Finding planar surfaces in knot- and link-manifolds

TL;DR

This paper presents algorithms for detecting embedded essential planar surfaces and punctured-disks in link-manifolds using advanced normal surface theory techniques, including properties of specialized triangulations and boundary length estimates.

Contribution

It introduces novel algorithms that decide the existence of embedded essential planar surfaces and punctured-disks in link-manifolds, utilizing non-classical normal surface methods and triangulation properties.

Findings

Algorithms for detecting embedded essential planar surfaces.

Algorithms for determining boundary slopes bounding punctured-disks.

Use of advanced triangulation techniques and boundary length estimates.

Abstract

It is shown that given any link-manifold, there is an algorithm to decide if the manifold contains an embedded, essential planar surface; if it does, the algorithm will construct one. If a slope on the boundary of the link-manifold is given, there is an algorithm to determine if the slope bounds an embedded punctured-disk; if a meridian slope is given, then it can be determined if a longitude bounds an embedded punctured-disk. The methods use normal surface theory but do not follow the classical approach. Properties of minimal vertex triangulations, layered-triangulations, 0--efficient triangulations and especially triangulated Dehn fillings are central to our methods. We also use an average length estimate for boundary curves of embedded normal surfaces; a version of the average length estimate with boundary conditions also is derived. An algorithm is given to construct precisely those slopes on the boundary of a given link-manifold that bound an embedded punctured-disk in the link-manifold.