Knot (mathematics)

In mathematics, a knot is an embedding of the circle (S¹) 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 Sⁿ, 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.