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

L10a115 L10a117

Knotscape

This page is passe. Go here instead! The 2-Component Link L10a116 Visit L10a116's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-25/2 + 2q-23/2 - 5q-21/2 + 7q-19/2 - 9q-17/2 + 10q-15/2 - 9q-13/2 + 7q-11/2 - 5q-9/2 + 2q-7/2 - q-5/2

A2 (sl(3)) Invariant:

q-38 + q-34 + 3q-32 + 2q-28 - q-24 + q-22 - 2q-20 + 2q-18 + 2q-12 - q-10 + q-8

HOMFLY-PT Polynomial:

- 2a5z - 3a5z3 - a5z5 - 4a7z - 6a7z3 - 2a7z5 - a9z-1 - a9z - 2a9z3 - a9z5 + a11z-1 + 2a11z + a11z3

Kauffman Polynomial:

- 2a5z + 3a5z3 - a5z5 - a6z2 + 4a6z4 - 2a6z6 + 4a7z - 7a7z3 + 7a7z5 - 3a7z7 - a8z2 + 2a8z4 + a8z6 - 2a8z8 + a9z-1 - 3a9z + 3a9z3 - a9z7 - a9z9 - a10 - 4a10z4 + 6a10z6 - 4a10z8 + a11z-1 - 3a11z + 3a11z3 - a11z7 - a11z9 - a12z2 + 2a12z4 + a12z6 - 2a12z8 + 4a13z - 7a13z3 + 7a13z5 - 3a13z7 - a14z2 + 4a14z4 - 2a14z6 - 2a15z + 3a15z3 - 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                   2 1 j = -8                 3     j = -10               4 2     j = -12             5 3       j = -14           5 4         j = -16         4 5           j = -18       3 5             j = -20     2 4               j = -22     3                 j = -24 1 2                   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, 116]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 116]]
Out[4]=	PD[X[12, 1, 13, 2], X[20, 9, 11, 10], X[14, 3, 15, 4], X[8, 15, 9, 16], > X[16, 5, 17, 6], X[18, 7, 19, 8], X[6, 17, 7, 18], X[4, 19, 5, 20], > X[2, 11, 3, 12], X[10, 13, 1, 14]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 3, -8, 5, -7, 6, -4, 2, -10}, > {9, -1, 10, -3, 4, -5, 7, -6, 8, -2}]
In[6]:=	Jones[L][q]
Out[6]=	-(25/2) 2 5 7 9 10 9 7 5 -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 2 -(5/2) > ---- - q 7/2 q
In[7]:=	A2Invariant[L][q]
Out[7]=	-38 -34 3 2 -24 -22 2 2 2 -10 -8 q + q + --- + --- - q + q - --- + --- + --- - q + q 32 28 20 18 12 q q q q q
In[8]:=	HOMFLYPT[Link[10, Alternating, 116]][a, z]
Out[8]=	9 11 a a 5 7 9 11 5 3 7 3 9 3 -(--) + --- - 2 a z - 4 a z - a z + 2 a z - 3 a z - 6 a z - 2 a z + z z 11 3 5 5 7 5 9 5 > a z - a z - 2 a z - a z
In[9]:=	Kauffman[Link[10, Alternating, 116]][a, z]
Out[9]=	9 11 10 a a 5 7 9 11 13 15 -a + -- + --- - 2 a z + 4 a z - 3 a z - 3 a z + 4 a z - 2 a z - z z 6 2 8 2 12 2 14 2 5 3 7 3 9 3 11 3 > a z - a z - a z - a z + 3 a z - 7 a z + 3 a z + 3 a z - 13 3 15 3 6 4 8 4 10 4 12 4 14 4 > 7 a z + 3 a z + 4 a z + 2 a z - 4 a z + 2 a z + 4 a z - 5 5 7 5 13 5 15 5 6 6 8 6 10 6 12 6 > a z + 7 a z + 7 a z - a z - 2 a z + a z + 6 a z + a z - 14 6 7 7 9 7 11 7 13 7 8 8 10 8 > 2 a z - 3 a z - a z - a z - 3 a z - 2 a z - 4 a z - 12 8 9 9 11 9 > 2 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	-6 -4 1 1 2 3 2 4 3 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 5 4 5 5 4 5 3 4 > ------ + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 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 2 3 2 > ------ + ----- + ---- 10 2 8 2 6 q t q t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L10a116

L10a115 L10a117