English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Tools annual

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Presents the access of premium content as premium feature

Premium Content

Global: Zendy Plus annual

Global: Zendy Free Plan

In this paper, we introduce the relative $\mathcal{L}$-invariant$r\mathcal{L}(X)$ of a smooth, orientable, compact 4-manifold $X$ withboundary. This invariant is defined by measuring the lengths of certain pathsin the cut complex of a trisection surface for $X$. This is motivated by thedefinition of the $\mathcal{L}$-invariant for smooth, orientable, closed4-manifolds by Kirby and Thompson. We show that if $X$ is a rational homologyball, then $r\mathcal{L}(X)=0$ if and only if $X\cong B^4$.  In order to better understand relative trisections, we also produce analgorithm to glue two relatively trisected 4-manifold by any Murasugi sum orplumbing in the boundary, and also prove that any two relative trisections of agiven 4-manifold $X$ are related by interior stabilization, relativestabilization, and the relative double twist, which we introduce in this paperas a trisection version of one of Piergallini and Zuddas's moves on open bookdecompositions. Previously, it was only known (by Gay and Kirby) that relativetrisections inducing equivalent open books on $X$ are related by interiorstabilizations.

The relative $\mathcal{L}$-invariant of a compact $4$-manifold