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L11n158 L11n160

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DrawMorseLink

PD Presentation:

X8192 X7,17,8,16 X10,4,11,3 X2,15,3,16 X14,10,15,9 X11,19,12,18 X5,13,6,12 X21,1,22,6 X20,14,21,13 X17,7,18,22 X4,20,5,19

Gauss Code:

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

Jones Polynomial:

- q1/2 + 2q3/2 - 3q5/2 + 2q7/2 - 4q9/2 + 2q11/2 - 2q13/2 + 2q15/2

A2 (sl(3)) Invariant:

q2 + q8 + 2q10 + 3q12 + 2q14 + 4q16 + q18 + q20 - q22 - 2q24 - 2q26 - 2q28 + q30

HOMFLY-PT Polynomial:

a-9z-1 - a-9z - 2a-7z-1 + a-7z + 4a-7z3 + a-7z5 - 2a-5z - 6a-5z3 - 5a-5z5 - a-5z7 + a-3z-1 + 3a-3z + 4a-3z3 + a-3z5

Kauffman Polynomial:

- 2a-10 + a-10z2 + a-9z-1 + a-9z - 5a-8 + 7a-8z2 - 7a-8z4 + 5a-8z6 - a-8z8 + 2a-7z-1 + 3a-7z - 5a-7z3 - a-7z5 + 4a-7z7 - a-7z9 - 3a-6 + 8a-6z2 - 19a-6z4 + 15a-6z6 - 3a-6z8 + 6a-5z - 12a-5z3 + 4a-5z5 + 3a-5z7 - a-5z9 + a-4 + 2a-4z2 - 12a-4z4 + 10a-4z6 - 2a-4z8 - a-3z-1 + 4a-3z - 7a-3z3 + 5a-3z5 - a-3z7

Khovanov Homology:

trqj r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 16               2 j = 14             1 1 j = 12           2 2   j = 10         2 1 1   j = 8       1 3       j = 6     2 2 1       j = 4   1 2           j = 2   1             j = 0 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, 159]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 159]]
Out[4]=	PD[X[8, 1, 9, 2], X[7, 17, 8, 16], X[10, 4, 11, 3], X[2, 15, 3, 16], > X[14, 10, 15, 9], X[11, 19, 12, 18], X[5, 13, 6, 12], X[21, 1, 22, 6], > X[20, 14, 21, 13], X[17, 7, 18, 22], X[4, 20, 5, 19]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -4, 3, -11, -7, 8}, > {-2, -1, 5, -3, -6, 7, 9, -5, 4, 2, -10, 6, 11, -9, -8, 10}]
In[6]:=	Jones[L][q]
Out[6]=	3/2 5/2 7/2 9/2 11/2 13/2 15/2 -Sqrt[q] + 2 q - 3 q + 2 q - 4 q + 2 q - 2 q + 2 q
In[7]:=	A2Invariant[L][q]
Out[7]=	2 8 10 12 14 16 18 20 22 24 26 q + q + 2 q + 3 q + 2 q + 4 q + q + q - q - 2 q - 2 q - 28 30 > 2 q + q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 159]][a, z]
Out[8]=	3 3 3 5 5 1 2 1 z z 2 z 3 z 4 z 6 z 4 z z 5 z ---- - ---- + ---- - -- + -- - --- + --- + ---- - ---- + ---- + -- - ---- + 9 7 3 9 7 5 3 7 5 3 7 5 a z a z a z a a a a a a a a a 5 7 z z > -- - -- 3 5 a a
In[9]:=	Kauffman[Link[11, NonAlternating, 159]][a, z]
Out[9]=	2 2 -2 5 3 -4 1 2 1 z 3 z 6 z 4 z z 7 z --- - -- - -- + a + ---- + ---- - ---- + -- + --- + --- + --- + --- + ---- + 10 8 6 9 7 3 9 7 5 3 10 8 a a a a z a z a z a a a a a a 2 2 3 3 3 4 4 4 5 5 8 z 2 z 5 z 12 z 7 z 7 z 19 z 12 z z 4 z > ---- + ---- - ---- - ----- - ---- - ---- - ----- - ----- - -- + ---- + 6 4 7 5 3 8 6 4 7 5 a a a a a a a a a a 5 6 6 6 7 7 7 8 8 8 9 9 5 z 5 z 15 z 10 z 4 z 3 z z z 3 z 2 z z z > ---- + ---- + ----- + ----- + ---- + ---- - -- - -- - ---- - ---- - -- - -- 3 8 6 4 7 5 3 8 6 4 7 5 a a a a a a a a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 4 4 6 -2 q q 6 8 6 2 8 2 10 2 2 q + 2 q + t + -- + -- + 2 q t + q t + q t + 3 q t + 2 q t + t t 10 3 12 3 10 4 12 4 14 4 14 5 16 5 > q t + 2 q t + q t + 2 q t + q t + q t + 2 q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n159

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