Mapping a knot by a continuous map

TL;DR

This paper investigates how a fixed continuous map on 3-space affects knot types, including the resulting knots from a single map and the sequences generated by repeated mappings.

Contribution

It introduces a framework for analyzing the transformation of knots under a fixed continuous map and explores the infinite sequences of knot types produced by iteration.

Findings

Classified possible knot types after a single continuous map

Identified conditions for infinite sequences of knot types

Provided insights into the dynamics of knot transformations

Abstract

By a fixed continuous map from a $3$-space to itself, a knot in the $3$-space may be mapped to another knot in the $3$-space. We analyze possible knot types of them. Then we map a knot repeatedly by a fixed continuous map and analyze possible infinite sequences of knot types.