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Knotscape

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DrawMorseLink

PD Presentation:

X6172 X2,9,3,10 X12,3,13,4 X10,5,11,6 X18,11,5,12 X4,17,1,18 X16,14,17,13 X14,8,15,7 X8,16,9,15

Gauss Code:

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

Jones Polynomial:

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

A2 (sl(3)) Invariant:

q-20 - q-18 + q-16 + 3q-14 - q-12 + 3q-10 + q-8 + q-6 + 2q-4 - 2q-2 + 2 - 2q2 + 2q6 - q8

HOMFLY-PT Polynomial:

a-1z3 - az3 - az5 - a3z-1 - 3a3z - 2a3z3 - a3z5 + a5z-1 + a5z + a5z3

Kauffman Polynomial:

- a-2z4 + 3a-1z3 - 4a-1z5 - 2z2 + 9z4 - 7z6 - az3 + 6az5 - 6az7 - 6a2z2 + 17a2z4 - 8a2z6 - 2a2z8 - a3z-1 + 3a3z - 10a3z3 + 18a3z5 - 10a3z7 + a4 - 7a4z2 + 13a4z4 - 4a4z6 - 2a4z8 - a5z-1 + 2a5z - 4a5z3 + 7a5z5 - 4a5z7 - 3a6z2 + 6a6z4 - 3a6z6 - a7z + 2a7z3 - 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               4 1   j = 0             5 3     j = -2           6 5       j = -4         5 4         j = -6       3 6           j = -8     3 5             j = -10   1 4               j = -12   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, 19]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[9, Alternating, 19]]
Out[4]=	PD[X[6, 1, 7, 2], X[2, 9, 3, 10], X[12, 3, 13, 4], X[10, 5, 11, 6], > X[18, 11, 5, 12], X[4, 17, 1, 18], X[16, 14, 17, 13], X[14, 8, 15, 7], > X[8, 16, 9, 15]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 3, -6}, {4, -1, 8, -9, 2, -4, 5, -3, 7, -8, 9, -7, 6, -5}]
In[6]:=	Jones[L][q]
Out[6]=	-(13/2) 3 6 8 11 10 9 3/2 -q + ----- - ---- + ---- - ---- + ---- - ------- + 7 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]=	-20 -18 -16 3 -12 3 -8 -6 2 2 2 6 2 + q - q + q + --- - q + --- + q + q + -- - -- - 2 q + 2 q - 14 10 4 2 q q q q 8 > q
In[8]:=	HOMFLYPT[Link[9, Alternating, 19]][a, z]
Out[8]=	3 5 3 a a 3 5 z 3 3 3 5 3 5 3 5 -(--) + -- - 3 a z + a z + -- - a z - 2 a z + a z - a z - a z z z a
In[9]:=	Kauffman[Link[9, Alternating, 19]][a, z]
Out[9]=	3 5 4 a a 3 5 7 2 2 2 4 2 6 2 a - -- - -- + 3 a z + 2 a z - a z - 2 z - 6 a z - 7 a z - 3 a z + z z 3 4 3 z 3 3 3 5 3 7 3 4 z 2 4 > ---- - a z - 10 a z - 4 a z + 2 a z + 9 z - -- + 17 a z + a 2 a 5 4 4 6 4 4 z 5 3 5 5 5 7 5 6 > 13 a z + 6 a z - ---- + 6 a z + 18 a z + 7 a z - a z - 7 z - a 2 6 4 6 6 6 7 3 7 5 7 2 8 > 8 a z - 4 a z - 3 a z - 6 a z - 10 a z - 4 a z - 2 a z - 4 8 > 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	5 1 2 1 4 3 5 3 6 5 + -- + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + 2 14 6 12 5 10 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 5 4 6 2 2 2 4 2 6 3 > ----- + ---- + ---- + 3 t + 4 q t + q t + 3 q t + q t 4 2 4 2 q t q t q t

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