A Revised Version of P. L. PROPER KNOT EQUIVALENCE CLASSES ARE GENERATED BY LOCALLY FLAT ISOTOPIES (Vol. 3, No. 4 (December 94), pp. 497–509)
A p. 1. proper knot is a proper p. 1. embedding of the real line into a p. 1. open 3-manifold. Two p. 1. proper knots are said to be equivalent if there is a p. 1. proper isotopy (possibly non-ambient) connecting them. A priori such an isotopy need not be locally flat: for example a local knot in the range could be shrunk to a point within a 3-cell. In this paper we show that, in essence, this is the only type of non-locally flat phenomenon. We show that such a local knot can always be “combed to infinity”. Thus if two p. 1. proper knots are connected by a p. 1. proper isotopy then the connecting isotopy can be modified to be a locally flat p. 1. proper isotopy. Hence, in proper knot theory, the smooth category and the p. 1. category are similar.