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 $d\ge 2$. We denote these spaces by ${\mathcal{O}}_d$, ${\mathcal{P}}_d$ and ${\mathcal{Q}}_d$. For $d\ge 3$, we show that the spaces ${\mathcal{O}}_d$ and ${\mathcal{P}}_d$ are path connected and the space ${\mathcal{O}}_d$ has the same homotopy type as $S^2$. Considering the space $\mathcal{P}={\bigcup }_{d\ge 2}{\mathcal{O}}_d$ of all polynomial knots with the inductive limit topology, we prove that it too has the same homotopy type as $S^2$. We also show that if two polynomial knots are path equivalent in ${\mathcal{Q}}_d$, then they are topologically equivalent. Furthermore, the number of path components in ${\mathcal{Q}}_d$ are in multiples of eight.