Counting n-gons

[Jonathan Post]

Good place to start is:
Ivars Peterson's *MathTrek*

November 3, 1997

 Knotted Walks http://www.maa.org/mathland/mathtrek_11_3.html

Oh, and nice illustrations in:

Discrete Knots
by Hew Wolff <http://www.hewwolff.org/>

http://www.jansteckel.com/Hew/WebSite/DiscreteKnots/index.html

An *orthogonal discrete knot* is a path through the lattice of
integer-valued points and orthogonal edges between them, which never visits
the same point twice and ends where it starts. Two knots are
*equivalent*if, when they are realized with pieces of stretchy string,
we can move one
around to look like the other. (Also, we consider mirror images to be the
same knot.) Naturally we are interested in knots which are not *trivial*,
that is, equivalent to a simple circle. Here are some questions about making
these knots as small as possible....


On 10/26/06, Jonathan Post <jvospost3 at gmail.com> wrote:
>
> Some references that I should give for knotted 3-D self-avoiding walks,
> meaning polygons embedded in Z^3, the 3-D integer lattice. Have any of these
> been in OEIS?
>
> (1) G. Buck, Four-thirds Power Law for Knots and Links, Nature, 392
> (1998), pp. 238-239.
> www.anselm.edu/academic/mathematics/Bucknature4-3F.pdf
>
> (2) ....to hold in cases where restrictions are placed on the number of
> edges per branch in a graph embedding. *Key* words. knots, graph
> embeddings, branched polymer, simple cubic lattice . *AMS*(MOS) subject
> classifications. 82B41,57M25,05C10,05C80,05C30. 1. *Introduction*. * In
> 1988 Sumners and Whittington [1] investigated questions about knottedness
> of a closed curve of given length embedded in the three dimensional
> integer lattice, Z 3 .* They and, independently, Pippenger [2] showed that
> sufficiently long closed curves embedded in Z 3 are almost surely knotted.
> *Soteros*, Sumners and Whittington ....
>
> ....than for Z 3 or the slab geometries. The second question posed above
> is addressed here by studying the probability that an embedding of a simple
> closed curve, i.e. a self avoiding polygon, confined to a particular
> lattice subset is knotted. * This probability has been studied previously
> in [1,4] and the results from these works are reviewed here.* It is shown
> that the probability is dependent on the lattice subset. The knottedness
> of embeddings of graphs in Z 3 has been previously investigated in [3,5] and
> the results from these works are reviewed and generalized further here.....
>
>
> D.W. Sumners and S.G. Whittington, *Knots in self-avoiding walks *, J.
> Physics A: Math. Gen., 21 (1988), pp. 1689--1694. KNOTS IN GRAPHS IN
> SUBSETS OF Z 3 33
>
>
>
> On 10/25/06, Jonathan Post <jvospost3 at gmail.com> wrote:
> >
> > I'm impressed by Richard Mathar's exposition.  Dr. George Hockney should
> > be flattered that he's been corrected by Richard Guy! I should be dubious
> > about anything asserted to me by a baseball fan during a World Series game.
> >
> > I've also played with self-avoiding walks, and wonder how Richard
> > Mathar's approach works for polygons embedded in the 3-D integer lattice,
> > some results of which are known in Knot Theory, both for short walks and for
> > sufficiently long walks.  For example, almost all self-avoiding walks in the
> > 3-D integer lattice which start and end at (0, 0, 0) are knotted. Of course,
> > polygons embedded in the 4-D integer lattice cannot be knotted.
> >
> > -- Jonathan Vos Post
> >
> > On 10/25/06, Richard Guy < rkg at cpsc.ucalgary.ca> wrote:
> > >
> > > max area of 1,1,2,2 quad is cyclic, a kite
> > > formed by 2 right triangles with legs 1 & 2,
> > > area 2.      R.
> > >
> > > On Tue, 24 Oct 2006, Jonathan Post wrote:
> > >
> > > > Dr. George Hockney (ex-Fermilab, now JPL)
> > > > responded to my suggestion that
> > > > the representative of each equivalence class of
> > > > deformed polygons with the
> > > > same edge-sequences be that with largest area
> > > > that: (1)  this might not be
> > > > unique, and (2) that the maximum area (1, 1, 2, 2)
> > > > quadrilateral is the
> > > > (1+1=2, 2, 2) equilateral triangle of area sqrt 3,
> > > > where the two edges of
> > > > length 1 make an angle of 180 degrees, and asks if
> > > > that equilateral triangle
> > > > with a vertex in the middle of its side is really
> > > > a quadrilateral, or is
> > > > that another ambiguity in equivalence of polygons
> > > > under geometrical
> > > > similarity? He is also dubious that the
> > > > integer-edged polygonal knots in R^3
> > > > are really polygons, as I've stated.
> > > >
> > > > I agree with David Wilson's sequence.  What about
> > > > a sequence table of
> > > > maximal areas of all simple integer-sided k-gons
> > > > of perimeter n?
> > > >
> > > > -- Jonathan Vos Post
> > > >
> > > > On 10/24/06, David Wilson
> > > > <davidwwilson at comcast.net> wrote:
> > > >>
> > > >> And we should have a sequence that counts all
> > > >> simple integer-sided
> > > >> polygons
> > > >> of perimeter n. Depending on exactly what we are
> > > >> counting, the sequence
> > > >> might go:
> > > >>
> > > >> a(3) = 1, the 1-1-1 triangle.
> > > >> a(4) = 1, the 1-1-1-1 quadrilateral.
> > > >> a(5) = 3, the 1-2-2 triangle, 1-1-1-2
> > > >> quadrilateral, the 1-1-1-1-1
> > > >> pentagon.
> > > >>
> > > >> etc.
> > > >>
> > > >>
> > > >
> > >
> >
> >
>
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