Trisections obtained by trivially regluing surface-knots

Let S be a P² P 2 -knot which is the connected sum of a 2-knot with normal Euler number 0 and an unknotted P² P 2 -knot with normal Euler number ± 2 ± 2 in a closed 4-manifold X with trisection TX T X . Then, we show that the trisection of X obtained by the trivial gluing of relative trisections of ν (S)̄ ν ( S ) ¯ and X-ν (S) X - ν ( S ) is diffeomorphic to a stabilization of TX 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-ν (S) X - ν ( S ) . As a corollary, if X=S⁴ X = S 4 and TX T X was the genus 0 trisection of S⁴ S 4 , the resulting trisection is diffeomorphic to a stabilization of the genus 0 trisection of S⁴ S 4 . This result is related to the conjecture that is a 4-dimensional analogue of Waldhausen’s theorem on Heegaard splittings.