Marithania Silvero

Mathematical tools for the future: Graph Theory and graphicable algebras

This study constitutes the continuation of innovative research in Discrete Mathematics introduced in earlier papers on algebras in general, regarding the use of graphs to study the particular case of graphicable algebras, which form a subset of evolution algebras. Evolution algebras are particularly interesting since they are intrinsically linked with other mathematical fields, such as group theory, stochastics processes, and dynamical systems, for instance. Our advances in this study are obtained by setting a natural correspondence between evolution algebras and direct graphs, in order to translate the general concepts of graphicable algebras: subalgebra, ideal, centralizer, normalizer...to the language of graphs. These translations will enable advances in the application of these algebras to various branches of Mathematics.

Alexander–Conway polynomial state model and link homology

This paper shows how the Formal Knot Theory state model for the Alexander–Conway polynomial is related to Knot Floer Homology. In particular, we prove a parity result about the states in this model that clarifies certain relationships of the model with Knot Floer Homology.

Strongly quasipositive links with braid index 3 have positive Conway polynomial

Strongly quasipositive links are those links which can be seen as closures of positive braids in terms of band generators. In this paper we give a necessary condition for a link with braid index 3 to be strongly quasipositive, by proving that in that case it has positive Conway polynomial (that is, all its coefficients are non-negative). We also show that this result cannot be extended to a higher number of strands, as we provide a strongly quasipositive braid whose closure has non-positive Conway polynomial.