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

L11n131 L11n133

Knotscape

This page is passe. Go here instead! The 2-Component Link L11n132 Visit L11n132's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

X8192 X18,11,19,12 X3,10,4,11 X17,3,18,2 X12,5,13,6 X6718 X9,16,10,17 X13,20,14,21 X15,22,16,7 X4,20,5,19 X21,14,22,15

Gauss Code:

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

Jones Polynomial:

q-21/2 - q-19/2 + q-17/2 - q-15/2 + q-13/2 - q-11/2 - q-7/2 - q-5/2

A2 (sl(3)) Invariant:

- q-32 - q-30 - 2q-28 - q-26 + q-20 + q-18 + 3q-16 + 3q-14 + 3q-12 + 2q-10 + q-8

HOMFLY-PT Polynomial:

- 3a5z-1 - 9a5z - 6a5z3 - a5z5 + 5a7z-1 + 11a7z + 6a7z3 + a7z5 - 2a9z-1 - 3a9z - a9z3

Kauffman Polynomial:

3a5z-1 - 10a5z + 6a5z3 - a5z5 - 5a6 + 10a6z2 - 6a6z4 + a6z6 + 5a7z-1 - 15a7z + 16a7z3 - 7a7z5 + a7z7 - 5a8 + 13a8z2 - 7a8z4 + a8z6 + 2a9z-1 - 4a9z + 4a9z3 - a9z5 - 3a10z2 + 4a10z4 - a10z6 + a11z - 6a11z3 + 5a11z5 - a11z7 + a12 - 6a12z2 + 5a12z4 - a12z6

Khovanov Homology:

trqj 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               1   1 j = -8             1 2     j = -10           1         j = -12         1 2 1       j = -14       1 1           j = -16       1 1           j = -18   1 1               j = -20                     j = -22 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, NonAlternating, 132]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 132]]
Out[4]=	PD[X[8, 1, 9, 2], X[18, 11, 19, 12], X[3, 10, 4, 11], X[17, 3, 18, 2], > X[12, 5, 13, 6], X[6, 7, 1, 8], X[9, 16, 10, 17], X[13, 20, 14, 21], > X[15, 22, 16, 7], X[4, 20, 5, 19], X[21, 14, 22, 15]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, 4, -3, -10, 5, -6}, > {6, -1, -7, 3, 2, -5, -8, 11, -9, 7, -4, -2, 10, 8, -11, 9}]
In[6]:=	Jones[L][q]
Out[6]=	-(21/2) -(19/2) -(17/2) -(15/2) -(13/2) -(11/2) -(7/2) q - q + q - q + q - q - q - -(5/2) > q
In[7]:=	A2Invariant[L][q]
Out[7]=	-32 -30 2 -26 -20 -18 3 3 3 2 -8 -q - q - --- - q + q + q + --- + --- + --- + --- + q 28 16 14 12 10 q q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 132]][a, z]
Out[8]=	5 7 9 -3 a 5 a 2 a 5 7 9 5 3 7 3 9 3 ----- + ---- - ---- - 9 a z + 11 a z - 3 a z - 6 a z + 6 a z - a z - z z z 5 5 7 5 > a z + a z
In[9]:=	Kauffman[Link[11, NonAlternating, 132]][a, z]
Out[9]=	5 7 9 6 8 12 3 a 5 a 2 a 5 7 9 11 -5 a - 5 a + a + ---- + ---- + ---- - 10 a z - 15 a z - 4 a z + a z + z z z 6 2 8 2 10 2 12 2 5 3 7 3 9 3 > 10 a z + 13 a z - 3 a z - 6 a z + 6 a z + 16 a z + 4 a z - 11 3 6 4 8 4 10 4 12 4 5 5 7 5 > 6 a z - 6 a z - 7 a z + 4 a z + 5 a z - a z - 7 a z - 9 5 11 5 6 6 8 6 10 6 12 6 7 7 11 7 > a z + 5 a z + a z + a z - a z - a z + a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	-6 -4 1 1 1 1 1 1 1 q + q + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 22 9 18 8 18 7 16 6 14 6 16 5 14 5 q t q t q t q t q t q t q t 1 2 1 1 1 2 1 > ------ + ------ + ------ + ------ + ----- + ----- + ----- 12 5 12 4 10 4 12 3 8 3 8 2 6 2 q t q t q t q t q t q t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n132

L11n131 L11n133