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.Ministerio de Ciencia e InnovaciónJunta de AndalucíaFondo Europeo de Desarrollo Regiona

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.Ministerio de Ciencia e InnovaciónJunta de Andalucía (Consejería de Innovación, Ciencia y Empresa)Fondo Europeo de Desarrollo Regiona

A geometric description of the extreme Khovanov cohomology

We prove that the hypothetical extreme Khovanov cohomology of a link is the cohomology of the independence simplicial complex of its Lando graph. We also provide a family of knots having as many non-trivial extreme Khovanov cohomology modules as desired, that is, examples of H-thick knots which are as far of being H-thin as desired.Ministerio de Economía y CompetitividadFondo Europeo de Desarrollo Regiona

A particular type of non-associative algebras and graph theory

Evolution algebras have many connections with other mathematical fields, like group theory, stochastics processes, dynamical systems and other related ones. The main goal of this paper is to introduce a novel non-usual research on Discrete Mathematics regarding the use of graphs to solve some open problems related to the theory of graphicable algebras, which constitute a subset of those algebras. We show as many our advances in this field as other non solved problems to be tackled in future