Trisections obtained by trivially regluing surface-knots

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 \({\pm }{2}\) in a closed 4-manifold X with trisection \(T_{X}\). Then, we show that the trisection of X obtained by the trivial gluing of 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 \(X=S^4\) and \(T_X\) was the genus 0 trisection of \(S^4\), 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.