192 research outputs found
The relative -invariant of a compact -manifold
In this paper, we introduce the relative -invariant
of a smooth, orientable, compact 4-manifold with
boundary. This invariant is defined by measuring the lengths of certain paths
in the cut complex of a trisection surface for . This is motivated by the
definition of the -invariant for smooth, orientable, closed
4-manifolds by Kirby and Thompson. We show that if is a rational homology
ball, then if and only if .
In order to better understand relative trisections, we also produce an
algorithm to glue two relatively trisected 4-manifold by any Murasugi sum or
plumbing in the boundary, and also prove that any two relative trisections of a
given 4-manifold are related by interior stabilization, relative
stabilization, and the relative double twist, which we introduce in this paper
as a trisection version of one of Piergallini and Zuddas's moves on open book
decompositions. Previously, it was only known (by Gay and Kirby) that relative
trisections inducing equivalent open books on are related by interior
stabilizations.Comment: 39 pages, 14 figures. v2: Significantly improved discussion of
relative double twist (Section 2), changed some statements in Section 4,
restructured Sections 3 and
Belnap-Dunn semantics for natural implicative expansions of Kleene's strong three-valued matrix with two designated values
27 p.A conditional is natural if it fulfils the three following conditions. (1) It coincides with the classical conditional when restricted to the classical values T and F; (2) it satisfies the Modus Ponens; and (3) it is assigned a designated value whenever the value assigned to its antecedent is less than or equal to the value assigned to its consequent. The aim of this paper is to provide a ‘bivalent’ Belnap-Dunn semantics for all natural implicative expansions of Kleene's strong 3-valued matrix with two designated elements. (We understand the notion ‘natural conditional’ according to N. Tomova, ‘A lattice of implicative extensions of regular Kleene's logics’, Reports on Mathematical Logic, 47, 173–182, 2012.)S
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