Braid and knot invariants via triangulations and Ptolemy relations I

TL;DR

This paper introduces a novel method for constructing invariants of braids, knots, and links by analyzing point dynamics in the plane and utilizing Ptolemy relations, bridging geometric and algebraic approaches.

Contribution

It presents a new framework combining geometric dynamics and Ptolemy relations to define invariants for braids, knots, and links, advancing topological invariant theory.

Findings

New invariants for braids, knots, and links derived

Application of Ptolemy relations in topological invariants

Potential connections to quantum topology and geometric structures

Abstract

In this paper, we construct invariants of braids, knots and links by studying dynamics of points in $\R^{2}$ and applying the Ptolemy relation $ac+bd=xy$.