Minimal Degree Sequence for Torus Knots

Abstract

It was observed by Arnold[6] and later proved by Shastri[5] that every non-compact knot-type can be represented by a polynomial embedding. For practical reasons it is important to construct real polynomials /(£), #(£), and h(t) such that the map t H-» (f(t),g(t),h(t)) from M to M is an embedding that represents a given knot type. If deg(f(t)) = Z, deg(g(t)) = m and deg(h(t)) = n, we say that the given knot-type can be represented by polynomials of degree /, m and n. We have tried this construction for a torus knot of type (p, q) which we denote by KPiq and obtained the following results [1],[4],