Trisections obtained by trivially regluing surface-knots

TL;DR

This paper demonstrates that certain trisections of 4-manifolds obtained by trivial regluing of surface-knots are equivalent to stabilized versions of the original trisection, extending understanding of 4-manifold decompositions.

Contribution

It introduces a method to obtain trisections via trivial regluing of surface-knots, linking to 4D analogues of classical 3D theorems.

Findings

Trisection after trivial regluing is diffeomorphic to a stabilization of the original.

Result applies to specific surface-knots with normal Euler numbers 0 and ±2.

In the case of S^4, the trisection is a stabilization of the genus 0 trisection.

Abstract

Let $S$ be a $P^2$-knot which is the connected sum of a 2-knot with normal Euler number 0 and an unknotted $P^2$-knot with normal Euler number $\pm2$ in a closed 4-manifold $X$ with trisection $T_{X}$. Then, we show that the trisection of $X$ obtained by the trivial gluing relative trisections of $\overline{\nu(S)}$ and $X-\nu(S)$ is diffeomorphic to a stabilization of $T_{X}$. It should be noted that this result is not obvious since boundary-stabilizations introduced by Kim and Miller are used to construct a relative trisection of $X-\nu(S)$. As a corollary, if $X=S^4$, the resulting trisection is diffeomorphic to a stabilization of the genus 0 trisection of $S^4$. This result is related to the conjecture that is a 4-dimensional analogue of Waldhausen's theorem on Heegaard splittings.