Some spaces of polynomial knots

Abstract

In this paper we study the topology of three different kinds of spaces associated to polynomial knots of degree at most d, for d >= 2. We denote these spaces by O-d,O- P-d and Q(d). For d >= 3, we show that the spaces O-d and T-d are path connected and the space O-d has the same homotopy type as S-2. Considering the space P = boolean OR(d >= 2) O-d of all polynomial knots with the inductive limit topology, we prove that it to has the same homotopy type as S-2. We also show that if two polynomial knots are path equivalent in Q(d), then they are topologically equivalent. Furthermore, the number of path components in Q(d) are in multiples of eight.