Intermediate links of plane curves

TL;DR

This paper explores the relationship between the topology of complex plane curves and their links, establishing a connection between Euler characteristics, rotation numbers, and braid properties through stereographic projections.

Contribution

It introduces a new link invariant relation for smooth complex curves and demonstrates how certain link properties influence braid positivity and fibredness.

Findings

Euler characteristic equals rotation number minus writhe for the projected link diagram.

If the diagram has no negative Seifert circles and the link is strongly quasipositive and fibred, then the Yamada-Vogel algorithm produces a quasipositive braid.

The results connect topological invariants of plane curves with braid theory and link positivity.

Abstract

For a smooth complex curve C, we consider the link L(r) intersection of C with the boundary of B(r), where B(r) denotes an Euclidean ball of radius r>0. We prove that the diagram D(r) obtained from L(r) by a complex stereographic projection satisfies that the Euler characteristic of the part of C in B(r) equals the rotation number of D(r) minus the writhe of D(r). As a consequence we show that if D(r) has no negative Seifert circles and L(r) is strongly quasipositive and fibred, then the Yamada-Vogel algorithm applied to D(r) yields a quasipositive braid.