On numerical semigroups with embedding dimension four

TL;DR

This paper introduces a geometric method for analyzing numerical semigroups with embedding dimension four, enabling computation of key invariants such as Frobenius numbers and Betti elements.

Contribution

The authors develop a novel geometric approach to compute various properties of four-generated numerical semigroups, including those generated by consecutive squares and triangular numbers.

Findings

Computed Frobenius numbers and genus for specific semigroups

Determined Betti elements and minimal presentations

Analyzed catenary degrees of semigroups with four generators

Abstract

We develop a geometric procedure for finding the Ap\'ery set of any numerical semigroup with embedding dimension four. We use this method to find the Frobenius numbers, genera, Betti elements, minimal presentations and catenary degrees of numerical semigroups generated by four consecutive squares and four consecutive triangular numbers.