Trisections obtained by trivially regluing surface-knots

2022/05/10 by Isoshima, Tsukasa
#FOS: Mathematics #Geometric Topology (math.GT)

paper · doi:10.48550/arxiv.2205.04817

Abstract

Let S be a P²-knot which is the connected sum of a 2-knot with normal Euler number 0 and an unknotted P²-knot with normal Euler number ±2 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 ν(S) and X-ν(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-ν(S). As a corollary, if X=S⁴, the resulting trisection is diffeomorphic to a stabilization of the genus 0 trisection of S⁴. This result is related to the conjecture that is a 4-dimensional analogue of Waldhausen's theorem on Heegaard splittings.

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