A linking invariant for algebraic curves

Abstract

We construct a topological invariant of algebraic plane curves, which is in some sense an adaptation of the linking number of knot theory. This invariant is shown to be a generalization of the I-invariant of line arrangements developed by the first author with Artal and Florens. We give two practical tools for computing this invariant, using a modification of the usual braid monodromy or using the connected numbers introduced by Shirane. As an application, we show that this invariant distinguishes several Zariski pairs, i.e. pairs of curves having same combinatorics, yet different topologies. The former is the well known Zariski pair found by Artal, composed of a smooth cubic with 3 tangent lines at its inflexion points. The latter is formed by a smooth quartic and 3 bitangents.

Connected Numbers and the Embedded Topology of Plane Curves

“…Some invariants different from the fundamental group are chosen in the step (i) above: Artal, Cogolludo and Carmona proved that the braid monodromy is a topological invariant of certain plane curves in [2], and there is a complement-equivalent Zariski pair whose embedded topologies are distinguished by the braid monodromy ( [3]). In [10], Guerville and Meilhan defined an invariant of the embedded topology of plane curves, called the linking set; and they discovered a Zariski pair of line arrangements. In this paper, we give a new topological invariant, called a connected number.…”

A note on the topology of arrangements for a smooth plane quartic and its bitangent lines

“…We refer to [2] for results on Zariski pairs before 2008. Within these several years, new approaches to study Zariski pairs for reducible plane curves have been introduced, such as (a) linking sets ( [9]), (b) splitting types ( [3]), (c) splitting and connected numbers ( [15,16]) and (d) the set of subarrangements of B ( [4,5]).…”

“…In [9,4], Zariski pairs for a smooth cubic and its k-inflectional tangents (k ≥ 4) were investigated based on the method (d) as above. This generalizes E. Artal's Zariski pair for a smooth cubic and its three inflectional tangents given in [1].…”

Nodal curves with a contact-conic and Zariski pairs

“…There are may results about Zariski pairs by using some topological invariants of plane curves (cf. [1,2,3,4,5,7,8,9,12,13,14,18]). A basic invariant is the fundamental group of the complement of a plane curve (cf.…”