On the classification of 2-plat 2-knots

TL;DR

This paper introduces a new invariant to classify 2-plat 2-knots, as traditional double branched covers cannot distinguish them, and explores its implications for invertibility of 2-knots.

Contribution

It develops a novel invariant for classifying 2-plat 2-knots, providing a tool analogous to torsion invariants and an obstruction to invertibility.

Findings

The new invariant distinguishes 2-plat 2-knots beyond double branched covers.

The invariant acts as an analogue of torsion invariants.

It serves as an obstruction to the invertibility of 2-knots.

Abstract

An $n$-plat 1-knot is one isotopic to the plat closure of some $2n$-braid, which is also called an $n$-bridge 1-knot. Schubert classified 2-bridge 1-knots by considering their double branched covers which are homeomorphic to lens spaces. A 2-knot is a 2-sphere smoothly embedded in 4-space or 4-sphere. An $n$-plat 2-knot is one isotopic to the plat closure of some 2-dimensional $2n$-braid. The aim of this paper is to classify 2-plat 2-knots. By a result of Montesinos, double branched covers do not distinguish 2-plat 2-knots. Thus, we introduce a new invariant to classify them. Our invariant serves as an analogue of a torsion invariant. Furthermore, it is an obstruction to invertibility of 2-knots.