Devashi Gulati | Search arXiv

arXiv2026

Pants distances of knotted surfaces in 4-manifolds

We define a pants distance for knotted surfaces in 4-manifolds, which generalizes the complexity studied by Blair-Campisi-Taylor-Tomova for surfaces in the 4-sphere. We determine that if the distance computed on a given diagram does not surpass a theoretical bound in terms of the multisection genus, then the pair (X, F) admits a standard form (i.e., has simple topology). Furthermore, we calculate the exact values of our invariants for many new examples, such as the spun lens spaces. We provide a characterization of genus two quadrisections with distance at most six.

math.GT↗

arXiv2024

Moduli Spaces of Lagrangian Surfaces in $\mathbb{CP}^2$ obtained from Triple Grid Diagrams

Links in $S^3$ as well as Legendrian links in the standard tight contact structure on $S^3$ can be encoded by grid diagrams. These consist of a collection of points on a toroidal grid, connected by vertical and horizontal edges. Blackwell, Gay and second author studied triple grid diagrams, a generalization where the points are connected by vertical, horizontal and diagonal edges. In certain cases, these determine Lagrangian surfaces in $\mathbb{CP}^2$. However, it was difficult to construct explicit examples of triple grid diagrams, either by an approximation method or combinatorial search. We give an elegant geometric construction that produces the moduli space of all triple grid diagrams. By conditioning on the abstract graph underlying the triple grid diagram, as opposed to the grid size, the problem reduces to linear algebra and can be solved quickly in polynomial time.

math.GT↗