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DrawMorseLink

PD Presentation:

X6172 X20,7,21,8 X4,21,1,22 X9,14,10,15 X3849 X5,13,6,12 X13,5,14,22 X15,18,16,19 X11,17,12,16 X17,11,18,10 X19,2,20,3

Gauss Code:

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

Jones Polynomial:

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

A2 (sl(3)) Invariant:

- q-24 + q-22 + 2q-18 + 2q-16 + q-14 + 3q-12 - q-10 + 2q-8 - q-6 - q-4 - 1 + q2 + q6 + q8

HOMFLY-PT Polynomial:

- a-1z-1 - 3a-1z - a-1z3 + 3az-1 + 8az + 8az3 + 2az5 - 4a3z-1 - 10a3z - 9a3z3 - 5a3z5 - a3z7 + 2a5z-1 + 4a5z + 4a5z3 + a5z5 - a7z

Kauffman Polynomial:

- a-1z-1 + 5a-1z - 8a-1z3 + 5a-1z5 - a-1z7 - 1 + 6z2 - 12z4 + 9z6 - 2z8 - 3az-1 + 17az - 31az3 + 19az5 - az7 - az9 - 3a2 + 13a2z2 - 27a2z4 + 24a2z6 - 6a2z8 - 4a3z-1 + 24a3z - 44a3z3 + 32a3z5 - 5a3z7 - a3z9 - 2a4 + 7a4z2 - 13a4z4 + 13a4z6 - 4a4z8 - 2a5z-1 + 14a5z - 25a5z3 + 18a5z5 - 5a5z7 - a6 - a6z2 + 2a6z4 - 2a6z6 + 2a7z - 4a7z3 - a8z2

Khovanov Homology:

trqj 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           4 3       j = -4         4 4         j = -6       3 3           j = -8     2 4             j = -10   2 3               j = -12   3                 j = -14 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, 32]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 32]]
Out[4]=	PD[X[6, 1, 7, 2], X[20, 7, 21, 8], X[4, 21, 1, 22], X[9, 14, 10, 15], > X[3, 8, 4, 9], X[5, 13, 6, 12], X[13, 5, 14, 22], X[15, 18, 16, 19], > X[11, 17, 12, 16], X[17, 11, 18, 10], X[19, 2, 20, 3]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, 11, -5, -3}, {-6, -1, 2, 5, -4, 10, -9, 6, -7, 4, -8, 9, -10, 8, > -11, -2, 3, 7}]
In[6]:=	Jones[L][q]
Out[6]=	-(13/2) 4 5 7 7 7 6 3/2 q - ----- + ---- - ---- + ---- - ---- + ------- - 4 Sqrt[q] + 2 q - 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q 5/2 > q
In[7]:=	A2Invariant[L][q]
Out[7]=	-24 -22 2 2 -14 3 -10 2 -6 -4 2 6 8 -1 - q + q + --- + --- + q + --- - q + -- - q - q + q + q + q 18 16 12 8 q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 32]][a, z]
Out[8]=	3 5 3 1 3 a 4 a 2 a 3 z 3 5 7 z -(---) + --- - ---- + ---- - --- + 8 a z - 10 a z + 4 a z - a z - -- + a z z z z a a 3 3 3 5 3 5 3 5 5 5 3 7 > 8 a z - 9 a z + 4 a z + 2 a z - 5 a z + a z - a z
In[9]:=	Kauffman[Link[11, NonAlternating, 32]][a, z]
Out[9]=	3 5 2 4 6 1 3 a 4 a 2 a 5 z 3 -1 - 3 a - 2 a - a - --- - --- - ---- - ---- + --- + 17 a z + 24 a z + a z z z z a 3 5 7 2 2 2 4 2 6 2 8 2 8 z > 14 a z + 2 a z + 6 z + 13 a z + 7 a z - a z - a z - ---- - a 3 3 3 5 3 7 3 4 2 4 4 4 > 31 a z - 44 a z - 25 a z - 4 a z - 12 z - 27 a z - 13 a z + 5 6 4 5 z 5 3 5 5 5 6 2 6 > 2 a z + ---- + 19 a z + 32 a z + 18 a z + 9 z + 24 a z + a 7 4 6 6 6 z 7 3 7 5 7 8 2 8 > 13 a z - 2 a z - -- - a z - 5 a z - 5 a z - 2 z - 6 a z - a 4 8 9 3 9 > 4 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	4 4 1 3 2 3 2 4 3 3 -- + -- + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ---- + 4 2 14 5 12 4 10 4 10 3 8 3 8 2 6 2 6 q q q t q t q t q t q t q t q t q t 4 3 t 2 2 2 2 3 4 3 6 4 > ---- + 3 t + --- + t + 3 q t + q t + q t + q t 4 2 q t q

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