Fibered knots and Property 2R

TL;DR

This paper investigates the properties of fibered knots in potential counterexamples to the Generalized Property R Conjecture, showing that minimal genus components in such counterexamples are not fibered, and analyzing their monodromy especially for genus two fibers.

Contribution

It demonstrates that minimal genus components in counterexamples are not fibered and explores the monodromy of fibered components, especially for genus two fibers, using sutured manifold theory.

Findings

Minimal genus components in counterexamples are not fibered.

The monodromy of genus two fibers is characterized.

Conditions on fibered knots in counterexamples are identified.

Abstract

It is shown, using sutured manifold theory, that if there are any 2-component counterexamples to the Generalized Property R Conjecture, then any knot of least genus among components of such counterexamples is not a fibered knot. The general question of what fibered knots might appear as a component of such a counterexample is further considered; much can be said about the monodromy of the fiber, particularly in the case in which the fiber is of genus two.