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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-21/2 - q-19/2 + q-17/2 - q-15/2 - q-11/2 - q-5/2

A2 (sl(3)) Invariant:

- q-34 - 2q-32 - q-30 - q-28 + q-26 + 2q-24 + 2q-22 + 2q-20 + q-18 + 2q-16 + q-14 + q-12 + q-10 + q-8

HOMFLY-PT Polynomial:

- a5z-1 - 5a5z - 5a5z3 - a5z5 + a7z + 2a9z-1 + 2a9z - a11z-1

Kauffman Polynomial:

a5z-1 - 5a5z + 5a5z3 - a5z5 - a6 + a6z2 - a7z + a7z3 + 3a8 - 8a8z2 + 6a8z4 - a8z6 - 2a9z-1 + 7a9z - 10a9z3 + 6a9z5 - a9z7 + 5a10 - 15a10z2 + 11a10z4 - 2a10z6 - a11z-1 + 3a11z - 6a11z3 + 5a11z5 - a11z7 + 2a12 - 6a12z2 + 5a12z4 - a12z6

Khovanov Homology:

trqj r = -9 r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 j = -4                   1 j = -6                   1 j = -8             1 1     j = -10           1         j = -12         1 3 1       j = -14       1             j = -16       1 1           j = -18   1 1               j = -20                     j = -22 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, NonAlternating, 9]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[9, NonAlternating, 9]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[7, 14, 8, 15], X[15, 18, 16, 5], > X[11, 16, 12, 17], X[17, 12, 18, 13], X[13, 8, 14, 9], X[2, 5, 3, 6], > X[4, 9, 1, 10]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -8, 2, -9}, {8, -1, -3, 7, 9, -2, -5, 6, -7, 3, -4, 5, -6, 4}]
In[6]:=	Jones[L][q]
Out[6]=	-(21/2) -(19/2) -(17/2) -(15/2) -(11/2) -(5/2) q - q + q - q - q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-34 2 -30 -28 -26 2 2 2 -18 2 -14 -12 -q - --- - q - q + q + --- + --- + --- + q + --- + q + q + 32 24 22 20 16 q q q q q -10 -8 > q + q
In[8]:=	HOMFLYPT[Link[9, NonAlternating, 9]][a, z]
Out[8]=	5 9 11 a 2 a a 5 7 9 5 3 5 5 -(--) + ---- - --- - 5 a z + a z + 2 a z - 5 a z - a z z z z
In[9]:=	Kauffman[Link[9, NonAlternating, 9]][a, z]
Out[9]=	5 9 11 6 8 10 12 a 2 a a 5 7 9 -a + 3 a + 5 a + 2 a + -- - ---- - --- - 5 a z - a z + 7 a z + z z z 11 6 2 8 2 10 2 12 2 5 3 7 3 > 3 a z + a z - 8 a z - 15 a z - 6 a z + 5 a z + a z - 9 3 11 3 8 4 10 4 12 4 5 5 9 5 > 10 a z - 6 a z + 6 a z + 11 a z + 5 a z - a z + 6 a z + 11 5 8 6 10 6 12 6 9 7 11 7 > 5 a z - a z - 2 a z - a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	-6 -4 1 1 1 1 1 1 1 q + q + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 22 9 18 8 18 7 16 6 14 6 16 5 12 5 q t q t q t q t q t q t q t 3 1 1 1 1 > ------ + ------ + ------ + ----- + ----- 12 4 10 4 12 3 8 3 8 2 q t q t q t q t q t

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