Topologies on sets of polynomial knots and the homotopy types of the respective spaces

Abstract

A polynomial knot in \(\mathbb {R}^n\) is a smooth embedding of \(\mathbb {R}\) in \(\mathbb {R}^n\) such that the component functions are real polynomials. In the earlier paper with Mishra, we have studied the space \(\mathcal {P}\) of polynomial knots in \(\mathbb {R}^3\) with the inductive limit topology coming from the spaces \(\mathcal {O}_d\) for \(d\ge 3\), where \(\mathcal {O}_d\) is the space of polynomial knots in \(\mathbb {R}^3\) with degree d and having some conditions on the degrees of the component polynomials. In the same paper, we have proved that the space of polynomial knots in \(\mathbb {R}^3\) has the same homotopy type as \(S^2\). The homotopy type of the space is the mere consequence of the topology chosen. If we have another topology on \(\mathcal {P}\), the homotopy type may change. With this in mind, we consider in general the set \(\mathcal {L}^n\) of polynomial knots in \(\mathbb {R}^n\) with various topologies on it and study the homotopy type of the respective spaces. Let \(\mathcal {L}\) be the union of the sets \(\mathcal {L}^n\) for \(n\ge 1\). We also explore the homotopy type of the space \(\mathcal {L}\) with some natural topologies on it.