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

L11n383 L11n385

Knotscape

This page is passe. Go here instead! The 3-Component Link L11n384 Visit L11n384's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

X6172 X14,7,15,8 X4,15,1,16 X10,6,11,5 X8493 X22,18,19,17 X11,20,12,21 X19,12,20,13 X18,22,5,21 X16,10,17,9 X2,14,3,13

Gauss Code:

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

Jones Polynomial:

- 2q-3 + 6q-2 - 7q-1 + 11 - 10q + 11q2 - 8q3 + 5q4 - 3q5 + q6

A2 (sl(3)) Invariant:

- 2q-10 + q-8 + 3q-6 + 2q-4 + 8q-2 + 5 + 6q2 + 4q4 + 2q8 - 3q10 + q14 - q16 + q18

HOMFLY-PT Polynomial:

a-4 + 2a-4z2 + a-4z4 + a-2z-2 - 3a-2 - 7a-2z2 - 4a-2z4 - a-2z6 - 2z-2 + 3 + 7z2 + 3z4 + a2z-2 - a2 - 2a2z2

Kauffman Polynomial:

a-6z2 - 3a-6z4 + a-6z6 - 2a-5z + 8a-5z3 - 10a-5z5 + 3a-5z7 + 2a-4 - 6a-4z2 + 12a-4z4 - 13a-4z6 + 4a-4z8 - 7a-3z + 21a-3z3 - 16a-3z5 + 2a-3z9 + a-2z-2 + 6a-2 - 26a-2z2 + 41a-2z4 - 31a-2z6 + 9a-2z8 - 2a-1z-1 - 9a-1z + 20a-1z3 - 13a-1z5 + a-1z7 + 2a-1z9 + 2z-2 + 7 - 28z2 + 32z4 - 16z6 + 5z8 - 2az-1 - 5az + 10az3 - 7az5 + 4az7 + a2z-2 + 2a2 - 9a2z2 + 6a2z4 + a2z6 - a3z + 3a3z3

Khovanov Homology:

trqj r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 j = 13                   1 j = 11                 2   j = 9               3 1   j = 7             5 2     j = 5           6 3       j = 3         4 5         j = 1       7 6           j = -1     4 8             j = -3   2 3               j = -5   4                 j = -7 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]=	3
In[3]:=	Show[DrawMorseLink[Link[11, NonAlternating, 384]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 384]]
Out[4]=	PD[X[6, 1, 7, 2], X[14, 7, 15, 8], X[4, 15, 1, 16], X[10, 6, 11, 5], > X[8, 4, 9, 3], X[22, 18, 19, 17], X[11, 20, 12, 21], X[19, 12, 20, 13], > X[18, 22, 5, 21], X[16, 10, 17, 9], X[2, 14, 3, 13]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -11, 5, -3}, {-8, 7, 9, -6}, > {4, -1, 2, -5, 10, -4, -7, 8, 11, -2, 3, -10, 6, -9}]
In[6]:=	Jones[L][q]
Out[6]=	2 6 7 2 3 4 5 6 11 - -- + -- - - - 10 q + 11 q - 8 q + 5 q - 3 q + q 3 2 q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	2 -8 3 2 8 2 4 8 10 14 16 18 5 - --- + q + -- + -- + -- + 6 q + 4 q + 2 q - 3 q + q - q + q 10 6 4 2 q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 384]][a, z]
Out[8]=	2 2 2 -4 3 2 2 1 a 2 2 z 7 z 2 2 4 3 + a - -- - a - -- + ----- + -- + 7 z + ---- - ---- - 2 a z + 3 z + 2 2 2 2 2 4 2 a z a z z a a 4 4 6 z 4 z z > -- - ---- - -- 4 2 2 a a a
In[9]:=	Kauffman[Link[11, NonAlternating, 384]][a, z]
Out[9]=	2 2 6 2 2 1 a 2 2 a 2 z 7 z 9 z 7 + -- + -- + 2 a + -- + ----- + -- - --- - --- - --- - --- - --- - 5 a z - 4 2 2 2 2 2 a z z 5 3 a a a z a z z a a 2 2 2 3 3 3 3 2 z 6 z 26 z 2 2 8 z 21 z 20 z > a z - 28 z + -- - ---- - ----- - 9 a z + ---- + ----- + ----- + 6 4 2 5 3 a a a a a a 4 4 4 5 3 3 3 4 3 z 12 z 41 z 2 4 10 z > 10 a z + 3 a z + 32 z - ---- + ----- + ----- + 6 a z - ----- - 6 4 2 5 a a a a 5 5 6 6 6 7 7 16 z 13 z 5 6 z 13 z 31 z 2 6 3 z z > ----- - ----- - 7 a z - 16 z + -- - ----- - ----- + a z + ---- + -- + 3 a 6 4 2 5 a a a a a a 8 8 9 9 7 8 4 z 9 z 2 z 2 z > 4 a z + 5 z + ---- + ---- + ---- + ---- 4 2 3 a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	8 2 4 2 3 4 3 3 2 - + 7 q + ----- + ----- + ----- + ---- + --- + 6 q t + 4 q t + 5 q t + q 7 3 5 2 3 2 3 q t q t q t q t q t 5 2 5 3 7 3 7 4 9 4 9 5 11 5 13 6 > 6 q t + 3 q t + 5 q t + 2 q t + 3 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n384

L11n383 L11n385