Generating family invariants for Legendrian links of unknots

TL;DR

This paper develops a theory for generating family invariants of Legendrian links of unknots in R^3, providing tools to distinguish Legendrian links beyond classical invariants and connecting to algebraic invariants.

Contribution

It introduces a unique linear-quadratic at infinity generating family for the maximal unknot and constructs invariant polynomials for Legendrian links, with computational techniques and comparisons to algebraic invariants.

Findings

Generated polynomials distinguish non-Legendrian equivalent links with same classical invariants.

Computed generating family polynomials for rational and twist Legendrian links.

Found agreement between generating family polynomials and linearized DGA polynomials.

Abstract

Theory is developed for linear-quadratic at infinity generating families for Legendrian knots in R^3. It is shown that the unknot with maximal Thurston--Bennequin invariant of -1 has a unique linear-quadratic at infinity generating family, up to fiber-preserving diffeomorphism and stabilization. From this, invariant generating family polynomials are constructed for 2-component Legendrian links where each component is a maximal unknot. Techniques are developed to compute these polynomials, and computations are done for two families of Legendrian links: rational links and twist links. The polynomials allow one to show that some topologically equivalent links with the same classical invariants are not Legendrian equivalent. It is also shown that for these families of links the generating family polynomials agree with the polynomials arising from a linearization of the differential graded algebra associated to the links.