Obtaining genus 2 Heegaard splittings from Dehn surgery

TL;DR

This paper investigates the conditions under which Dehn surgery on hyperbolic knots in S^3 produces 3-manifolds of genus 2, revealing constraints on the knot’s bridge number and tunnel number based on embedded surfaces.

Contribution

It establishes new relationships between Dehn surgery outcomes and the topological properties of the resulting 3-manifolds, especially regarding embedded surfaces and knot bridge/tunnel numbers.

Findings

If the manifold lacks an embedded Dyck's surface, the dual knot is 0- or 1-bridge.

Presence of a Dyck's surface leads to similar constraints on the knot.

Without an incompressible genus 2 surface, the knot's tunnel number is at most 2.

Abstract

Let K' be a hyperbolic knot in S^3 and suppose that some Dehn surgery on K' with distance at least 3 from the meridian yields a 3-manifold M of Heegaard genus 2. We show that if M does not contain an embedded Dyck's surface (the closed non-orientable surface of Euler characteristic -1), then the knot dual to the surgery is either 0-bridge or 1-bridge with respect to a genus 2 Heegaard splitting of M. In the case M does contain an embedded Dyck's surface, we obtain similar results. As a corollary, if M does not contain an incompressible genus 2 surface, then the tunnel number of K' is at most 2.