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L10n85 L10n87

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DrawMorseLink

PD Presentation:

X6172 X10,3,11,4 X13,20,14,15 X16,8,17,7 X8,16,9,15 X11,18,12,19 X19,12,20,13 X17,14,18,5 X2536 X4,9,1,10

Gauss Code:

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

Jones Polynomial:

q-8 - 2q-7 + 4q-6 - 3q-5 + 5q-4 - 3q-3 + 3q-2 - 2q-1 + 1

A2 (sl(3)) Invariant:

2q-24 + 2q-22 + 4q-20 + 4q-18 + 4q-16 + 5q-14 + 2q-12 + 3q-10 + 1

HOMFLY-PT Polynomial:

a2 + 3a2z2 + a2z4 + a4z-2 + a4 - 3a4z2 - 4a4z4 - a4z6 - 2a6z-2 - 2a6 + 2a6z2 + a6z4 + a8z-2

Kauffman Polynomial:

- a2 + 4a2z2 - 4a2z4 + a2z6 + 7a3z3 - 8a3z5 + 2a3z7 - a4z-2 + 2a4 + a4z2 - 2a4z4 - 2a4z6 + a4z8 + 2a5z-1 - 5a5z + 9a5z3 - 10a5z5 + 3a5z7 - 2a6z-2 + 6a6 - 10a6z2 + 6a6z4 - 3a6z6 + a6z8 + 2a7z-1 - 5a7z + 4a7z3 - 2a7z5 + a7z7 - a8z-2 + 4a8 - 6a8z2 + 4a8z4 + 2a9z3 + a10z2

Khovanov Homology:

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 j = 1                 1 j = -1               1   j = -3             2 1   j = -5           2 2     j = -7         3 1       j = -9       1 3         j = -11     3 2           j = -13     2             j = -15 1 2               j = -17 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]=	3
In[3]:=	Show[DrawMorseLink[Link[10, NonAlternating, 86]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, NonAlternating, 86]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[13, 20, 14, 15], X[16, 8, 17, 7], > X[8, 16, 9, 15], X[11, 18, 12, 19], X[19, 12, 20, 13], X[17, 14, 18, 5], > X[2, 5, 3, 6], X[4, 9, 1, 10]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 2, -10}, {5, -4, -8, 6, -7, 3}, > {9, -1, 4, -5, 10, -2, -6, 7, -3, 8}]
In[6]:=	Jones[L][q]
Out[6]=	-8 2 4 3 5 3 3 2 1 + q - -- + -- - -- + -- - -- + -- - - 7 6 5 4 3 2 q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	2 2 4 4 4 5 2 3 1 + --- + --- + --- + --- + --- + --- + --- + --- 24 22 20 18 16 14 12 10 q q q q q q q q
In[8]:=	HOMFLYPT[Link[10, NonAlternating, 86]][a, z]
Out[8]=	4 6 8 2 4 6 a 2 a a 2 2 4 2 6 2 2 4 a + a - 2 a + -- - ---- + -- + 3 a z - 3 a z + 2 a z + a z - 2 2 2 z z z 4 4 6 4 4 6 > 4 a z + a z - a z
In[9]:=	Kauffman[Link[10, NonAlternating, 86]][a, z]
Out[9]=	4 6 8 5 7 2 4 6 8 a 2 a a 2 a 2 a 5 7 -a + 2 a + 6 a + 4 a - -- - ---- - -- + ---- + ---- - 5 a z - 5 a z + 2 2 2 z z z z z 2 2 4 2 6 2 8 2 10 2 3 3 5 3 > 4 a z + a z - 10 a z - 6 a z + a z + 7 a z + 9 a z + 7 3 9 3 2 4 4 4 6 4 8 4 3 5 > 4 a z + 2 a z - 4 a z - 2 a z + 6 a z + 4 a z - 8 a z - 5 5 7 5 2 6 4 6 6 6 3 7 5 7 > 10 a z - 2 a z + a z - 2 a z - 3 a z + 2 a z + 3 a z + 7 7 4 8 6 8 > a z + a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	2 2 1 1 2 2 3 2 1 3 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 5 3 17 6 15 6 15 5 13 4 11 4 11 3 9 3 9 2 q q q t q t q t q t q t q t q t q t 3 1 2 t t 2 > ----- + ---- + ---- + -- + - + q t 7 2 7 5 3 q q t q t q t q

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