On Round Surgery Diagrams For 3-Manifolds

TL;DR

This paper introduces round surgery diagrams as a new way to represent 3-manifolds, establishes a calculus for manipulating them, and applies this to prove the existence of certain geometric structures on these manifolds.

Contribution

It defines round surgery diagrams for 3-manifolds, develops a Kirby Calculus for them, and applies this to demonstrate the existence of taut foliations and tight contact structures.

Findings

Established a correspondence between round surgery and Dehn surgery diagrams.

Proved that any two diagrams of the same 3-manifold are related by a finite sequence of moves.

Applied the framework to show existence of taut foliations and tight contact structures.

Abstract

We introduce the notion of round surgery diagrams in $S^3$ for representing 3-manifolds similar to Dehn surgery diagrams. We give a correspondence between a certain class of round surgery diagrams and Dehn surgery diagrams for 3-manifolds. As a consequence, we recover Asimov's result, stating that any closed connected oriented 3-manifold can be obtained by a round surgery on a framed link in $S^3$. There may be more than one round surgery diagram giving rise to the same 3-manifold. Thus, it is natural to ask whether there is a version of Kirby Calculus for round surgery diagrams, similar to the case of Dehn surgery diagrams with integral framings. In this direction, we define four types of moves on round surgery diagrams such that any two round surgery diagrams corresponding to the same 3-manifold can be obtained one from another by a finite sequence of these moves, thereby establishing a version of Kirby Calculus. As an application, we prove the existence of taut foliations, hence the existence of tight contact structures on 3-manifolds obtained by round 1-surgery on fibred links with two components on $S^3$.