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DrawMorseLink

PD Presentation:

X8192 X2,9,3,10 X10,3,11,4 X6718 X16,11,17,12 X14,6,15,5 X4,16,5,15 X20,17,7,18 X18,14,19,13 X12,20,13,19

Gauss Code:

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

Jones Polynomial:

- q-15/2 + 3q-13/2 - 5q-11/2 + 7q-9/2 - 9q-7/2 + 8q-5/2 - 8q-3/2 + 6q-1/2 - 4q1/2 + 2q3/2 - q5/2

A2 (sl(3)) Invariant:

q-22 - q-20 + q-18 + 2q-12 - q-10 + 3q-8 + q-4 + q-2 - 1 + q2 + q6 + q8

HOMFLY-PT Polynomial:

- a-1z-1 - 3a-1z - a-1z3 + 2az-1 + 8az + 8az3 + 2az5 - 2a3z-1 - 8a3z - 9a3z3 - 5a3z5 - a3z7 + a5z-1 + 2a5z + 3a5z3 + a5z5

Kauffman Polynomial:

- a-1z-1 + 5a-1z - 8a-1z3 + 5a-1z5 - a-1z7 + 5z2 - 12z4 + 9z6 - 2z8 - 2az-1 + 14az - 29az3 + 18az5 - az7 - az9 - a2 + 7a2z2 - 22a2z4 + 22a2z6 - 6a2z8 - 2a3z-1 + 12a3z - 31a3z3 + 29a3z5 - 6a3z7 - a3z9 + 7a4z6 - 4a4z8 - a5z-1 + 2a5z - 5a5z3 + 11a5z5 - 6a5z7 - a6z2 + 7a6z4 - 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 r = 4 j = 6                     1 j = 4                   1   j = 2                 3 1   j = 0               3 1     j = -2             5 3       j = -4           4 4         j = -6         5 4           j = -8       3 5             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[10, Alternating, 66]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 66]]
Out[4]=	PD[X[8, 1, 9, 2], X[2, 9, 3, 10], X[10, 3, 11, 4], X[6, 7, 1, 8], > X[16, 11, 17, 12], X[14, 6, 15, 5], X[4, 16, 5, 15], X[20, 17, 7, 18], > X[18, 14, 19, 13], X[12, 20, 13, 19]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 3, -7, 6, -4}, > {4, -1, 2, -3, 5, -10, 9, -6, 7, -5, 8, -9, 10, -8}]
In[6]:=	Jones[L][q]
Out[6]=	-(15/2) 3 5 7 9 8 8 6 -q + ----- - ----- + ---- - ---- + ---- - ---- + ------- - 4 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 > 2 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-22 -20 -18 2 -10 3 -4 -2 2 6 8 -1 + q - q + q + --- - q + -- + q + q + q + q + q 12 8 q q
In[8]:=	HOMFLYPT[Link[10, Alternating, 66]][a, z]
Out[8]=	3 5 3 1 2 a 2 a a 3 z 3 5 z 3 -(---) + --- - ---- + -- - --- + 8 a z - 8 a z + 2 a z - -- + 8 a z - a z z z z a a 3 3 5 3 5 3 5 5 5 3 7 > 9 a z + 3 a z + 2 a z - 5 a z + a z - a z
In[9]:=	Kauffman[Link[10, Alternating, 66]][a, z]
Out[9]=	3 5 2 1 2 a 2 a a 5 z 3 5 7 2 -a - --- - --- - ---- - -- + --- + 14 a z + 12 a z + 2 a z - a z + 5 z + a z z z z a 3 2 2 6 2 8 2 8 z 3 3 3 5 3 7 3 > 7 a z - a z + a z - ---- - 29 a z - 31 a z - 5 a z + 4 a z - a 5 9 3 4 2 4 6 4 8 4 5 z 5 3 5 > a z - 12 z - 22 a z + 7 a z - 3 a z + ---- + 18 a z + 29 a z + a 7 5 5 7 5 6 2 6 4 6 6 6 z 7 > 11 a z - 5 a z + 9 z + 22 a z + 7 a z - 6 a z - -- - a z - a 3 7 5 7 8 2 8 4 8 9 3 9 > 6 a z - 6 a z - 2 z - 6 a z - 4 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	4 5 1 2 1 3 2 4 3 5 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 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 4 3 t 2 2 2 2 3 4 3 6 4 > ----- + ---- + ---- + 3 t + --- + t + 3 q t + q t + q t + q t 6 2 6 4 2 q t q t q t q

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