Let S be a $$P^2$$P 2 -knot which is the connected sum of a 2-knot with normal Euler number 0 and an unknotted $$P^2$$P 2 -knot with normal Euler number $${\pm }{2}$$± 2 in a closed 4-manifold X with trisection $$T_{X}$$T X . Then, we show that the trisection of X obtained by the trivial gluing of relative trisections of $$\overline{\nu (S)}$$ν ( S ) ¯ and $$X-\nu (S)$$X - ν ( S ) is diffeomorphic to a stabilization of $$T_{X}$$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 $$X-\nu (S)$$X - ν ( S ) . As a corollary, if $$X=S^4$$X = S 4 and $$T_X$$T X was the genus 0 trisection of $$S^4$$S 4 , the resulting trisection is diffeomorphic to a stabilization of the genus 0 trisection of $$S^4$$S 4 . This result is related to the conjecture that is a 4-dimensional analogue of Waldhausen’s theorem on Heegaard splittings.

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.

Gay and Meier asked whether or not a trisection diagram obtained by the Gluck twist on a spun or a twist spun 2-knot obtained from some method is standard. In this paper, we depict the trisection diagrams explicitly when the 2- knot is the spun $(2n + 1, -2)$-torus knot, where $n\geq1$, and show that the trisection diagram is standard when $n = 1$. Moreover, we introduce a notion of homologically standard for trisection diagrams and show that the trisection diagram is homologically standard for all $n$.

We show that certain two kinds of trisection diagrams of the doubles of the Mazur type 4-manifolds introduced by Akbulut and Kirby are standard. One is constructed by doubling a certain relative trisection diagram of the Mazur type. The other is constructed by using an algorithm taking Kirby diagrams to trisection diagrams.

Gay and Meier asked if a trisection diagram for the Gluck twist on a spun or twist-spun 2-knot in $S^4$ obtained by a certain method is standard. In this paper, we show that the trisection diagram for the Gluck twist on the spun $(p+1,p)$-torus knot is standard, where $p$ is any integer greater than or equal to 2.