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L11n217 L11n219

Knotscape

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DrawMorseLink

PD Presentation:

X10,1,11,2 X8,9,1,10 X3,12,4,13 X22,16,9,15 X17,3,18,2 X21,4,22,5 X5,15,6,14 X13,21,14,20 X16,12,17,11 X19,7,20,6 X7,19,8,18

Gauss Code:

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

Jones Polynomial:

- q-3/2 + q-1/2 - q1/2 - q5/2 - q7/2 + q9/2 - q11/2 + q13/2

A2 (sl(3)) Invariant:

q-6 + 1 + q2 + 3q4 + 2q6 + 3q8 + q10 - q16 - q18 - q20

HOMFLY-PT Polynomial:

a-5z-1 + 3a-5z + a-5z3 - 3a-3z-1 - 6a-3z - 5a-3z3 - a-3z5 + 2a-1z-1 + 2a-1z + az

Kauffman Polynomial:

- a-6 + 11a-6z2 - 15a-6z4 + 7a-6z6 - a-6z8 + a-5z-1 - 6a-5z + 13a-5z3 - 15a-5z5 + 7a-5z7 - a-5z9 - 3a-4 + 18a-4z2 - 28a-4z4 + 14a-4z6 - 2a-4z8 + 3a-3z-1 - 11a-3z + 15a-3z3 - 15a-3z5 + 7a-3z7 - a-3z9 - 3a-2 + 10a-2z2 - 14a-2z4 + 7a-2z6 - a-2z8 + 2a-1z-1 - 3a-1z + a-1z3 + 3z2 - z4 + 2az - az3

Khovanov Homology:

trqj r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 j = 14                   1 j = 12                     j = 10               1 1   j = 8           1 1       j = 6         1   1       j = 4         2 1         j = 2     2 1             j = 0   1 2 1             j = -2                     j = -4 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, 218]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 218]]
Out[4]=	PD[X[10, 1, 11, 2], X[8, 9, 1, 10], X[3, 12, 4, 13], X[22, 16, 9, 15], > X[17, 3, 18, 2], X[21, 4, 22, 5], X[5, 15, 6, 14], X[13, 21, 14, 20], > X[16, 12, 17, 11], X[19, 7, 20, 6], X[7, 19, 8, 18]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, 5, -3, 6, -7, 10, -11, -2}, > {2, -1, 9, 3, -8, 7, 4, -9, -5, 11, -10, 8, -6, -4}]
In[6]:=	Jones[L][q]
Out[6]=	-(3/2) 1 5/2 7/2 9/2 11/2 13/2 -q + ------- - Sqrt[q] - q - q + q - q + q Sqrt[q]
In[7]:=	A2Invariant[L][q]
Out[7]=	-6 2 4 6 8 10 16 18 20 1 + q + q + 3 q + 2 q + 3 q + q - q - q - q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 218]][a, z]
Out[8]=	3 3 5 1 3 2 3 z 6 z 2 z z 5 z z ---- - ---- + --- + --- - --- + --- + a z + -- - ---- - -- 5 3 a z 5 3 a 5 3 3 a z a z a a a a a
In[9]:=	Kauffman[Link[11, NonAlternating, 218]][a, z]
Out[9]=	2 -6 3 3 1 3 2 6 z 11 z 3 z 2 11 z -a - -- - -- + ---- + ---- + --- - --- - ---- - --- + 2 a z + 3 z + ----- + 4 2 5 3 a z 5 3 a 6 a a a z a z a a a 2 2 3 3 3 4 4 4 18 z 10 z 13 z 15 z z 3 4 15 z 28 z 14 z > ----- + ----- + ----- + ----- + -- - a z - z - ----- - ----- - ----- - 4 2 5 3 a 6 4 2 a a a a a a a 5 5 6 6 6 7 7 8 8 8 9 9 15 z 15 z 7 z 14 z 7 z 7 z 7 z z 2 z z z z > ----- - ----- + ---- + ----- + ---- + ---- + ---- - -- - ---- - -- - -- - -- 5 3 6 4 2 5 3 6 4 2 5 3 a a a a a a a a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 1 2 4 2 6 2 4 3 8 3 6 4 2 + 2 q + ----- + - + t + q t + 2 q t + q t + q t + q t + q t + 4 2 t q t 8 4 10 5 10 6 14 7 > q t + q t + q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n218

L11n217 L11n219