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DrawMorseLink

PD Presentation:

X6172 X12,3,13,4 X14,8,15,7 X16,10,5,9 X8,16,9,15 X10,14,11,13 X2536 X4,11,1,12

Gauss Code:

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

Jones Polynomial:

- q-7/2 + q-5/2 - 4q-3/2 + 4q-1/2 - 5q1/2 + 5q3/2 - 4q5/2 + 3q7/2 - q9/2

A2 (sl(3)) Invariant:

q-12 + 2q-10 + 2q-8 + 4q-6 + 2q-4 + q-2 + 1 - 2q2 - 2q6 - q12 + q14

HOMFLY-PT Polynomial:

- a-3z - a-3z3 + a-1z-1 + 3a-1z + 3a-1z3 + a-1z5 - 3az-1 - 5az - 2az3 + 2a3z-1 + a3z

Kauffman Polynomial:

- a-5z3 + 2a-4z2 - 3a-4z4 - a-3z + 4a-3z3 - 4a-3z5 - a-2 + a-2z2 + 2a-2z4 - 3a-2z6 + a-1z-1 - 2a-1z + 6a-1z3 - 3a-1z5 - a-1z7 - 3 + 2z2 + 6z4 - 4z6 + 3az-1 - 6az + 5az3 - az7 - 3a2 + 3a2z2 + a2z4 - a2z6 + 2a3z-1 - 5a3z + 4a3z3 - a3z5

Khovanov Homology:

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 10                 1 j = 8               2   j = 6             2 1   j = 4           3 2     j = 2         2 2       j = 0       3 4         j = -2     1 1           j = -4     3             j = -6 1 1               j = -8 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[8, Alternating, 5]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[8, Alternating, 5]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 3, 13, 4], X[14, 8, 15, 7], X[16, 10, 5, 9], > X[8, 16, 9, 15], X[10, 14, 11, 13], X[2, 5, 3, 6], X[4, 11, 1, 12]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -7, 2, -8}, {7, -1, 3, -5, 4, -6, 8, -2, 6, -3, 5, -4}]
In[6]:=	Jones[L][q]
Out[6]=	-(7/2) -(5/2) 4 4 3/2 5/2 7/2 -q + q - ---- + ------- - 5 Sqrt[q] + 5 q - 4 q + 3 q - 3/2 Sqrt[q] q 9/2 > q
In[7]:=	A2Invariant[L][q]
Out[7]=	-12 2 2 4 2 -2 2 6 12 14 1 + q + --- + -- + -- + -- + q - 2 q - 2 q - q + q 10 8 6 4 q q q q
In[8]:=	HOMFLYPT[Link[8, Alternating, 5]][a, z]
Out[8]=	3 3 3 5 1 3 a 2 a z 3 z 3 z 3 z 3 z --- - --- + ---- - -- + --- - 5 a z + a z - -- + ---- - 2 a z + -- a z z z 3 a 3 a a a a
In[9]:=	Kauffman[Link[8, Alternating, 5]][a, z]
Out[9]=	3 2 -2 2 1 3 a 2 a z 2 z 3 2 2 z -3 - a - 3 a + --- + --- + ---- - -- - --- - 6 a z - 5 a z + 2 z + ---- + a z z z 3 a 4 a a 2 3 3 3 4 4 z 2 2 z 4 z 6 z 3 3 3 4 3 z 2 z > -- + 3 a z - -- + ---- + ---- + 5 a z + 4 a z + 6 z - ---- + ---- + 2 5 3 a 4 2 a a a a a 5 5 6 7 2 4 4 z 3 z 3 5 6 3 z 2 6 z 7 > a z - ---- - ---- - a z - 4 z - ---- - a z - -- - a z 3 a 2 a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 1 1 3 1 3 1 2 4 4 + 2 q + ----- + ----- + ----- + ----- + ----- + - + ---- + 2 q t + 3 q t + 8 4 6 4 6 3 4 2 2 2 t 2 q t q t q t q t q t q t 4 2 6 2 6 3 8 3 10 4 > 2 q t + 2 q t + q t + 2 q t + q t

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