Convex Embeddability and Knot Theory

TL;DR

This paper explores the complexity of convex embeddability in linear and circular orders, applying these findings to analyze the combinatorial properties and classification challenges of arcs and knots.

Contribution

It introduces new combinatorial and set-theoretic results on convex embeddability and applies them to knot theory, establishing lower bounds for the complexity of relations between knots.

Findings

Established combinatorial properties of convex embeddability

Extended results to circular orders

Provided lower bounds for complexity of knot relations

Abstract

We consider countable linear orders and study the quasi-order of convex embeddability and its induced equivalence relation. We obtain both combinatorial and descriptive set-theoretic results, and further extend our research to the case of circular orders. These results are then applied to the study of arcs and knots, establishing combinatorial properties and lower bounds (in terms of Borel reducibility) for the complexity of some natural relations between these geometrical objects.