arXiv subjects

Laura Marino

Publications and source records attributed to Laura Marino.

4 recordsLinked to original sources

arXiv2026

Monoidal 2-categories from foam evaluation

In this paper we describe a general framework for constructing examples of locally linear semistrict monoidal 2-categories covering many examples appearing in link homology theory. The main input datum is a closed foam evaluation formula. As examples, we rigorously construct semistrict monoidal 2-categories based on gl(N)-foams, which underlie the general linear link homology theories, and further examples based on Bar-Natan's decorated cobordisms, related to Khovanov homology. These monoidal 2-categories are typically non-semisimple, have duals for all objects, adjoints for all 1-morphisms, and carry a canonical spatial duality structure expressing oriented 3-dimensional pivotality and sphericality.

math.QA↗

arXiv2025

Khovanov homology and rational unknotting

Building on work by Alishahi-Dowlin, we extract a new knot invariant $λ\ge 0$ from universal Khovanov homology. While $λ$ is a lower bound for the unknotting number, in fact more is true: $λ$ is a lower bound for the proper rational unknotting number (the minimal number of rational tangle replacements preserving connectivity necessary to relate a knot to the unknot). Moreover, we show that for all $n \ge 0$, there exists a knot K with $λ(K) = n$. Along the way, following Thompson, we compute the Bar-Natan complexes of rational tangles.

math.GT↗

arXiv2024

Khovanov homology and refined bounds for Gordian distances

From Khovanov homology, we extract a new lower bound for the Gordian distance of knots, which combines and strengthens the previously existing bounds coming from Rasmussen invariants and from torsion invariants. We also improve the bounds for the proper rational Gordian distance.

math.GT↗

arXiv2023

Computing the symmetric $\mathfrak{gl}_1$-homology

The symmetric $\mathfrak{gl}_n$-homologies, introduced by Robert and Wagner, provide a categorification of the Reshetikhin--Turaev invariants corresponding to symmetric powers of the standard representation of quantum $\mathfrak{gl}_n$. Unlike in the exterior setting, these homologies are already non-trivial when $n=1$. Moreover, in this case, their construction can be greatly simplified. Our first aim is giving a down-to-earth description of the non-equivariant symmetric $\mathfrak{gl}_1$-homology, together with relations that hold in this setting. We then find a basis for the state spaces of graphs, and use it to construct an algorithm and a program computing the invariant for uncolored links.

math.GT↗