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Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-19/2 + 2q-17/2 - 3q-15/2 + 4q-13/2 - 5q-11/2 + 4q-9/2 - 4q-7/2 + 2q-5/2 - q-3/2

A2 (sl(3)) Invariant:

q-30 - q-26 + q-24 + q-20 + 2q-18 + q-16 + 2q-14 + q-10 + q-8 - q-6 + q-4

HOMFLY-PT Polynomial:

- a3z - a3z3 - a5z-1 - 3a5z - 2a5z3 + a7z-1 - a7z3 + a9z

Kauffman Polynomial:

a3z - a3z3 + a4z2 - 2a4z4 + a5z-1 - 4a5z + 4a5z3 - 3a5z5 - a6 + 2a6z2 - 2a6z6 + a7z-1 - 4a7z + 6a7z3 - 2a7z5 - a7z7 - 3a8z2 + 8a8z4 - 4a8z6 - a9z + 4a9z3 - a9z7 - 4a10z2 + 6a10z4 - 2a10z6 - 2a11z + 3a11z3 - a11z5

Khovanov Homology:

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

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