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

L10a156 L10a158

Knotscape

This page is passe. Go here instead! The 3-Component Link L10a157 Visit L10a157's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

X8192 X18,12,19,11 X10,4,11,3 X2,20,3,19 X16,8,17,7 X20,9,13,10 X12,18,7,17 X4,13,5,14 X14,5,15,6 X6,15,1,16

Gauss Code:

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

Jones Polynomial:

- q-6 + 3q-5 - 5q-4 + 8q-3 - 9q-2 + 10q-1 - 8 + 8q - 4q2 + 3q3 - q4

A2 (sl(3)) Invariant:

- q-18 + q-16 + 2q-10 - q-8 + 3q-6 + 2q-4 + 4q-2 + 6 + 3q2 + 5q4 + 2q6 + q8 + q10 - q12

HOMFLY-PT Polynomial:

a-2z-2 + a-2 - 2a-2z2 - a-2z4 - 2z-2 - 2 + z2 + 3z4 + z6 + a2z-2 + a2 + 2a2z2 + 3a2z4 + a2z6 - 2a4z2 - a4z4

Kauffman Polynomial:

4a-3z3 - 4a-3z5 + a-3z7 + a-2z-2 - 2a-2 - 8a-2z2 + 20a-2z4 - 14a-2z6 + 3a-2z8 - 2a-1z-1 + 2a-1z + 6a-1z3 - 2a-1z5 - 5a-1z7 + 2a-1z9 + 2z-2 - 3 - 10z2 + 32z4 - 30z6 + 8z8 - 2az-1 + 2az + 11az3 - 16az5 + az7 + 2az9 + a2z-2 - 2a2 + 3a2z2 - a2z4 - 9a2z6 + 5a2z8 + 5a3z3 - 13a3z5 + 7a3z7 + 4a4z2 - 10a4z4 + 7a4z6 - 3a5z3 + 5a5z5 - a6z2 + 3a6z4 + a7z3

Khovanov Homology:

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 9                     1 j = 7                   2   j = 5                 2 1   j = 3               6 2     j = 1             3 3       j = -1           7 5         j = -3         4 5           j = -5       4 5             j = -7     2 5               j = -9   1 3                 j = -11   2                   j = -13 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]=	3
In[3]:=	Show[DrawMorseLink[Link[10, Alternating, 157]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 157]]
Out[4]=	PD[X[8, 1, 9, 2], X[18, 12, 19, 11], X[10, 4, 11, 3], X[2, 20, 3, 19], > X[16, 8, 17, 7], X[20, 9, 13, 10], X[12, 18, 7, 17], X[4, 13, 5, 14], > X[14, 5, 15, 6], X[6, 15, 1, 16]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -4, 3, -8, 9, -10}, {5, -1, 6, -3, 2, -7}, > {8, -9, 10, -5, 7, -2, 4, -6}]
In[6]:=	Jones[L][q]
Out[6]=	-6 3 5 8 9 10 2 3 4 -8 - q + -- - -- + -- - -- + -- + 8 q - 4 q + 3 q - q 5 4 3 2 q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-18 -16 2 -8 3 2 4 2 4 6 8 10 12 6 - q + q + --- - q + -- + -- + -- + 3 q + 5 q + 2 q + q + q - q 10 6 4 2 q q q q
In[8]:=	HOMFLYPT[Link[10, Alternating, 157]][a, z]
Out[8]=	2 2 4 -2 2 2 1 a 2 2 z 2 2 4 2 4 z -2 + a + a - -- + ----- + -- + z - ---- + 2 a z - 2 a z + 3 z - -- + 2 2 2 2 2 2 z a z z a a 2 4 4 4 6 2 6 > 3 a z - a z + z + a z
In[9]:=	Kauffman[Link[10, Alternating, 157]][a, z]
Out[9]=	2 2 2 2 2 1 a 2 2 a 2 z 2 8 z -3 - -- - 2 a + -- + ----- + -- - --- - --- + --- + 2 a z - 10 z - ---- + 2 2 2 2 2 a z z a 2 a z a z z a 3 3 2 2 4 2 6 2 4 z 6 z 3 3 3 5 3 > 3 a z + 4 a z - a z + ---- + ---- + 11 a z + 5 a z - 3 a z + 3 a a 4 5 5 7 3 4 20 z 2 4 4 4 6 4 4 z 2 z > a z + 32 z + ----- - a z - 10 a z + 3 a z - ---- - ---- - 2 3 a a a 6 7 5 3 5 5 5 6 14 z 2 6 4 6 z > 16 a z - 13 a z + 5 a z - 30 z - ----- - 9 a z + 7 a z + -- - 2 3 a a 7 8 9 5 z 7 3 7 8 3 z 2 8 2 z 9 > ---- + a z + 7 a z + 8 z + ---- + 5 a z + ---- + 2 a z a 2 a a
In[10]:=	Kh[L][q, t]
Out[10]=	5 7 1 2 1 3 2 5 4 5 -- + - + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ---- + 3 q 13 5 11 4 9 4 9 3 7 3 7 2 5 2 5 q q t q t q t q t q t q t q t q t 4 5 t 2 3 2 3 3 5 3 5 4 > ---- + --- + 3 q t + 3 q t + 6 q t + 2 q t + 2 q t + q t + 3 q q t 7 4 9 5 > 2 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L10a157

L10a156 L10a158