Jorge L. Ramirez Alfonsin

arXiv2025

Strong Geometry : Knots

In this paper, we introduce the notion of strong geometry, a structure composed by both the chirotope of a set of points X in the d-dimensional space and the wedge chirotope which is the specific adjoint chirotope induced by the hyperplanes spanned by X. We present various properties relating these two chirotopes, for instance, by introducing the witness chirotope, we are able to give a formula expressing the wedge chirotope in terms of the usual chirotope. With this on hand, we answer positively a strong geometry version of a question due to M. Las Vergnas about reconstructing polygonal knots via chirotopes. Moreover, we also show that linear spatial graphs are determined by their corresponding strong geometries.

math.CO↗

arXiv2020

Primes in numerical semigroups

Let 0 < a < b be two relatively prime integers and let be the numerical semigroup generated by a and b with Frobenius number g(a,b)=ab-a-b. In this note, we prove that there exists a prime number p in with p < g(a,b) when the product ab is sufficiently large. Two related conjectures are posed and discussed as well.

math.NT↗