Skein theory in knot theory and beyond

John Conway invented skein theory as a tool for manipulating knots and links and in particular calculating the Alexander polynomial. It was extended somewhat in the 1980's when new knot polynomials were discovered. More recently it has proved useful in the theory of subfactors as a tool to construct and analyse certain exotic structures. These structures remain related to low dimensional topology as they have topological quantum field theories attached. The unifying language is that of planar algebras.