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

L11a170 L11a172

Knotscape

This page is passe. Go here instead! The 2-Component Link L11a171 Visit L11a171's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

X8192 X20,9,21,10 X14,5,15,6 X18,12,19,11 X10,4,11,3 X12,7,13,8 X16,13,17,14 X22,17,7,18 X6,15,1,16 X4,21,5,22 X2,20,3,19

Gauss Code:

{{1, -11, 5, -10, 3, -9}, {6, -1, 2, -5, 4, -6, 7, -3, 9, -7, 8, -4, 11, -2, 10, -8}}

Jones Polynomial:

- q-19/2 + 5q-17/2 - 12q-15/2 + 20q-13/2 - 27q-11/2 + 31q-9/2 - 31q-7/2 + 26q-5/2 - 20q-3/2 + 11q-1/2 - 5q1/2 + q3/2

A2 (sl(3)) Invariant:

q-28 - 3q-26 + 3q-24 - 4q-20 + 6q-18 - 5q-16 + 3q-14 - 2q-10 + 7q-8 - 4q-6 + 6q-4 + q-2 - 2 + 3q2 - q4

HOMFLY-PT Polynomial:

- az-1 - az + az3 + az5 + a3z-1 - a3z3 - 2a3z5 - a3z7 - a5z3 - 2a5z5 - a5z7 + a7z3 + a7z5

Kauffman Polynomial:

z4 - z6 + az-1 - az - 5az3 + 9az5 - 5az7 - a2 + 2a2z2 - 11a2z4 + 19a2z6 - 10a2z8 + a3z-1 - a3z - 8a3z3 + 9a3z5 + 10a3z7 - 10a3z9 + 6a4z2 - 34a4z4 + 54a4z6 - 19a4z8 - 4a4z10 - 6a5z3 - a5z5 + 34a5z7 - 23a5z9 + 6a6z2 - 38a6z4 + 64a6z6 - 26a6z8 - 4a6z10 - 8a7z3 + 14a7z5 + 7a7z7 - 13a7z9 + 2a8z2 - 13a8z4 + 25a8z6 - 17a8z8 - 5a9z3 + 14a9z5 - 12a9z7 + 3a10z4 - 5a10z6 - a11z5

Khovanov Homology:

trqj r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 j = 4                       1 j = 2                     4   j = 0                   7 1   j = -2                 13 4     j = -4               14 8       j = -6             17 12         j = -8           15 15           j = -10         12 16             j = -12       8 15               j = -14     4 12                 j = -16   1 8                   j = -18   4                     j = -20 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[11, Alternating, 171]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 171]]
Out[4]=	PD[X[8, 1, 9, 2], X[20, 9, 21, 10], X[14, 5, 15, 6], X[18, 12, 19, 11], > X[10, 4, 11, 3], X[12, 7, 13, 8], X[16, 13, 17, 14], X[22, 17, 7, 18], > X[6, 15, 1, 16], X[4, 21, 5, 22], X[2, 20, 3, 19]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -11, 5, -10, 3, -9}, > {6, -1, 2, -5, 4, -6, 7, -3, 9, -7, 8, -4, 11, -2, 10, -8}]
In[6]:=	Jones[L][q]
Out[6]=	-(19/2) 5 12 20 27 31 31 26 20 -q + ----- - ----- + ----- - ----- + ---- - ---- + ---- - ---- + 17/2 15/2 13/2 11/2 9/2 7/2 5/2 3/2 q q q q q q q q 11 3/2 > ------- - 5 Sqrt[q] + q Sqrt[q]
In[7]:=	A2Invariant[L][q]
Out[7]=	-28 3 3 4 6 5 3 2 7 4 6 -2 -2 + q - --- + --- - --- + --- - --- + --- - --- + -- - -- + -- + q + 26 24 20 18 16 14 10 8 6 4 q q q q q q q q q q 2 4 > 3 q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 171]][a, z]
Out[8]=	3 a a 3 3 3 5 3 7 3 5 3 5 5 5 -(-) + -- - a z + a z - a z - a z + a z + a z - 2 a z - 2 a z + z z 7 5 3 7 5 7 > a z - a z - a z
In[9]:=	Kauffman[Link[11, Alternating, 171]][a, z]
Out[9]=	3 2 a a 3 2 2 4 2 6 2 8 2 3 -a + - + -- - a z - a z + 2 a z + 6 a z + 6 a z + 2 a z - 5 a z - z z 3 3 5 3 7 3 9 3 4 2 4 4 4 > 8 a z - 6 a z - 8 a z - 5 a z + z - 11 a z - 34 a z - 6 4 8 4 10 4 5 3 5 5 5 7 5 > 38 a z - 13 a z + 3 a z + 9 a z + 9 a z - a z + 14 a z + 9 5 11 5 6 2 6 4 6 6 6 8 6 > 14 a z - a z - z + 19 a z + 54 a z + 64 a z + 25 a z - 10 6 7 3 7 5 7 7 7 9 7 2 8 > 5 a z - 5 a z + 10 a z + 34 a z + 7 a z - 12 a z - 10 a z - 4 8 6 8 8 8 3 9 5 9 7 9 > 19 a z - 26 a z - 17 a z - 10 a z - 23 a z - 13 a z - 4 10 6 10 > 4 a z - 4 a z
In[10]:=	Kh[L][q, t]
Out[10]=	8 13 1 4 1 8 4 12 8 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 4 2 20 8 18 7 16 7 16 6 14 6 14 5 12 5 q q q t q t q t q t q t q t q t 15 12 16 15 15 17 12 14 > ------ + ------ + ------ + ----- + ----- + ----- + ---- + ---- + 7 t + 12 4 10 4 10 3 8 3 8 2 6 2 6 4 q t q t q t q t q t q t q t q t 4 t 2 2 2 4 3 > --- + t + 4 q t + q t 2 q

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a171

L11a170 L11a172