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

TL;DR

This paper investigates how different topologies on spaces of polynomial knots in various dimensions affect their homotopy types, extending previous results from three-dimensional knots to higher dimensions and unions.

Contribution

It generalizes the study of polynomial knot spaces by analyzing the impact of various topologies on their homotopy types across different dimensions.

Findings

Homotopy type of polynomial knot spaces can vary with topology.

The space of polynomial knots in $ ext{R}^3$ has the same homotopy type as $S^2$.

Homotopy types of higher-dimensional polynomial knot spaces are characterized.

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 $d\geq3$, 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 $n\geq1$. We also explore the homotopy type of the space $\mathcal{L}$ with some natural topologies on it.