arXiv subjects

David Gay

Publications and source records attributed to David Gay.

5 recordsLinked to original sources

arXiv2026

Morse diagrams, Murasugi sums, and the mapping class group

A combinatorial Morse structure encodes a mapping class for a surface with boundary, and the data may be efficiently represented via a Morse diagram. This diagram determines an open book decomposition of a 3-manifold, and hence, a contact structure on that 3-manifold. We examine how combinatorial Morse structures behave under the connect sum of open books, with particular attention paid to the case of negative stabilisation. This leads to a diagrammatic criterion for detecting overtwisted contact structures. Finally, in the case of open books with one-holed torus pages, we classify all the Morse diagrams associated to a fixed open book decomposition.

math.GT↗

arXiv2018

On Dividing a Rectangle

This paper deals with the history of the following problem: "Can an arbitrary rectangle be dissected into 3 non-rectangular congruent regions?" We present a new elementary proof that the answer is indeed no.

math.HO↗

arXiv2018

Doubly pointed trisection diagrams and surgery on 2-knots

We study embedded spheres in 4-manifolds (2-knots) via doubly pointed trisection diagrams, showing that such descriptions are unique up to stabilization and handleslides, and we describe how to obtain trisection diagrams for certain cut-and-paste operations along 2-knots directly from doubly pointed trisection diagrams. The operations described are classical surgery, Gluck surgery, blowdown, and $(\pm4)$-rational blowdown, and we illustrate our techniques and results with many examples.

math.GT↗

arXiv2011

Convex plumbings and Lefschetz fibrations

We show that under appropriate hypotheses, a plumbing of symplectic surfaces in a symplectic 4-manifold admits strongly convex neighborhoods. Moreover the neighborhoods are Lefschetz fibered with an easily-described open book on the boundary supporting the induced contact structure. We point out some applications to cut-and-paste constructions of symplectic 4-manifolds.

math.SG↗

arXiv2010

Discretized configurations and partial partitions

We show that the discretized configuration space of $k$ points in the $n$-simplex is homotopy equivalent to a wedge of spheres of dimension $n-k+1$. This space is homeomorphic to the order complex of the poset of ordered partial partitions of $\{1,...,n+1\}$ with exactly $k$ parts. We compute the exponential generating function for the Euler characteristic of this space in two different ways, thereby obtaining a topological proof of a combinatorial recurrence satisfied by the Stirling numbers of the second kind.

math.GT↗