Abstract
A relative trisection of a compact [Formula: see text]-manifold with connected boundary is said to be a Trivially Extendable (TE)-relative trisection if the boundary monodromy of the relative trisection is the identity map. We call a relative trisection diagram associated to a [Formula: see text]-relative trisection of a compact [Formula: see text]-manifold, a [Formula: see text]-relative trisection diagram. Due to a work of Castro and Ozbagci, there is a natural way to associate a trisection diagram to a [Formula: see text]-relative trisection diagram. In fact, every closed oriented [Formula: see text]-manifold can be realized as a [Formula: see text]-relative trisection. In this paper, we study the notion of Murasugi sum of [Formula: see text]-relative trisection diagrams. We show that a trisection diagram associated to a Murasugi sum of two [Formula: see text]-relative trisection diagrams [Formula: see text] and [Formula: see text] is diffeomorphism and handle slide equivalent to the connected sum of the trisection diagrams associated to [Formula: see text] and [Formula: see text]. This implies that the closed oriented [Formula: see text]-manifold associated to the Murasugi sum of [Formula: see text]-relative trisection diagrams [Formula: see text] and [Formula: see text] is the connected sum of the closed [Formula: see text]-manifolds associated to [Formula: see text] and [Formula: see text]. Thus, the notion of Murasugi sum can be considered for closed [Formula: see text]-manifolds in terms of [Formula: see text]-relative trisections. Finally, we discuss an external stabilization of a [Formula: see text]-relative trisection of a closed oriented [Formula: see text]-manifold. We prove that the trisection diagram associated to a stabilization of a [Formula: see text]-relative trisection diagram is diffeomorphism and handle slide equivalent to a stabilization of the trisection diagram associated to the [Formula: see text]-relative trisection diagram.
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