Some spaces of polynomial knots

TL;DR

This paper investigates the topology of spaces of polynomial knots, revealing their homotopy types, connectivity, and relationships between different spaces, with implications for knot classification.

Contribution

It establishes the homotopy type of polynomial knot spaces and their connectivity properties, providing new insights into their topological structure.

Findings

Spaces  and  are path connected for d3

Space  has the same homotopy type as S^2

Number of path components in  are multiples of eight

Abstract

In this paper we study the topology of three different kinds 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 the same homotopy type as $S^2$. 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 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.