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DrawMorseLink

PD Presentation:

X8192 X2,9,3,10 X10,3,11,4 X16,12,17,11 X12,6,13,5 X4,17,5,18 X14,7,15,8 X18,13,7,14 X6,16,1,15

Gauss Code:

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

Jones Polynomial:

- q-15/2 + 3q-13/2 - 5q-11/2 + 7q-9/2 - 8q-7/2 + 7q-5/2 - 7q-3/2 + 4q-1/2 - 3q1/2 + q3/2

A2 (sl(3)) Invariant:

q-22 - q-20 + q-18 - q-16 - q-14 + q-12 - q-10 + 4q-8 + q-6 + 3q-4 + 2q-2 + q2 - q4

HOMFLY-PT Polynomial:

- az-1 + az + 3az3 + az5 + a3z-1 - 4a3z - 8a3z3 - 5a3z5 - a3z7 + 2a5z + 3a5z3 + a5z5

Kauffman Polynomial:

- 2z2 + 3z4 - z6 + az-1 + 2az - 11az3 + 11az5 - 3az7 - a2 - 5a2z2 + 5a2z4 + 3a2z6 - 2a2z8 + a3z-1 + 4a3z - 19a3z3 + 24a3z5 - 8a3z7 - 7a4z2 + 13a4z4 - 2a4z6 - 2a4z8 + a5z - 3a5z3 + 8a5z5 - 5a5z7 - 3a6z2 + 8a6z4 - 6a6z6 - a7z + 4a7z3 - 5a7z5 + a8z2 - 3a8z4 - a9z3

Khovanov Homology:

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

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