Associated Topics || Dr. Math Home || Search Dr. Math

### Topology

```
Date: 05/10/97 at 13:47:01
From: tabitha gray
Subject: Topology

What is topology? What is knot theory?

I have looked in encyclopedias, but they don't explain everything so
that I can understand it.

Thank you!
```

```
Date: 05/10/97 at 14:42:09
From: Doctor Sarah
Subject: Re: Topology

Hi,

Great questions!  A lot of what you read in encyclopedias on this
topic can be hard to understand, but there are some less advanced
sites on the web that I think you'll enjoy.

The Math Forum has a collection of internet sites on topology:

http://mathforum.org/library/browse/static/topic/topology.html

Most of them are advanced so you may want to skim through the
descriptions before you choose what to select.  To do this, choose the
"Annotated version" link at the top of the outline page listed above.

Here are a couple of sites I've chosen for you but you can certainly
find more.

Geometry and the Imagination has some explanations meant to be
accessible to high school students:

Topology - Geometry and the Imagination

http://www.geom.uiuc.edu/docs/doyle/mpls/handouts/node13.html

Topology is the theory of shapes which are allowed to stretch,
compress, flex and bend, but without tearing or gluing. For example, a
square is topologically equivalent to a circle, since a square can be
continously deformed into a circle. As another example, a doughnut and
a coffee cup with a handle are topologically equivalent, since a
doughnut can be reshaped into a coffee cup without tearing or gluing.

Knots - Geometry and the Imagination

http://www.geom.uiuc.edu/docs/doyle/mpls/handouts/node7.html

A mathematical knot is a knotted loop. For example, you might take an
extension cord from a drawer and plug one end into the other: this
makes a mathematical knot. It might or might not be possible to unknot
it without unplugging the cord. A knot which can be unknotted is
called an unknot. Two knots are considered equivalent if it is
possible to rearrange one to the form of the other, without cutting
the loop and without allowing it to pass through itself. The reason
for using loops of string in the mathematical definition is that knots
in a length of string can always be undone by pulling the ends
through, so any two lengths of string are equivalent in this sense.

If you drop a knotted loop of string on a table, it crosses over
itself in a certain number of places. Possibly, there are ways to
rearrange it with fewer crossings. The minimum possible number of
crossings is the crossing number of the knot. (There's a neat diagram
here that you'll want to look at.)

The KnotPlot site

http://www.cs.ubc.ca/nest/imager/contributions/scharein/knot-theory/knot-theory.html

Knot theory is a branch of algebraic topology where one studies what
is known as the placement problem, or the embedding of one topological
space into another. The simplest form of knot theory involves the
embedding of the unit circle into three-dimensional space. For the
purposes of this document a knot is defined to be a closed piecewise
linear curve in three-dimensional Euclidean space R^3. Two or more
knots together are called a link. Thus a mathematical knot is somewhat
different from the usual idea of a knot, that is, a piece of string
with free ends. The knots studied in knot theory are (almost) always
considered to be closed loops.

Two knots or links are considered equivalent if one can be smoothly
deformed into the other, or equivalently, if there exists a
homeomorphism on R^3 which maps the image of the first knot onto the
second. Cutting the knot or allowing it to pass through itself are not
permitted. In general it is a very difficult problem to decide if two
given knots are equivalent, and much of knot theory is devoted to
developing techniques to aid in answering this question. Knots that
are equivalent to polygonal paths in three-dimensional space are
called tame. All other knots are known as wild. Most of knot theory
concerns only tame knots, and these are the only knots examined here.
Knots that are equivalent to the unit circle are considered to be
unknotted or trivial.

Try Frank Potter's Science Gems for some less complex topology sites:

http://www-sci.lib.uci.edu/SEP/math.html#20

Hope this helps!

-Doctor Sarah,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```
Associated Topics:
High School Definitions
High School Geometry
High School Higher-Dimensional Geometry
Middle School Definitions
Middle School Geometry
Middle School Higher-Dimensional Geometry

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search