Incompressible one-sided surfaces in even fillings of Figure 8 knot space

TL;DR

This paper classifies geometrically incompressible one-sided surfaces in certain hyperbolic 3-manifolds obtained from Figure 8 knot space via Dehn fillings, revealing their uniqueness based on filling ratios.

Contribution

It provides a complete classification of incompressible one-sided surfaces in even fillings of Figure 8 knot space, linking their existence to filling ratios and establishing their uniqueness.

Findings

Incompressible one-sided surfaces are uniquely determined by the filling ratio p/q.

Classification applies to hyperbolic, non-Haken manifolds obtained from Figure 8 knot space.

No similar classification exists for two-sided Heegaard splittings.

Abstract

In the closed, non-Haken, hyperbolic class of examples generated by (2p,q) Dehn fillings of Figure 8 knot space, the geometrically incompressible one-sided surfaces are identified by the filling ratio p/q and determined to be unique in all cases. When applied to one-sided Heegaard splittings, this can be used to classify all geometrically incompressible splittings in this class of closed, hyperbolic examples; no analogous classification exists for two-sided Heegaard splittings.