Fibered knots and Property 2R, II

TL;DR

This paper investigates Property 2R for knots, especially the square knot, characterizing related links and exploring whether handle slides can reduce certain links to unlinks, with implications for 4-manifold theory.

Contribution

It characterizes all two-component links containing the square knot that surger to connected sums of S^1 x S^2 and argues the square knot likely lacks Property 2R, introducing a weaker property involving Hopf pairs.

Findings

Characterization of links containing the square knot that surger to S^1 x S^2 # S^1 x S^2

Evidence suggesting the square knot probably does not have Property 2R

Discussion of a weaker property involving addition and removal of Hopf pairs

Abstract

A knot K in the 3-sphere is said to have Property nR if, whenever K is a component of an n-component link L and some integral surgery on L produces the connected sum of n copies of S^1 x S^2, there is a sequence of handle slides on L that converts L into a 0-framed unlink. The Generalized Property R Conjecture is that all knots have Property nR for all n. The simplest plausible counterexample could be the square knot. Exploiting the remarkable symmetry of the square knot, we characterize all two-component links that contain it and which surger to S^1 x S^2 # S^1 x S^2. We argue that at least one such link probably cannot be reduced to the unlink by a series of handle-slides, so the square knot probably does not have Property 2R. This example is based on a classic construction of the first author. On the other hand, the square knot may well satisfy a somewhat weaker property, which is still useful in 4-manifold theory. For the weaker property, copies of canceling Hopf pairs may be added to the link before the handle slides and then removed after the handle slides. Beyond our specific example, we discuss the mechanics of how addition and later removal of a Hopf pair can be used to reduce the complexity of a surgery description.