Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In mathematics, a knot is an embedding of a circle in 3-dimensional Euclidean space, R3, considered up to continuous deformations (isotopies). 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 Sj in Sn, especially in the case j = n – 2. The branch of mathematics that studies knots is known as knot theory.

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