Berge duals and universally tight contact structures

TL;DR

This paper investigates the properties of dual knots arising from lens space surgeries on knots in S^3, focusing on the contact structures they support, especially universally tight ones, and their homological relations.

Contribution

It establishes that duals supporting universally tight contact structures are homologous to duals of torus knots, revealing a new link between contact topology and knot theory in lens spaces.

Findings

Duals of knots with lens space surgeries can support contact structures.

Universally tight contact structures imply the dual knot is homologous to a torus knot dual.

The homology class of the dual is constrained by the tightness of the contact structure.

Abstract

Dehn surgery on a knot determines a dual knot in the surgered manifold, the core of the filling torus. We consider duals of knots in $S^3$ that have a lens space surgery. Each dual supports a contact structure. We show that if a universally tight contact structure is supported, then the dual is in the same homology class as the dual of a torus knot.