Surgery on a knot in (Surface x I)

TL;DR

This paper investigates how surgeries on knots in surface cross interval manifolds affect the topology, revealing conditions under which the resulting manifold becomes reducible or contains specific annuli.

Contribution

It establishes a relationship between surgeries on knots in surface x I and the existence of annuli, providing criteria for reducibility and knot placement in fibered 3-manifolds.

Findings

Existence of an annulus linking the knot to an essential curve in the surface.

Conditions under which surgery yields a reducible manifold.

Characterization of knots leading to reducible surgeries in fibered manifolds.

Abstract

Suppose F is a compact orientable surface, K is a knot in F x I, and N is the 3-manifold obtained by some non-trivial surgery on K. If F x {0} compresses in N, then there is an annulus in F x I with one end K and the other end an essential simple closed curve in F x {0}. Moreover, the end of the annulus at K determines the surgery slope. An application: suppose M is a compact orientable 3-manifold that fibers over the circle. If surgery on a knot K in M yields a reducible manifold, then either: the projection of K to S^1 has non-trivial winding number; or K lies in a ball; or K lies in a fiber; or K is a cabled knot.