Some spaces of polynomial knots

Hitesh Raundal, Rama Mishra

Published 2016-03-30Version 1

In this paper we study the topology of three different kind of spaces associated to polynomial knots of degree at most $d$, for $d\geq2$. We denote these spaces by $\mathcal{O}_d$, $\mathcal{P}_d$ and $\mathcal{Q}_d$. For $d\geq3$, we show that the spaces $\mathcal{O}_d$ and $\mathcal{P}_d$ are path connected and the space $\mathcal{O}_d$ has homotopy type of $S^2$. Considering the space $\mathcal{P}=\bigcup_{d\geq2}\mathcal{O}_d$ 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 the number of path components of the space $\mathcal{Q}_d$, for $d\geq 2$, are in multiple of eight. Furthermore, we prove that the path components of the space $\mathcal{Q}_d$ are contractible.