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<DIV> </DIV>
<DIV>Don't know enough about homeomorphisms to tell you.</DIV>
<DIV> </DIV>
<DIV>The basic idea is that if two "good" polygons are equivalent, I can deform
one</DIV>
<DIV>into the other continuously, with all the intermediates being good polygons
as</DIV>
<DIV>well. Because we stay within the space of good polygons, the
transformation</DIV>
<DIV>will not allow, say, a vertex to pass through an edge or three edges
to</DIV>
<DIV>intersect at a point during the transformation. Is this a
homeomorphism?</DIV>
<DIV> </DIV>
<DIV>About the graph observation, I don't think the polygon equivalence is the
same</DIV>
<DIV>as the graph equivalance. For example, consider the following two
pentagons:</DIV>
<DIV> </DIV>
<DIV>X: (0,0) (4,0) (2,3) (3,2) (0,4)</DIV>
<DIV>Y: (0,0) (4,0) (1,2) (2,1) (0,4)</DIV>
<DIV> </DIV>
<DIV>I believe that X and Y are distinct polygons according to my definition,
but</DIV>
<DIV>their graphs are isomorphic.</DIV>
<DIV> </DIV>
<DIV>The six pentagons I found were the two above plus</DIV>
<DIV> </DIV>
<DIV>(0,0) (1,0) (2,1) (1,2) (0,1) (simple pentagon)</DIV>
<DIV>(0,0) (2,1) (0,1) (1,0) (1,2) (pentagram)</DIV>
<DIV>(1,0) (1,1) (0,1) (3,0) (0,3)</DIV>
<DIV>(2,0) (2,2) (0,2) (0,3) (3,0)</DIV>
<DIV> </DIV>
<BLOCKQUOTE dir=ltr
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<DIV style="FONT: 10pt arial">----- Original Message ----- </DIV>
<DIV
style="BACKGROUND: #e4e4e4; FONT: 10pt arial; font-color: black"><B>From:</B>
<A title=Sterten@aol.com href="mailto:Sterten@aol.com">Sterten@aol.com</A>
</DIV>
<DIV style="FONT: 10pt arial"><B>To:</B> <A title=ham
href="mailto:ham">ham</A> ; <A title=davidwwilson@comcast.net
href="mailto:davidwwilson@comcast.net">davidwwilson@comcast.net</A> </DIV>
<DIV style="FONT: 10pt arial"><B>Sent:</B> Friday, April 22, 2005 9:15
AM</DIV>
<DIV style="FONT: 10pt arial"><B>Subject:</B> Re: Counting Self-intersecting
n-gons</DIV>
<DIV><BR></DIV><FONT id=role_document color=#000000>
<DIV> >define two polygons in P as equivalent if there is a
continuous<BR> >tranformation between them in P.</DIV>
<DIV> </DIV>
<DIV>homeomorphism ?</DIV>
<DIV> </DIV>
<DIV> >Assuming this definition means anything,</DIV>
<DIV> </DIV>
<DIV>you mean: "means something" ?</DIV>
<DIV> </DIV>
<DIV> >I am pretty sure that<BR> >equivalent polygons will
dissect the plane into the same number<BR> >of regions and
corresponding regions will have the same number<BR> >of edges.</DIV>
<DIV> </DIV>
<DIV>a polygon X in P makes a planar graph G(X) by adding all
intersection-points<BR>as new vertices. X-vertices have degree 2 in G(X), but
there can also be <BR>(maximum (n-2)*(n-3)/2, I think) other vertices of
degree 4.</DIV>
<DIV> </DIV>
<DIV>When polygons X and Y are equivalent, then G(X) and G(Y) are isomorphic
?<BR>When G(X) and G(Y) are isomorphic, then polygons X and Y are equivalent
?</DIV>
<DIV> </DIV>
<DIV> >The question is, according to this definition, how many
distinct<BR> >self-intersecting n-gons are
there?<BR> ><BR> >Clearly, a(3) = 1, since all triangles are
equivalent.<BR> ><BR> >I believe a(4) = 2, all
self-intersecting quadrilaterals begin equivalent<BR> >to either a
square or a bowtie.<BR> ><BR> >I have found 6 inequivalent
pentagons, so a(5) >= 6.</DIV>
<DIV> </DIV>
<DIV>I only found 5, they have 0,1,2,3,5 crossings.</DIV>
<DIV> </DIV>
<DIV><BR>-Guenter.</DIV></BLOCKQUOTE></FONT></BODY></HTML>