Lamination links in 3-manifolds

TL;DR

This paper introduces a new class of lamination links in 3-manifolds, generalizing classical links, and studies their bounding Seifert laminations, including the existence of taut laminations with maximum Euler characteristic.

Contribution

It defines oriented framed measured lamination links and proves the existence of taut Seifert laminations that maximize Euler characteristic within this framework.

Findings

Lamination links generalize classical links in 3-manifolds.

Taut Seifert laminations maximize Euler characteristic.

Continuity of maximum Euler characteristic function on parametrized families.

Abstract

We introduce and define "oriented framed measured lamination links" in a 3-manifold $M$. These generalize oriented framed links in 3-manifolds, and are confined to 2-dimensional improperly embedded subsurfaces of the 3-manifold. Just as some framed links bound Seifert surfaces, so also some framed lamination links bound 2-dimensional measured and oriented "Seifert laminations." We show that any lamination link which bounds a 2-dimensional Seifert lamination, bounds a "taut" Seifert lamination, i.e. one of maximum Euler characteristic, subject to the condition that the Seifert lamination is carried by an aspherical branched surface. This maximum Euler characteristic function is continuous on certain parametrized families of lamination links carried by a train track neighborhood. Taut Seifert laminations generalize minimal genus Seifert surfaces.