Asymptotic link invariants for ergodic vector fields

TL;DR

This paper investigates the long-term behavior of a class of link invariants associated with ergodic vector fields, revealing that their asymptotic properties are dominated by a single invariant, the asymptotic signature.

Contribution

It introduces the concept of asymptotic linear saddle invariants and shows they form a 1-dimensional space generated by the asymptotic signature, connecting it to the asymptotic slice genus.

Findings

Asymptotic linear saddle invariants form a 1-dimensional space.

The asymptotic signature is the generator of this space.

A relationship between asymptotic slice genus and asymptotic signature is established.

Abstract

We study the asymptotics of a family of link invariants on the orbits of a smooth volume-preserving ergodic vector field on a compact domain of the 3-space. These invariants, called linear saddle invariants, include many concordance invariants and generate an infinite-dimensional vector space of link invariants. In contrast, the vector space of asymptotic linear saddle invariants is 1-dimensional, generated by the asymptotic signature. We also relate the asymptotic slice genus to the asymptotic signature.