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

L11n187 L11n189

Knotscape

This page is passe. Go here instead! The 2-Component Link L11n188 Visit L11n188's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

X8192 X12,3,13,4 X16,6,17,5 X22,13,7,14 X18,15,19,16 X14,21,15,22 X9,20,10,21 X4,18,5,17 X19,10,20,11 X2738 X6,11,1,12

Gauss Code:

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

Jones Polynomial:

2q-17/2 - 5q-15/2 + 8q-13/2 - 11q-11/2 + 12q-9/2 - 12q-7/2 + 9q-5/2 - 7q-3/2 + 3q-1/2 - q1/2

A2 (sl(3)) Invariant:

- 2q-26 + q-24 - q-22 - 2q-20 + 3q-18 - q-16 + 3q-14 + q-12 + q-10 + 4q-8 - q-6 + 3q-4 - 1 + q2

HOMFLY-PT Polynomial:

- az - az3 - 2a3z-1 - 4a3z + a3z5 + 3a5z-1 + 7a5z + 6a5z3 + 2a5z5 - a7z-1 - 3a7z - 2a7z3

Kauffman Polynomial:

- az + 2az3 - az5 - a2z2 + 5a2z4 - 3a2z6 - 2a3z-1 + 7a3z - 8a3z3 + 10a3z5 - 5a3z7 + 3a4 - 5a4z2 + 3a4z4 + 4a4z6 - 4a4z8 - 3a5z-1 + 14a5z - 28a5z3 + 26a5z5 - 9a5z7 - a5z9 + 3a6 - 10a6z2 + 3a6z4 + 7a6z6 - 6a6z8 - a7z-1 + 5a7z - 12a7z3 + 12a7z5 - 5a7z7 - a7z9 + a8 - 2a8z2 + 2a8z4 - 2a8z8 - a9z + 6a9z3 - 3a9z5 - a9z7 + 4a10z2 - 3a10z4

Khovanov Homology:

trqj r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 j = 2                   1 j = 0                 2   j = -2               5 1   j = -4             5 3     j = -6           7 4       j = -8         6 6         j = -10       5 6           j = -12     3 6             j = -14   2 5               j = -16   3                 j = -18 2

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, NonAlternating, 188]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 188]]
Out[4]=	PD[X[8, 1, 9, 2], X[12, 3, 13, 4], X[16, 6, 17, 5], X[22, 13, 7, 14], > X[18, 15, 19, 16], X[14, 21, 15, 22], X[9, 20, 10, 21], X[4, 18, 5, 17], > X[19, 10, 20, 11], X[2, 7, 3, 8], X[6, 11, 1, 12]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -8, 3, -11}, > {10, -1, -7, 9, 11, -2, 4, -6, 5, -3, 8, -5, -9, 7, 6, -4}]
In[6]:=	Jones[L][q]
Out[6]=	2 5 8 11 12 12 9 7 3 ----- - ----- + ----- - ----- + ---- - ---- + ---- - ---- + ------- - Sqrt[q] 17/2 15/2 13/2 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	2 -24 -22 2 3 -16 3 -12 -10 4 -6 3 -1 - --- + q - q - --- + --- - q + --- + q + q + -- - q + -- + 26 20 18 14 8 4 q q q q q q 2 > q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 188]][a, z]
Out[8]=	3 5 7 -2 a 3 a a 3 5 7 3 5 3 7 3 ----- + ---- - -- - a z - 4 a z + 7 a z - 3 a z - a z + 6 a z - 2 a z + z z z 3 5 5 5 > a z + 2 a z
In[9]:=	Kauffman[Link[11, NonAlternating, 188]][a, z]
Out[9]=	3 5 7 4 6 8 2 a 3 a a 3 5 7 9 3 a + 3 a + a - ---- - ---- - -- - a z + 7 a z + 14 a z + 5 a z - a z - z z z 2 2 4 2 6 2 8 2 10 2 3 3 3 > a z - 5 a z - 10 a z - 2 a z + 4 a z + 2 a z - 8 a z - 5 3 7 3 9 3 2 4 4 4 6 4 8 4 > 28 a z - 12 a z + 6 a z + 5 a z + 3 a z + 3 a z + 2 a z - 10 4 5 3 5 5 5 7 5 9 5 2 6 > 3 a z - a z + 10 a z + 26 a z + 12 a z - 3 a z - 3 a z + 4 6 6 6 3 7 5 7 7 7 9 7 4 8 > 4 a z + 7 a z - 5 a z - 9 a z - 5 a z - a z - 4 a z - 6 8 8 8 5 9 7 9 > 6 a z - 2 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	3 5 2 3 2 5 3 6 5 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 4 2 18 7 16 6 14 6 14 5 12 5 12 4 10 4 q q q t q t q t q t q t q t q t 6 6 6 7 4 5 t 2 2 > ------ + ----- + ----- + ----- + ---- + ---- + 2 t + -- + q t 10 3 8 3 8 2 6 2 6 4 2 q t q t q t q t q t q t q

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n188

L11n187 L11n189