Vincent Florens

A topological invariant of line arrangements

We define a new topological invariant of line arrangements in the complex projective plane. This invariant is a root of unity defined under some combinatorial restrictions for arrangements endowed with some special torsion character on the fundamental group of their complements. It is derived from the peripheral structure on the group induced by the inclusion map of the boundary of a tubular neigborhood in the exterior of the arrangement. By similarity with knot theory, it can be viewed as an analogue of linking numbers. This is an orientation-preserving invariant for ordered arrangements. We give an explicit method to compute the invariant from the equations of the arrangement, by using wiring diagrams introduced by Arvola, that encode the braid monodromy. Moreover, this invariant is a crucial ingredient to compute the depth of a character satisfying some resonant conditions, and complete the existent methods by Libgober and the first author. Finally, we compute the invariant for extended MacLane arrangements with an additional line and observe that it takes different values for the deformation classes.

On the slice genus of links

We define Casson-Gordon σ-invariants for links and give a lower bound of the slice genus of a link in terms of these invariants. We study as an example a family of two component links of genus h and show that their slice genus is h, whereas the Murasugi-Tristram inequality does not obstruct this link from bounding an annulus in the 4-ball. AMS Classification 57M25; 57M27

On the Fox-Milnor theorem for the Alexander polynomial of links

We show that the Alexander polynomial Δ L (t 1 ,…,t μ ) of a link L in S 3 bounding a smooth compact oriented surface in B 4 with μ connected components and Euler characteristic 1 is of the form p(t1,…,tμ)p(t1−1,…,tμ−1) for p ∈ ℤ[t 1 ,…t μ ], up to a unit and factors t k −1.

The Signature of a Splice

We study the behavior of the signature of colored links [Flo05, CF08] under the splice operation. We extend the construction to colored links in integral homology spheres and show that the signature is almost additive, with a correction term independent of the links. We interpret this correction term as the signature of a generalized Hopf link and give a simple closed formula to compute it.

ON COMPLEX LINE ARRANGEMENTS AND THEIR BOUNDARY MANIFOLDS

Let A be a line arrangement in the complex projective plane P2. We consider the boundary manifold, defined as the boundary of a close regular neighborhood of A in P2 and study the inclusion map on the complement. We give an explicit method to compute the map induced on the fundamental groups. This extends the work of E.Hironaka on the homotopy type of the complement of (complexified) real arrangements to any complex arrangement.