Taming Wild Knots with Mosaics

TL;DR

This paper introduces a novel mosaic-based framework to represent and analyze wild knots with isolated wild points, extending existing knot mosaic theory to complex topological structures.

Contribution

It extends knot mosaic theory to include wild knots with isolated wild points using infinite rooted trees and embedding functions.

Findings

Represented wild knots using infinite rooted trees with mosaics.

Introduced mosaic tangles and mosaic rigid vertex spatial graphs.

Provided a new framework for classifying complex wild knots.

Abstract

Wild knots--knots with infinite knotting behavior--have resisted traditional methods of knot classification, making them more of a curiosity in topology than a subject of sustained investigation. In this paper, we present a new way to investigate these objects. We extend Lomonaco and Kauffman's knot mosaic theory to represent a substantial subclass of wild knots that have isolated wild points. Our mosaics consist of infinite rooted trees with mosaics assigned to vertices and embedding functions governing connections. In developing this framework, we also introduce a notion of mosaic tangles as well as mosaic rigid vertex spatial graphs of which mosaic singular knots are a special case.