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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/   
    
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