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Concept# Knot (mathematics)

Summary

In mathematics, a knot is an embedding of the circle S^1 into three-dimensional Euclidean space, R3 (also known as E3). Often two knots are considered equivalent if they are ambient isotopic, that is, if there exists a continuous deformation of R3 which takes one knot to the other.
A crucial difference between the standard mathematical and conventional notions of a knot is that mathematical knots are closed — there are no ends to tie or untie on a mathematical knot. Physical properties such as friction and thickness also do not apply, although there are mathematical definitions of a knot that take such properties into account. The term knot is also applied to embeddings of S^ j in S^n, especially in the case j = n − 2. The branch of mathematics that studies knots is known as knot theory and has many relations to graph theory.
A knot is an embedding of the circle (S^1) into three-dimensional Euclidean space (R3), or the 3-sphere (S^3), since the 3-sphere is compact. Two knots are defined to be equivalent if there is an ambient isotopy between them.
A knot in R3 (or alternatively in the 3-sphere, S^3), can be projected onto a plane R2 (respectively a sphere S^2). This projection is almost always regular, meaning that it is injective everywhere, except at a finite number of crossing points, which are the projections of only two points of the knot, and these points are not collinear. In this case, by choosing a projection side, one can completely encode the isotopy class of the knot by its regular projection by recording a simple over/under information at these crossings. In graph theory terms, a regular projection of a knot, or knot diagram is thus a quadrivalent planar graph with over/under-decorated vertices. The local modifications of this graph which allow to go from one diagram to any other diagram of the same knot (up to ambient isotopy of the plane) are called Reidemeister moves.
Image:Reidemeister_move_1.png|Reidemeister move 1
Image:Reidemeister_move_2.png|Reidemeister move 2
Image:Reidemeister_move_3.

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