Constructing real rational knots by gluing

TL;DR

This paper reduces the problem of constructing low-degree real rational knots to an algebraic problem involving pure braid groups and predicts their existence based on polygonal edge counts.

Contribution

It introduces an algebraic approach using pure braid groups to construct real rational knots and predicts their degrees based on polygonal representations.

Findings

Reduction of knot construction to pure braid group algebra

Prediction of knot degrees from polygonal edge counts

Framework for constructing low-degree real rational knots

Abstract

We show that the problem of constructing a real rational knot of a reasonably low degree can be reduced to an algebraic problem involving the pure braid group: expressing an associated element of the pure braid group in terms of the standard generators of the pure braid group. We also predict the existence of a real rational knot in a degree that is expressed in terms of the edge number of its polygonal representation.