Incompressible one-sided surfaces in filled link spaces

TL;DR

This paper proves the uniqueness of boundary incompressible positions of geometrically incompressible one-sided surfaces in filled link spaces and classifies certain one-sided splittings in specific knot space fillings.

Contribution

It introduces a boundary incompressibility result for one-sided surfaces in Dehn filled link manifolds and completes the classification of one-sided splittings in (2p, q) fillings of the Figure 8 knot space.

Findings

Unique boundary incompressible position for the surfaces

Classifies (2p, q) fillings with |2p/q|<3 as having two non-isotopic surfaces

Uses sweep-out technique from Heegaard splitting theory

Abstract

When a Dehn filled link manifold contains a geometrically incompressible one-sided surface, it is shown there is a unique boundary incompressible position that the surface can take in the link space. The proof uses a version of the sweep-out technique from two-sided Heegaard splitting theory. When applied to one-sided Heegaard splittings, this result can be used to complete the classification of one-sided splittings of (2p, q) fillings of Figure 8 knot space: determining that fillings with |2p/q|<3 have two non-isotopic geometrically incompressible one-sided splitting surfaces.