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

L11a383 L11a385

Knotscape

This page is passe. Go here instead! The 2-Component Link L11a384 Visit L11a384's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

X12,1,13,2 X14,8,15,7 X16,4,17,3 X20,6,21,5 X18,10,19,9 X10,18,1,17 X4,20,5,19 X22,14,11,13 X6,16,7,15 X2,11,3,12 X8,22,9,21

Gauss Code:

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

Jones Polynomial:

- q-1/2 + 3q1/2 - 8q3/2 + 14q5/2 - 19q7/2 + 22q9/2 - 23q11/2 + 19q13/2 - 15q15/2 + 9q17/2 - 4q19/2 + q21/2

A2 (sl(3)) Invariant:

q-2 - 1 + 4q4 - 4q6 + 2q8 - 4q12 + 4q14 - q16 + 6q18 + 2q20 - q22 + 4q24 - 4q26 + 2q30 - q32

HOMFLY-PT Polynomial:

a-9z3 - a-7z-1 + 2a-7z + a-7z3 - a-7z5 + a-5z-1 - 3a-5z - 6a-5z3 - 3a-5z5 + 3a-3z + a-3z3 - a-3z5 + a-1z + a-1z3

Kauffman Polynomial:

- a-12z2 + 2a-12z4 - a-12z6 + a-11z - 6a-11z3 + 9a-11z5 - 4a-11z7 + a-10z2 - 8a-10z4 + 15a-10z6 - 7a-10z8 + a-9z - 5a-9z3 + 7a-9z5 + 6a-9z7 - 6a-9z9 + 5a-8z2 - 21a-8z4 + 33a-8z6 - 12a-8z8 - 2a-8z10 + a-7z-1 - 5a-7z + 2a-7z3 - 4a-7z5 + 18a-7z7 - 12a-7z9 - a-6 + 10a-6z2 - 28a-6z4 + 33a-6z6 - 13a-6z8 - 2a-6z10 + a-5z-1 - a-5z - 8a-5z3 + 8a-5z5 + 2a-5z7 - 6a-5z9 + 6a-4z2 - 13a-4z4 + 13a-4z6 - 8a-4z8 + 3a-3z - 7a-3z3 + 9a-3z5 - 6a-3z7 - a-2z2 + 4a-2z4 - 3a-2z6 - a-1z + 2a-1z3 - a-1z5

Khovanov Homology:

trqj r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 r = 8 r = 9 j = 22                       1 j = 20                     3   j = 18                   6 1   j = 16                 9 3     j = 14               11 7       j = 12             12 8         j = 10           10 11           j = 8         9 12             j = 6       5 10               j = 4     3 9                 j = 2   1 6                   j = 0   2                     j = -2 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, 384]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 384]]
Out[4]=	PD[X[12, 1, 13, 2], X[14, 8, 15, 7], X[16, 4, 17, 3], X[20, 6, 21, 5], > X[18, 10, 19, 9], X[10, 18, 1, 17], X[4, 20, 5, 19], X[22, 14, 11, 13], > X[6, 16, 7, 15], X[2, 11, 3, 12], X[8, 22, 9, 21]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 3, -7, 4, -9, 2, -11, 5, -6}, > {10, -1, 8, -2, 9, -3, 6, -5, 7, -4, 11, -8}]
In[6]:=	Jones[L][q]
Out[6]=	1 3/2 5/2 7/2 9/2 11/2 -(-------) + 3 Sqrt[q] - 8 q + 14 q - 19 q + 22 q - 23 q + Sqrt[q] 13/2 15/2 17/2 19/2 21/2 > 19 q - 15 q + 9 q - 4 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-2 4 6 8 12 14 16 18 20 22 -1 + q + 4 q - 4 q + 2 q - 4 q + 4 q - q + 6 q + 2 q - q + 24 26 30 32 > 4 q - 4 q + 2 q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 384]][a, z]
Out[8]=	3 3 3 3 3 5 5 5 1 1 2 z 3 z 3 z z z z 6 z z z z 3 z z -(----) + ---- + --- - --- + --- + - + -- + -- - ---- + -- + -- - -- - ---- - -- 7 5 7 5 3 a 9 7 5 3 a 7 5 3 a z a z a a a a a a a a a a
In[9]:=	Kauffman[Link[11, Alternating, 384]][a, z]
Out[9]=	2 2 2 2 -6 1 1 z z 5 z z 3 z z z z 5 z 10 z -a + ---- + ---- + --- + -- - --- - -- + --- - - - --- + --- + ---- + ----- + 7 5 11 9 7 5 3 a 12 10 8 6 a z a z a a a a a a a a a 2 2 3 3 3 3 3 3 4 4 4 6 z z 6 z 5 z 2 z 8 z 7 z 2 z 2 z 8 z 21 z > ---- - -- - ---- - ---- + ---- - ---- - ---- + ---- + ---- - ---- - ----- - 4 2 11 9 7 5 3 a 12 10 8 a a a a a a a a a a 4 4 4 5 5 5 5 5 5 6 28 z 13 z 4 z 9 z 7 z 4 z 8 z 9 z z z > ----- - ----- + ---- + ---- + ---- - ---- + ---- + ---- - -- - --- + 6 4 2 11 9 7 5 3 a 12 a a a a a a a a a 6 6 6 6 6 7 7 7 7 7 15 z 33 z 33 z 13 z 3 z 4 z 6 z 18 z 2 z 6 z > ----- + ----- + ----- + ----- - ---- - ---- + ---- + ----- + ---- - ---- - 10 8 6 4 2 11 9 7 5 3 a a a a a a a a a a 8 8 8 8 9 9 9 10 10 7 z 12 z 13 z 8 z 6 z 12 z 6 z 2 z 2 z > ---- - ----- - ----- - ---- - ---- - ----- - ---- - ----- - ----- 10 8 6 4 9 7 5 8 6 a a a a a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 2 4 1 2 q 4 6 6 2 8 2 6 q + 3 q + ----- + - + -- + 9 q t + 5 q t + 10 q t + 9 q t + 2 2 t t q t 8 3 10 3 10 4 12 4 12 5 14 5 > 12 q t + 10 q t + 11 q t + 12 q t + 8 q t + 11 q t + 14 6 16 6 16 7 18 7 18 8 20 8 22 9 > 7 q t + 9 q t + 3 q t + 6 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a384

L11a383 L11a385