Intersecting Braids and Intersecting Knot Theory

TL;DR

This paper extends braid and knot theory to include intersecting braids with double and triple intersections, proving an extended Alexander theorem, constructing invariants, and exploring potential applications in quantum gravity.

Contribution

It introduces new operators for intersecting braids, proves an extended Alexander theorem for certain intersections, and develops intersecting knot invariants, advancing the mathematical framework of knot theory.

Findings

Extended Alexander theorem for double and triple intersections

Counterexample for quadruple intersections

Construction of intersecting knot invariants

Abstract

An extension of the Artin Braid Group with new operators that generate double and triple intersections is considered. The extended Alexander theorem, relating intersecting closed braids and intersecting knots is proved for double and triple intersections, and a counter example is given for the case of quadruple intersections. Intersecting knot invariants are constructed via Markov traces defined on intersecting braid algebra representations, and the extended Turaev representation is discussed as an example. Possible applications of the formalism to quantum gravity are discussed.