Grid invariants in universally tight lens spaces

TL;DR

This paper introduces combinatorial invariants for Legendrian and transverse links in universally tight lens spaces using grid diagrams, establishing their equivalence to existing invariants and exploring applications to knot surgeries and the Berge conjecture.

Contribution

It generalizes grid invariants to lens spaces and proves their equivalence to known invariants, with potential applications to knot surgery classifications.

Findings

Invariants are equivalent to existing Legendrian and transverse invariants.

Characterization of index one grid diagrams for knots with surgeries to S^3.

Discussion of applications to the Berge conjecture.

Abstract

We define combinatorial invariants of Legendrian and transverse links in universally tight lens spaces using grid diagrams, generalizing [OST08] and prove that they are equivalent to the invariants defined in [BVVV13] and [LOSS09]. We use these combinatorial invariants to characterize index one grid diagrams for knots in lens spaces which admit surgeries to the 3-sphere and discuss a potential application to the Berge conjecture.