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L9a46 L9a48

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-8 + 3q-7 - 5q-6 + 8q-5 - 8q-4 + 10q-3 - 7q-2 + 6q-1 - 3 + q

A2 (sl(3)) Invariant:

- q-26 - q-24 + 2q-22 + 2q-18 + 5q-16 + 2q-14 + 5q-12 + 3q-10 + 4q-8 + 4q-6 + 3q-2 - 1 - q2 + q4

HOMFLY-PT Polynomial:

z2 + a2z-2 + 3a2 + a2z2 - a2z4 - 2a4z-2 - 5a4 - 4a4z2 - 2a4z4 + a6z-2 + 3a6 + 3a6z2 - a8

Kauffman Polynomial:

- z2 + z4 - 3az3 + 3az5 + a2z-2 - 4a2 + 7a2z2 - 8a2z4 + 5a2z6 - 2a3z-1 + 3a3z - a3z3 - 3a3z5 + 4a3z7 + 2a4z-2 - 9a4 + 24a4z2 - 26a4z4 + 10a4z6 + a4z8 - 2a5z-1 + 5a5z - 10a5z5 + 7a5z7 + a6z-2 - 8a6 + 21a6z2 - 24a6z4 + 8a6z6 + a6z8 + 3a7z - 4a7z3 - 3a7z5 + 3a7z7 - 2a8 + 5a8z2 - 7a8z4 + 3a8z6 + a9z - 2a9z3 + a9z5

Khovanov Homology:

trqj r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 j = 3                   1 j = 1                 2   j = -1               4 1   j = -3             4 3     j = -5           6 3       j = -7         4 6         j = -9       4 4           j = -11     2 5             j = -13   1 3               j = -15   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[9, Alternating, 47]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[9, Alternating, 47]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 3, 13, 4], X[10, 13, 5, 14], X[18, 15, 11, 16], > X[14, 7, 15, 8], X[8, 18, 9, 17], X[16, 10, 17, 9], X[2, 5, 3, 6], > X[4, 11, 1, 12]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -8, 2, -9}, {8, -1, 5, -6, 7, -3}, {9, -2, 3, -5, 4, -7, 6, -4}]
In[6]:=	Jones[L][q]
Out[6]=	-8 3 5 8 8 10 7 6 -3 - q + -- - -- + -- - -- + -- - -- + - + q 7 6 5 4 3 2 q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-26 -24 2 2 5 2 5 3 4 4 3 2 4 -1 - q - q + --- + --- + --- + --- + --- + --- + -- + -- + -- - q + q 22 18 16 14 12 10 8 6 2 q q q q q q q q q
In[8]:=	HOMFLYPT[Link[9, Alternating, 47]][a, z]
Out[8]=	2 4 6 2 4 6 8 a 2 a a 2 2 2 4 2 6 2 3 a - 5 a + 3 a - a + -- - ---- + -- + z + a z - 4 a z + 3 a z - 2 2 2 z z z 2 4 4 4 > a z - 2 a z
In[9]:=	Kauffman[Link[9, Alternating, 47]][a, z]
Out[9]=	2 4 6 3 5 2 4 6 8 a 2 a a 2 a 2 a 3 5 -4 a - 9 a - 8 a - 2 a + -- + ---- + -- - ---- - ---- + 3 a z + 5 a z + 2 2 2 z z z z z 7 9 2 2 2 4 2 6 2 8 2 3 > 3 a z + a z - z + 7 a z + 24 a z + 21 a z + 5 a z - 3 a z - 3 3 7 3 9 3 4 2 4 4 4 6 4 8 4 > a z - 4 a z - 2 a z + z - 8 a z - 26 a z - 24 a z - 7 a z + 5 3 5 5 5 7 5 9 5 2 6 4 6 > 3 a z - 3 a z - 10 a z - 3 a z + a z + 5 a z + 10 a z + 6 6 8 6 3 7 5 7 7 7 4 8 6 8 > 8 a z + 3 a z + 4 a z + 7 a z + 3 a z + a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	3 4 1 2 1 3 2 5 4 4 -- + - + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 3 q 17 7 15 6 13 6 13 5 11 5 11 4 9 4 9 3 q q t q t q t q t q t q t q t q t 4 6 6 3 4 t 3 2 > ----- + ----- + ----- + ---- + ---- + - + 2 q t + q t 7 3 7 2 5 2 5 3 q q t q t q t q t q t

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