Summer school on Alexander polynomials: abstracts
May 30–June 3, 2022
University of Nantes

Schedule: All lectures will be held at the Campus Sciences of the University of Nantes, in Building 26 (Bâtiment 26), Room 121. Lunch will be at the university cafeteria, but you're free to look for alternatives (you can ask me or the locals once you're here). The conference dinner will be on Wednesday evening.

	Monday	Tuesday	Wednesday	Thursday	Friday
9:30-10:30	Ben Aribi	Ben Aribi	Florens	Florens	Florens
10:30-11:00	Coffee break	Coffee break	Coffee break	Coffee break	Coffee break
11:00-12:00	Florens	Florens	Ben Aribi		Ben Aribi
12:00-14:30	Lunch	Lunch	Lunch	Lunch	Lunch
14:00-15:00	Ben Aribi	Exercises BA		Short talks
15:30-16:30				Exercises F

Mini-courses:

Fathi Ben Aribi (UCLouvain)
Twisted Alexander invariants for knots
Since its discovery one century ago, the Alexander polynomial of knots has admitted various alternate definitions.
Several of these definitions can be adapted to construct Twisted Alexander Polynomials, each one associated to a knot and a representation of the knot group, in order to detect more topological information of the knot.
In this mini-course, we will review the theories of Fox calculus and Reidemeister torsions, and the definitions of (twisted) Alexander polynomials they provide.
We also hope to tackle L²-Alexander invariants of knots, which are associated to regular infinite-dimensional representations of the knot groups.
Along the way, we will study several connections between these twisted invariants and other knot invariants (genus, hyperbolic volume...), and we will describe some computation techniques.

Vincent Florens (Université de Pau)
Alexander invariants of plane algebraic curves
The study of the topology of plane algebraic curves was initiated by Enriques and Zariski in the 20s. The fundamental group of their complement is a natural invariant, strong enough to show that the description of the singularities of a curve may not determine its embedding. Since there is no classification of fundamental groups of curves, and the isomorphism problem is undecidable, one cannot directly use this invariant in an effective way. The Alexander polynomial provides an accessible invariant of the group, still very sensitive to the topology of the embedding.
The aim of this lecture is to survey the method of construction and calculation of the group and the Alexander module of a curve complement, and their main properties.
We will start with an overview on the topology of singularities and the classification of algebraic links, in relation with historical methods of Newton and Puiseux, and Milnor's theory. Then we will describe how to compute the fundamental group of a curve complement, in terms of the braid monodromy, introduced by Chisini and Moishezon. From this calculation, we will introduce the Alexander polynomial of an affine or projective curve and its main properties, according to the results of Libgober (and many others). We will also present different generalisations derived from the Alexander module and the twisted homology of a curve complement, in relation with finite abelian coverings. Finally, we will survey some recent developments on characteristic varieties and existence of pencils, on twisted Alexander polynomials, and invariants derived from the boundary manifold of the exterior of a curve.
Along the way, we will emphasize the several parallels of this subject with knot theory.
An exercice session will be devoted to basic examples and computations, with a focus on the particular case of line arrangements (i.e. curves whose components have all degree one).

Short talks:

Vitalijs Brejevs (University of Glasgow)
Computing twisted Alexander polynomials with SageMath
I will give a quick practical overview of how one can compute twisted Alexander polynomials of knots with a view towards applying the Herald–Kirk–Livingston sliceness obstruction, based on the algorithm of Aceto–Meier–Miller–Miller–Park–Stipsicz. I'll also discuss the uses and limitations of the built-in method in SnapPy for computing this obstruction.

Lisa Lokteva (University of Glasgow)
Surgeries on torus knots bounding rational homology balls
In 2020, Aceto, Golla, Larson and Lecuona classified the positive integral surgeries on positive torus knots that bound rational homology balls. We want to generalise this classification to include rational surgeries. We construct multiple new families of rational surgeries on torus knots that bound rational homology 4-balls.

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