In
topology,
knot theory is the study of
mathematical knots. While inspired by knots which appear in daily life in shoelaces and rope, a mathematician's knot differs in that the ends are joined together so that it cannot be undone. In mathematical language, a knot is an
embedding of a
circle in 3-dimensional
Euclidean space,
R3 (in topology, a circle isn't bound to the classical geometric concept, but to all of its
homeomorphisms). Two mathematical knots are equivalent if one can be transformed into the other via a deformation of
R3 upon itself (known as an
ambient isotopy); these transformations correspond to manipulations of a knotted string that do not involve cutting the string or passing the string through itself.