Knot (mathematics)

The triangle is associated with the trefoil knot.

In mathematics, a knot is an embedding of the circle ( $S 1$ ) into three-dimensional Euclidean space, $R 3$ (also known as $E 3$ ). Often two knots are considered equivalent if they are ambient isotopic, that is, if there exists a continuous deformation of $R 3$ 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.