 7, 11, 8], X[18, 9, 19, 10], X[10, 17, 1, 18], > X[8, 19, 9, 20], X[4, 15, 5, 16]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 3, -10, 4, -5, 6, -9, 7, -8}, > {5, -1, 2, -3, 10, -4, 8, -7, 9, -6}]
In[6]:=	Jones[L][q]
Out[6]=	-(25/2) -(23/2) 2 3 4 4 4 3 2 -q + q - ----- + ----- - ----- + ----- - ----- + ----- - ---- + 21/2 19/2 17/2 15/2 13/2 11/2 9/2 q q q q q q q -(7/2) -(5/2) > q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-38 -36 -34 -32 -28 -26 -24 -22 -20 -12 -8 q + q + q + q + q + q + q + q - q + q + q
In[8]:=	HOMFLYPT[Link[10, Alternating, 120]][a, z]
Out[8]=	9 11 a a 5 7 9 11 5 3 7 3 9 3 -(--) + --- - 3 a z - a z - 4 a z + 3 a z - 4 a z - 3 a z - 4 a z + z z 11 3 5 5 7 5 9 5 > a z - a z - a z - a z
In[9]:=	Kauffman[Link[10, Alternating, 120]][a, z]
Out[9]=	9 11 10 a a 5 7 9 11 13 15 6 2 -a + -- + --- - 3 a z + a z - 7 a z - 7 a z + a z - 3 a z - a z + z z 8 2 10 2 12 2 14 2 5 3 7 3 9 3 > 3 a z + 8 a z + 3 a z - a z + 4 a z - 2 a z + 15 a z + 11 3 13 3 15 3 6 4 8 4 10 4 > 15 a z - 2 a z + 4 a z + 3 a z - 6 a z - 18 a z - 12 4 14 4 5 5 7 5 9 5 11 5 13 5 > 6 a z + 3 a z - a z + 3 a z - 15 a z - 15 a z + 3 a z - 15 5 6 6 8 6 10 6 12 6 14 6 7 7 > a z - a z + 4 a z + 10 a z + 4 a z - a z - a z + 9 7 11 7 13 7 8 8 10 8 12 8 9 9 11 9 > 6 a z + 6 a z - a z - a z - 2 a z - a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	-6 -4 1 1 1 1 1 2 1 q + q + ------- + ------- + ------ + ------ + ------ + ------ + ------ + 26 10 24 10 24 9 22 8 20 8 20 7 18 7 q t q t q t q t q t q t q t 2 2 2 2 2 2 1 2 > ------ + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 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 1 1 1 > ------ + ----- + ---- 10 2 8 2 6 q t q t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L10a120

L10a119 L10a121