TopologyDate: 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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