Trisections obtained by trivially regluing surface-knots

Abstract

AbstractLet 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.