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.

设S是一个\(P^2\) -knot，它是具有正常欧拉数 0 的 2-knot 和具有正常欧拉数\({\pm }的无结 \ (P^2\) -knot的连通和{2}\)在一个三等分的封闭 4 流形X中\(T_{X}\)。然后，我们证明通过对\(\overline{\nu (S)}\)和\(X-\nu (S)\)的相对三等分进行简单粘合而获得的X的三等分对于稳定化是微分同胚的\ (T_{X}\)。应该注意的是，这个结果并不明显，因为 Kim 和 Miller 引入的边界稳定用于构造\(X-\nu (S)\)的相对三等分。作为推论，如果\(X=S^4\)和\(T_X\)是\(S^4\)的属 0 三等分，则所得三等分与\(的属 0 三等分的稳定性是微分同胚的S^4\)。该结果与瓦尔德豪森 Heegaard 分裂定理的 4 维模拟猜想相关。