Counting n-gons

[Jonathan Post]

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
latticesubset 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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