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DrawMorseLink

PD Presentation:

X10,1,11,2 X12,3,13,4 X14,6,15,5 X16,7,17,8 X18,16,19,15 X20,13,9,14 X6,17,7,18 X4,20,5,19 X2,9,3,10 X8,11,1,12

Gauss Code:

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

Jones Polynomial:

q-15/2 - 3q-13/2 + 5q-11/2 - 9q-9/2 + 11q-7/2 - 13q-5/2 + 12q-3/2 - 10q-1/2 + 7q1/2 - 4q3/2 + q5/2

A2 (sl(3)) Invariant:

- q-24 + 2q-20 - q-18 + 2q-16 + 3q-14 - q-12 + 3q-10 + q-6 + q-4 - 2q-2 + 3 - 2q2 + 2q6 - q8

HOMFLY-PT Polynomial:

a-1z3 - az - az3 - az5 - a3z-1 - 2a3z - a3z3 - a3z5 + a5z-1 + 2a5z + 2a5z3 - a7z

Kauffman Polynomial:

- a-2z4 + 3a-1z3 - 4a-1z5 - 2z2 + 8z4 - 7z6 - az3 + 7az5 - 7az7 - 6a2z2 + 13a2z4 - 3a2z6 - 4a2z8 - a3z-1 + 7a3z - 20a3z3 + 27a3z5 - 11a3z7 - a3z9 + a4 - 5a4z2 + a4z4 + 11a4z6 - 7a4z8 - a5z-1 + 10a5z - 26a5z3 + 26a5z5 - 7a5z7 - a5z9 - 3a6z2 + 6a6z6 - 3a6z8 + 3a7z - 10a7z3 + 10a7z5 - 3a7z7 - 2a8z2 + 3a8z4 - a8z6

Khovanov Homology:

trqj r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 j = 6                     1 j = 4                   3   j = 2                 4 1   j = 0               6 3     j = -2             7 5       j = -4           6 5         j = -6         5 7           j = -8       4 6             j = -10     2 6               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[10, Alternating, 88]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 88]]
Out[4]=	PD[X[10, 1, 11, 2], X[12, 3, 13, 4], X[14, 6, 15, 5], X[16, 7, 17, 8], > X[18, 16, 19, 15], X[20, 13, 9, 14], X[6, 17, 7, 18], X[4, 20, 5, 19], > X[2, 9, 3, 10], X[8, 11, 1, 12]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 2, -8, 3, -7, 4, -10}, > {9, -1, 10, -2, 6, -3, 5, -4, 7, -5, 8, -6}]
In[6]:=	Jones[L][q]
Out[6]=	-(15/2) 3 5 9 11 13 12 10 q - ----- + ----- - ---- + ---- - ---- + ---- - ------- + 7 Sqrt[q] - 13/2 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q q 3/2 5/2 > 4 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-24 2 -18 2 3 -12 3 -6 -4 2 2 6 3 - q + --- - q + --- + --- - q + --- + q + q - -- - 2 q + 2 q - 20 16 14 10 2 q q q q q 8 > q
In[8]:=	HOMFLYPT[Link[10, Alternating, 88]][a, z]
Out[8]=	3 5 3 a a 3 5 7 z 3 3 3 5 3 -(--) + -- - a z - 2 a z + 2 a z - a z + -- - a z - a z + 2 a z - z z a 5 3 5 > a z - a z
In[9]:=	Kauffman[Link[10, Alternating, 88]][a, z]
Out[9]=	3 5 4 a a 3 5 7 2 2 2 4 2 6 2 a - -- - -- + 7 a z + 10 a z + 3 a z - 2 z - 6 a z - 5 a z - 3 a z - z z 3 4 8 2 3 z 3 3 3 5 3 7 3 4 z > 2 a z + ---- - a z - 20 a z - 26 a z - 10 a z + 8 z - -- + a 2 a 5 2 4 4 4 8 4 4 z 5 3 5 5 5 > 13 a z + a z + 3 a z - ---- + 7 a z + 27 a z + 26 a z + a 7 5 6 2 6 4 6 6 6 8 6 7 > 10 a z - 7 z - 3 a z + 11 a z + 6 a z - a z - 7 a z - 3 7 5 7 7 7 2 8 4 8 6 8 3 9 5 9 > 11 a z - 7 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]=	5 1 2 1 3 2 6 4 6 6 + -- + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 2 16 7 14 6 12 6 12 5 10 5 10 4 8 4 8 3 q q t q t q t q t q t q t q t q t 5 7 6 5 7 2 2 2 4 2 6 3 > ----- + ----- + ----- + ---- + ---- + 3 t + 4 q t + q t + 3 q t + q t 6 3 6 2 4 2 4 2 q t q t q t q t q t

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