Some spaces of polynomial knots

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 Od, Pd and Qd. For d≥3, we show that the spaces Od and Pd are path connected and the space Od has the same homotopy type as S2. Considering the space P=⋃d≥2Od of all polynomial knots with the inductive limit topology, we prove that it too has the same homotopy type as S2. We also show that if two polynomial knots are path equivalent in Qd, then they are topologically equivalent. Furthermore, the number of path components in Qd are in multiples of eight.