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DrawMorseLink

PD Presentation:

X6172 X12,4,13,3 X18,8,5,7 X16,10,17,9 X14,12,15,11 X10,16,11,15 X8,18,9,17 X2536 X4,14,1,13

Gauss Code:

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

Jones Polynomial:

- q-3/2 + q-1/2 - 3q1/2 + 3q3/2 - 4q5/2 + 4q7/2 - 3q9/2 + 2q11/2 - 2q13/2 + q15/2

A2 (sl(3)) Invariant:

q-6 + q-4 + q-2 + 3 + q2 + q4 + q6 - q12 + q16 + q20 - q24

HOMFLY-PT Polynomial:

a-7z - a-5z - a-5z3 - a-3z3 - a-1z-1 - a-1z - a-1z3 + az-1 + az

Kauffman Polynomial:

- 3a-8z2 + 4a-8z4 - a-8z6 + 2a-7z - 10a-7z3 + 9a-7z5 - 2a-7z7 - a-6z4 + 3a-6z6 - a-6z8 + 2a-5z - 10a-5z3 + 11a-5z5 - 3a-5z7 + 3a-4z2 - 4a-4z4 + 3a-4z6 - a-4z8 + a-3z5 - a-3z7 - a-2z6 - a-1z-1 + 2a-1z - a-1z3 - a-1z5 + 1 - z4 - az-1 + 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 = 16                   1 j = 14                 1   j = 12               1 1   j = 10             2 1     j = 8           2 1       j = 6         2 2         j = 4       1 2           j = 2     2 2             j = 0   1 3               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[9, Alternating, 18]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[9, Alternating, 18]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 4, 13, 3], X[18, 8, 5, 7], X[16, 10, 17, 9], > X[14, 12, 15, 11], X[10, 16, 11, 15], X[8, 18, 9, 17], X[2, 5, 3, 6], > X[4, 14, 1, 13]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -8, 2, -9}, {8, -1, 3, -7, 4, -6, 5, -2, 9, -5, 6, -4, 7, -3}]
In[6]:=	Jones[L][q]
Out[6]=	-(3/2) 1 3/2 5/2 7/2 9/2 11/2 -q + ------- - 3 Sqrt[q] + 3 q - 4 q + 4 q - 3 q + 2 q - Sqrt[q] 13/2 15/2 > 2 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-6 -4 -2 2 4 6 12 16 20 24 3 + q + q + q + q + q + q - q + q + q - q
In[8]:=	HOMFLYPT[Link[9, Alternating, 18]][a, z]
Out[8]=	3 3 3 1 a z z z z z z -(---) + - + -- - -- - - + a z - -- - -- - -- a z z 7 5 a 5 3 a a a a a
In[9]:=	Kauffman[Link[9, Alternating, 18]][a, z]
Out[9]=	2 2 3 3 3 1 a 2 z 2 z 2 z 3 z 3 z 10 z 10 z z 1 - --- - - + --- + --- + --- + 2 a z - ---- + ---- - ----- - ----- - -- - a z z 7 5 a 8 4 7 5 a a a a a a a 4 4 4 5 5 5 5 6 6 6 3 4 4 z z 4 z 9 z 11 z z z z 3 z 3 z > a z - z + ---- - -- - ---- + ---- + ----- + -- - -- - -- + ---- + ---- - 8 6 4 7 5 3 a 8 6 4 a a a a a a a a a 6 7 7 7 8 8 z 2 z 3 z z z z > -- - ---- - ---- - -- - -- - -- 2 7 5 3 6 4 a a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 1 2 4 4 2 6 2 6 3 8 3 3 + 2 q + ----- + - + 2 q t + q t + 2 q t + 2 q t + 2 q t + 2 q t + 4 2 t q t 8 4 10 4 10 5 12 5 12 6 14 6 16 7 > q t + 2 q t + q t + q t + q t + q t + q t

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