Regular isotopy
Encyclopedia
In the mathematical
subject of knot theory
, a regular isotopy of a link diagram is the equivalence relation generated by using the 2nd and 3rd Reidemeister move
s only. The notion of regular isotopy was introduced by Louis Kauffman
(Kauffman 1990). It can be thought of as an isotopy of a ribbon pressed flat against the plane which keeps the ribbon flat. For diagrams in the plane this is a finer equivalence relation than ambient isotopy of a framed link, since the 2nd and 3rd Reidemeister moves preserve the winding number
of the diagram (Kauffman 1990, pp. 450ff.). However, for diagrams in the sphere (considered as the plane plus infinity), the two notions are equivalent, due to the extra freedom of passing a strand through infinity.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
subject of knot theory
Knot theory
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 precise mathematical language, a knot is an embedding of a...
, a regular isotopy of a link diagram is the equivalence relation generated by using the 2nd and 3rd Reidemeister move
Reidemeister move
In the mathematical area of knot theory, a Reidemeister move refers to one of three local moves on a link diagram. In 1926, Kurt Reidemeister and independently, in 1927, J.W. Alexander and G.B...
s only. The notion of regular isotopy was introduced by Louis Kauffman
Louis Kauffman
Louis H. Kauffman is an American mathematician, topologist, and professor of Mathematics in the Department of Mathematics, Statistics, and Computer science at the University of Illinois at Chicago...
(Kauffman 1990). It can be thought of as an isotopy of a ribbon pressed flat against the plane which keeps the ribbon flat. For diagrams in the plane this is a finer equivalence relation than ambient isotopy of a framed link, since the 2nd and 3rd Reidemeister moves preserve the winding number
Winding number
In mathematics, the winding number of a closed curve in the plane around a given point is an integer representing the total number of times that curve travels counterclockwise around the point...
of the diagram (Kauffman 1990, pp. 450ff.). However, for diagrams in the sphere (considered as the plane plus infinity), the two notions are equivalent, due to the extra freedom of passing a strand through infinity.