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DrawMorseLink

PD Presentation:

X6172 X10,3,11,4 X11,19,12,18 X7,17,8,16 X17,9,18,8 X13,21,14,20 X15,5,16,22 X19,13,20,12 X21,15,22,14 X2536 X4,9,1,10

Gauss Code:

{{1, -10, 2, -11}, {10, -1, -4, 5, 11, -2, -3, 8, -6, 9, -7, 4, -5, 3, -8, 6, -9, 7}}

Jones Polynomial:

- q-3/2 - q1/2 - q3/2 + q5/2 - 2q7/2 + 3q9/2 - 3q11/2 + 2q13/2 - q15/2 + q17/2

A2 (sl(3)) Invariant:

q-4 + 2q-2 + 3 + 3q2 + 4q4 + 2q6 + 2q8 - 2q12 - q14 - 2q16 - q20 - q22 - q26

HOMFLY-PT Polynomial:

2a-7z + a-7z3 + 3a-5z-1 + 5a-5z + 2a-5z3 - 7a-3z-1 - 19a-3z - 20a-3z3 - 8a-3z5 - a-3z7 + 4a-1z-1 + 10a-1z + 6a-1z3 + a-1z5

Kauffman Polynomial:

- a-10 + 3a-10z2 - a-10z4 + 2a-9z3 - a-9z5 - a-8z2 + 2a-8z4 - a-8z6 + 2a-7z - 7a-7z3 + 4a-7z5 - a-7z7 - 3a-6z2 + 2a-6z4 - a-6z6 + 3a-5z-1 - 7a-5z + 3a-5z3 - a-5z5 - 7a-4 + 26a-4z2 - 26a-4z4 + 9a-4z6 - a-4z8 + 7a-3z-1 - 30a-3z + 49a-3z3 - 34a-3z5 + 10a-3z7 - a-3z9 - 7a-2 + 25a-2z2 - 25a-2z4 + 9a-2z6 - a-2z8 + 4a-1z-1 - 21a-1z + 37a-1z3 - 28a-1z5 + 9a-1z7 - a-1z9

Khovanov Homology:

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 j = 18                       1 j = 16                         j = 14                   2 1   j = 12                 1       j = 10               2 2       j = 8           1 1 1         j = 6           1 2           j = 4       1 2 1             j = 2         2               j = 0     1                   j = -2 1                       j = -4 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[11, NonAlternating, 74]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 74]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[11, 19, 12, 18], X[7, 17, 8, 16], > X[17, 9, 18, 8], X[13, 21, 14, 20], X[15, 5, 16, 22], X[19, 13, 20, 12], > X[21, 15, 22, 14], X[2, 5, 3, 6], X[4, 9, 1, 10]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, -4, 5, 11, -2, -3, 8, -6, 9, -7, 4, -5, 3, > -8, 6, -9, 7}]
In[6]:=	Jones[L][q]
Out[6]=	-(3/2) 3/2 5/2 7/2 9/2 11/2 13/2 -q - Sqrt[q] - q + q - 2 q + 3 q - 3 q + 2 q - 15/2 17/2 > q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-4 2 2 4 6 8 12 14 16 20 22 26 3 + q + -- + 3 q + 4 q + 2 q + 2 q - 2 q - q - 2 q - q - q - q 2 q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 74]][a, z]
Out[8]=	3 3 3 3 5 3 7 4 2 z 5 z 19 z 10 z z 2 z 20 z 6 z 8 z ---- - ---- + --- + --- + --- - ---- + ---- + -- + ---- - ----- + ---- - ---- + 5 3 a z 7 5 3 a 7 5 3 a 3 a z a z a a a a a a a 5 7 z z > -- - -- a 3 a
In[9]:=	Kauffman[Link[11, NonAlternating, 74]][a, z]
Out[9]=	2 2 -10 7 7 3 7 4 2 z 7 z 30 z 21 z 3 z z -a - -- - -- + ---- + ---- + --- + --- - --- - ---- - ---- + ---- - -- - 4 2 5 3 a z 7 5 3 a 10 8 a a a z a z a a a a a 2 2 2 3 3 3 3 3 4 4 3 z 26 z 25 z 2 z 7 z 3 z 49 z 37 z z 2 z > ---- + ----- + ----- + ---- - ---- + ---- + ----- + ----- - --- + ---- + 6 4 2 9 7 5 3 a 10 8 a a a a a a a a a 4 4 4 5 5 5 5 5 6 6 6 2 z 26 z 25 z z 4 z z 34 z 28 z z z 9 z > ---- - ----- - ----- - -- + ---- - -- - ----- - ----- - -- - -- + ---- + 6 4 2 9 7 5 3 a 8 6 4 a a a a a a a a a a 6 7 7 7 8 8 9 9 9 z z 10 z 9 z z z z z > ---- - -- + ----- + ---- - -- - -- - -- - -- 2 7 3 a 4 2 3 a a a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	4 2 4 1 1 -2 q 4 6 8 6 2 8 2 2 q + 2 q + ----- + ----- + t + -- + q t + q t + q t + 2 q t + q t + 4 4 2 4 t q t q t 8 3 10 3 10 4 12 4 14 5 14 6 18 7 > q t + 2 q t + 2 q t + q t + 2 q t + q t + q t

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