In
mathematics, a
knot is an
embedding of a
circle in 3-dimensional
Euclidean space,
R3 (also known as
E3), 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
in
, especially in the case
. The branch of mathematics that studies knots is known as
knot theory, and has many simple relations to
graph theory.