An approach to the polygonal knot problem using projections and isotopies

Abstract

The author extends earlier work of Tait, Gauss, Nagy, and Penney in defining and developing properties of (1) the boundary collection of a knot function, and (2) simple sequences of knot functions or boundary collections. The main results are (1) if two knot functions have isomorphic boundary collections then the knots they determine are equivalent, and (2) if two knot functions determine equivalent knots, then the given functions (their boundary collections) are the ends of a simple sequence of knot functions (boundary collections). Matrices are also defined for knot functions.