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

L10a5 L10a7

Knotscape

This page is passe. Go here instead! The 2-Component Link L10a6 Visit L10a6's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

X6172 X18,7,19,8 X4,19,1,20 X12,6,13,5 X8493 X16,13,17,14 X14,9,15,10 X10,15,11,16 X20,12,5,11 X2,18,3,17

Gauss Code:

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

Jones Polynomial:

- q-13/2 + 4q-11/2 - 8q-9/2 + 13q-7/2 - 17q-5/2 + 17q-3/2 - 17q-1/2 + 13q1/2 - 9q3/2 + 4q5/2 - q7/2

A2 (sl(3)) Invariant:

q-20 - q-18 - q-16 + 2q-14 - 4q-12 + 2q-10 + q-8 + 5q-4 - 2q-2 + 5 - q2 + 3q6 - 2q8 + q10

HOMFLY-PT Polynomial:

- a-1z-1 - 2a-1z - 2a-1z3 - a-1z5 + az-1 + 5az + 7az3 + 4az5 + az7 - 4a3z - 5a3z3 - 2a3z5 + a5z + a5z3

Kauffman Polynomial:

a-3z3 - a-3z5 - 2a-2z2 + 5a-2z4 - 4a-2z6 - a-1z-1 + 4a-1z - 10a-1z3 + 14a-1z5 - 8a-1z7 + 1 - 4z2 + 6z4 + 5z6 - 7z8 - az-1 + 10az - 31az3 + 42az5 - 17az7 - 2az9 - 7a2z2 + 7a2z4 + 14a2z6 - 13a2z8 + 8a3z - 28a3z3 + 39a3z5 - 16a3z7 - 2a3z9 - 8a4z2 + 12a4z4 + a4z6 - 6a4z8 + 2a5z - 7a5z3 + 11a5z5 - 7a5z7 - 3a6z2 + 6a6z4 - 4a6z6 + a7z3 - a7z5

Khovanov Homology:

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 8                     1 j = 6                   3   j = 4                 6 1   j = 2               7 3     j = 0             10 6       j = -2           9 9         j = -4         8 8           j = -6       5 9             j = -8     3 8               j = -10   1 5                 j = -12   3                   j = -14 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, 6]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 6]]
Out[4]=	PD[X[6, 1, 7, 2], X[18, 7, 19, 8], X[4, 19, 1, 20], X[12, 6, 13, 5], > X[8, 4, 9, 3], X[16, 13, 17, 14], X[14, 9, 15, 10], X[10, 15, 11, 16], > X[20, 12, 5, 11], X[2, 18, 3, 17]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 5, -3}, {4, -1, 2, -5, 7, -8, 9, -4, 6, -7, 8, -6, 10, -2, > 3, -9}]
In[6]:=	Jones[L][q]
Out[6]=	-(13/2) 4 8 13 17 17 17 3/2 -q + ----- - ---- + ---- - ---- + ---- - ------- + 13 Sqrt[q] - 9 q + 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q 5/2 7/2 > 4 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-20 -18 -16 2 4 2 -8 5 2 2 6 8 5 + q - q - q + --- - --- + --- + q + -- - -- - q + 3 q - 2 q + 14 12 10 4 2 q q q q q 10 > q
In[8]:=	HOMFLYPT[Link[10, Alternating, 6]][a, z]
Out[8]=	3 1 a 2 z 3 5 2 z 3 3 3 5 3 -(---) + - - --- + 5 a z - 4 a z + a z - ---- + 7 a z - 5 a z + a z - a z z a a 5 z 5 3 5 7 > -- + 4 a z - 2 a z + a z a
In[9]:=	Kauffman[Link[10, Alternating, 6]][a, z]
Out[9]=	2 1 a 4 z 3 5 2 2 z 2 2 1 - --- - - + --- + 10 a z + 8 a z + 2 a z - 4 z - ---- - 7 a z - a z z a 2 a 3 3 4 2 6 2 z 10 z 3 3 3 5 3 7 3 > 8 a z - 3 a z + -- - ----- - 31 a z - 28 a z - 7 a z + a z + 3 a a 4 5 5 4 5 z 2 4 4 4 6 4 z 14 z 5 > 6 z + ---- + 7 a z + 12 a z + 6 a z - -- + ----- + 42 a z + 2 3 a a a 6 3 5 5 5 7 5 6 4 z 2 6 4 6 6 6 > 39 a z + 11 a z - a z + 5 z - ---- + 14 a z + a z - 4 a z - 2 a 7 8 z 7 3 7 5 7 8 2 8 4 8 9 > ---- - 17 a z - 16 a z - 7 a z - 7 z - 13 a z - 6 a z - 2 a z - a 3 9 > 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	9 1 3 1 5 3 8 5 9 10 + -- + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + 2 14 6 12 5 10 5 10 4 8 4 8 3 6 3 6 2 q q t q t q t q t q t q t q t q t 8 8 9 2 2 2 4 2 4 3 6 3 > ----- + ---- + ---- + 6 t + 7 q t + 3 q t + 6 q t + q t + 3 q t + 4 2 4 2 q t q t q t 8 4 > q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L10a6

L10a5 L10a7