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Abstract

In this paper, we introduce the relative $\mathcal{L}$-invariant
$r\mathcal{L}(X)$ of a smooth, orientable, compact 4-manifold $X$ with
boundary. This invariant is defined by measuring the lengths of certain paths
in the cut complex of a trisection surface for $X$. This is motivated by the
definition of the $\mathcal{L}$-invariant for smooth, orientable, closed
4-manifolds by Kirby and Thompson. We show that if $X$ is a rational homology
ball, then $r\mathcal{L}(X)=0$ if and only if $X\cong B^4$.
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 $X$ 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 $X$ are related by interior
stabilizations.