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

L11n254 L11n256

Knotscape

This page is passe. Go here instead! The 3-Component Link L11n255 Visit L11n255's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

X6172 X3,11,4,10 X11,16,12,17 X21,18,22,19 X13,20,14,21 X19,12,20,13 X17,22,18,9 X15,8,16,5 X7,14,8,15 X2536 X9,1,10,4

Gauss Code:

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

Jones Polynomial:

- 3q-9 + 6q-8 - 9q-7 + 11q-6 - 11q-5 + 12q-4 - 7q-3 + 6q-2 - 2q-1 + 1

A2 (sl(3)) Invariant:

- q-32 - q-30 - 4q-28 - 2q-26 - q-24 - 3q-22 + 3q-20 + q-18 + 6q-16 + 7q-14 + 5q-12 + 8q-10 + 2q-8 + 4q-6 + 2q-4 + 1

HOMFLY-PT Polynomial:

a2z-2 + 3a2 + 3a2z2 + a2z4 - a4z-2 - 2a4 - 3a4z2 - 3a4z4 - a4z6 - 2a6z-2 - 5a6 - 4a6z2 - 3a6z4 - a6z6 + 3a8z-2 + 5a8 + 3a8z2 + a8z4 - a10z-2 - a10

Kauffman Polynomial:

a2z-2 - 4a2 + 6a2z2 - 4a2z4 + a2z6 - 2a3z-1 + 4a3z + a3z3 - 5a3z5 + 2a3z7 + a4z-2 - 3a4 + 9a4z2 - 9a4z4 - a4z6 + 2a4z8 + 2a5z-1 - 10a5z + 18a5z3 - 19a5z5 + 5a5z7 + a5z9 - 2a6z-2 + 4a6 - 3a6z2 + 8a6z4 - 15a6z6 + 7a6z8 + 10a7z-1 - 34a7z + 46a7z3 - 34a7z5 + 10a7z7 + a7z9 - 3a8z-2 + 5a8 - 8a8z2 + 13a8z4 - 10a8z6 + 5a8z8 + 8a9z-1 - 27a9z + 35a9z3 - 20a9z5 + 7a9z7 - a10z-2 + a10 - 2a10z2 + 3a10z6 + 2a11z-1 - 7a11z + 6a11z3

Khovanov Homology:

trqj r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 j = 1                   1 j = -1                 1   j = -3               5 1   j = -5             5 4     j = -7           7 2       j = -9         4 5         j = -11       7 7           j = -13     3 5             j = -15   3 6               j = -17   3                 j = -19 3

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[11, NonAlternating, 255]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 255]]
Out[4]=	PD[X[6, 1, 7, 2], X[3, 11, 4, 10], X[11, 16, 12, 17], X[21, 18, 22, 19], > X[13, 20, 14, 21], X[19, 12, 20, 13], X[17, 22, 18, 9], X[15, 8, 16, 5], > X[7, 14, 8, 15], X[2, 5, 3, 6], X[9, 1, 10, 4]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, -2, 11}, {10, -1, -9, 8}, > {-11, 2, -3, 6, -5, 9, -8, 3, -7, 4, -6, 5, -4, 7}]
In[6]:=	Jones[L][q]
Out[6]=	3 6 9 11 11 12 7 6 2 1 - -- + -- - -- + -- - -- + -- - -- + -- - - 9 8 7 6 5 4 3 2 q q q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-32 -30 4 2 -24 3 3 -18 6 7 5 8 1 - q - q - --- - --- - q - --- + --- + q + --- + --- + --- + --- + 28 26 22 20 16 14 12 10 q q q q q q q q 2 4 2 > -- + -- + -- 8 6 4 q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 255]][a, z]
Out[8]=	2 4 6 8 10 2 4 6 8 10 a a 2 a 3 a a 2 2 3 a - 2 a - 5 a + 5 a - a + -- - -- - ---- + ---- - --- + 3 a z - 2 2 2 2 2 z z z z z 4 2 6 2 8 2 2 4 4 4 6 4 8 4 4 6 > 3 a z - 4 a z + 3 a z + a z - 3 a z - 3 a z + a z - a z - 6 6 > a z
In[9]:=	Kauffman[Link[11, NonAlternating, 255]][a, z]
Out[9]=	2 4 6 8 10 3 5 2 4 6 8 10 a a 2 a 3 a a 2 a 2 a -4 a - 3 a + 4 a + 5 a + a + -- + -- - ---- - ---- - --- - ---- + ---- + 2 2 2 2 2 z z z z z z z 7 9 11 10 a 8 a 2 a 3 5 7 9 11 > ----- + ---- + ----- + 4 a z - 10 a z - 34 a z - 27 a z - 7 a z + z z z 2 2 4 2 6 2 8 2 10 2 3 3 5 3 > 6 a z + 9 a z - 3 a z - 8 a z - 2 a z + a z + 18 a z + 7 3 9 3 11 3 2 4 4 4 6 4 8 4 > 46 a z + 35 a z + 6 a z - 4 a z - 9 a z + 8 a z + 13 a z - 3 5 5 5 7 5 9 5 2 6 4 6 6 6 > 5 a z - 19 a z - 34 a z - 20 a z + a z - a z - 15 a z - 8 6 10 6 3 7 5 7 7 7 9 7 4 8 > 10 a z + 3 a z + 2 a z + 5 a z + 10 a z + 7 a z + 2 a z + 6 8 8 8 5 9 7 9 > 7 a z + 5 a z + a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	4 5 3 3 3 6 3 5 7 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 5 3 19 7 17 6 15 6 15 5 13 5 13 4 11 4 q q q t q t q t q t q t q t q t 7 4 5 7 2 5 t t 2 > ------ + ----- + ----- + ----- + ---- + ---- + -- + - + q t 11 3 9 3 9 2 7 2 7 5 3 q 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 L11n255

L11n254 L11n256