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DrawMorseLink

PD Presentation:

X8192 X12,3,13,4 X18,10,7,9 X10,14,11,13 X16,5,17,6 X14,18,15,17 X2738 X4,11,5,12 X6,15,1,16

Gauss Code:

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

Jones Polynomial:

- q-13/2 + 2q-11/2 - 6q-9/2 + 7q-7/2 - 9q-5/2 + 10q-3/2 - 8q-1/2 + 6q1/2 - 4q3/2 + q5/2

A2 (sl(3)) Invariant:

q-22 + 2q-20 + 2q-16 + 4q-14 + 2q-10 - q-8 - 2q-6 - 2q-2 + 3 - q2 + 2q6 - q8

HOMFLY-PT Polynomial:

a-1z3 - 3az - 2az3 - az5 + 3a3z + 3a3z3 - a5z-1 - 3a5z + a7z-1

Kauffman Polynomial:

- a-2z4 + 4a-1z3 - 4a-1z5 - 3z2 + 8z4 - 6z6 - 2az + 8az3 - az5 - 4az7 - 14a2z2 + 24a2z4 - 11a2z6 - a2z8 + 2a3z - 2a3z3 + 9a3z5 - 7a3z7 - 11a4z2 + 18a4z4 - 7a4z6 - a4z8 + a5z-1 + a5z - 3a5z3 + 5a5z5 - 3a5z7 - a6 + 3a6z4 - 2a6z6 + a7z-1 - 3a7z + 3a7z3 - a7z5

Khovanov Homology:

trqj 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               3 1   j = 0             5 3     j = -2           6 4       j = -4         3 4         j = -6       4 6           j = -8     2 3             j = -10     4               j = -12 1 2                 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[9, Alternating, 33]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[9, Alternating, 33]]
Out[4]=	PD[X[8, 1, 9, 2], X[12, 3, 13, 4], X[18, 10, 7, 9], X[10, 14, 11, 13], > X[16, 5, 17, 6], X[14, 18, 15, 17], X[2, 7, 3, 8], X[4, 11, 5, 12], > X[6, 15, 1, 16]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -7, 2, -8, 5, -9}, {7, -1, 3, -4, 8, -2, 4, -6, 9, -5, 6, -3}]
In[6]:=	Jones[L][q]
Out[6]=	-(13/2) 2 6 7 9 10 8 3/2 -q + ----- - ---- + ---- - ---- + ---- - ------- + 6 Sqrt[q] - 4 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]=	-22 2 2 4 2 -8 2 2 2 6 8 3 + q + --- + --- + --- + --- - q - -- - -- - q + 2 q - q 20 16 14 10 6 2 q q q q q q
In[8]:=	HOMFLYPT[Link[9, Alternating, 33]][a, z]
Out[8]=	5 7 3 a a 3 5 z 3 3 3 5 -(--) + -- - 3 a z + 3 a z - 3 a z + -- - 2 a z + 3 a z - a z z z a
In[9]:=	Kauffman[Link[9, Alternating, 33]][a, z]
Out[9]=	5 7 6 a a 3 5 7 2 2 2 4 2 -a + -- + -- - 2 a z + 2 a z + a z - 3 a z - 3 z - 14 a z - 11 a z + z z 3 4 4 z 3 3 3 5 3 7 3 4 z 2 4 > ---- + 8 a z - 2 a z - 3 a z + 3 a z + 8 z - -- + 24 a z + a 2 a 5 4 4 6 4 4 z 5 3 5 5 5 7 5 6 > 18 a z + 3 a z - ---- - a z + 9 a z + 5 a z - a z - 6 z - a 2 6 4 6 6 6 7 3 7 5 7 2 8 4 8 > 11 a z - 7 a z - 2 a z - 4 a z - 7 a z - 3 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	4 1 1 2 4 2 3 4 6 5 + -- + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + 2 14 6 12 6 12 5 10 4 8 4 8 3 6 3 6 2 q q t q t q t q t q t q t q t q t 3 4 6 2 2 2 4 2 6 3 > ----- + ---- + ---- + 3 t + 3 q t + q t + 3 q t + q t 4 2 4 2 q t q t q t

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