Date: Thu, 14 Jan 1999 13:52:11 -0500
From: "Charles E. McManus" <fiero@netrax.net>
Subject: Re: question abt knots

> Mallory,

    I am also a High School Student from Pennsylvania, but I have never
participated in the PJAS.  One good book that covers a  lot of basic topology,
including some knot theory is ``Topology'' by John G. Hocking and Gail S. Young.
It is out on Dover
as a paperback, and is fairly inexpensive(about $11, paperback).  It can be
ordered from www.amazon.com.  There are also
sets of notes on algebraic topology that can be downloaded from the math
department at Cornell.

    Knot theory involves algebraic topology, which can be hard to learn, but is
very interesting.  I would study homotopy theory
and then get into some basic imbedding problems with knots.  You have to know a
bit of point-set topology before you can get into this kind of knot theory, but
this can be found in the book I mentioned above.  Since you pobably only have a
month or two in which to do the project, one idea would be to use knots to
understand basic homotopy.

    Formally, a knot K is a simple closed curve(like a piece of string tied
    together at the ends) that can not be turned into a circle by an
    ``orientation preserving homeomorphism."  Topologists treat knots as
    manifolds.  One of the things that this means is that a two-dimensional
    curve that is a knot exists in three-dimensional space (E^3).  We call
    this kind of knot a 2-manifold.  This makes sense because if the knot was
    only allowed to live in a plane, we couldn't twist it around, because
    this involves moving in three-dimensions.  A 2-knot is a kind of
    two-dimensional curve that has been imbedded in E^3.

I'll stop with that definition, and remark that the point behind knot theory
is to classify knots by using the theory of invariants.  This may sound like
Greek to you, or it may make sense.  In case you were wondering, a
homeomorphism is a kind of machine that can transform a curve into another
curve by bending and stretching it, but not by sharply bending or stretching
it.  It is a kind of  ``function.'' Homotopy is a bit difficult for me to
concisely describe.  My advice would be to talk to an expert, but if you ever
need anything I can be reached at fiero@netrax.net.  What I have descibed to
you is subject matter from a rather advanced  mathematics course, but if you
want to learn it, you can do a project on it in a few weeks.

My best advice is to make your work conform to some kind of problem and
hypothesis/scientific method.  At my High School Science fair this week, my
teacher gave me a bad grade for my project on ``Elliptic Curves Over Finite
Fields'' because he thought that my proofs of several theorems about
something called the Hasse-Weil Zeta function didn't count as a scientific
experiement. At any rate, I've just learned from experience that math
research doesn't always get recognition at science fairs.  Don't let this
keep you from doing a project on math.  It is very rewarding regardless of
what recognition you get.

Good luck,
Andy McManus