The KnotPlot Site – trefoil

Trefoil Knot“Here you will find a [browsable] collection of knots and links, viewed from a (mostly) mathematical perspective. Nearly all of the images here were created with KnotPlot, a fairly elaborate program to visualize and manipulate mathematical knots in three and four dimensions.” The diagrams are in color and are exceptionally clear.

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 R3. 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 R3 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 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.

The simplest non-trivial knot is the trefoil knot which comes in a left and a right handed form.

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