Relate the topological and geometric asymptotic intersection terms

Determine the relation between the topological asymptotic term defined through linking numbers for broken Floer cylinders and the classical geometric asymptotic term defined by Siefring through winding numbers near limiting orbits.

Background

The paper introduces a topological intersection number for Floer cylinders whose limiting orbits agree at a breaking orbit. In this degenerate situation, the intersection number contains an additional asymptotic contribution defined as the difference between linking numbers of two-component links in a local three-sphere neighborhood of the breaking orbit.

Siefring’s theory instead defines an asymptotic intersection contribution for pseudoholomorphic curves using winding numbers near their limiting orbits. The unresolved problem is to establish how the paper’s linking-number-based invariant corresponds to Siefring’s geometric invariant.