Cobordism of domes over curves

TL;DR

This paper investigates the cobordism relationships of polyhedral surfaces called domes over integral curves, disproving a conjecture for oriented cases and establishing a cobordism to unions of unit rhombi.

Contribution

It demonstrates that not all integral curves can be oriented domed by a single rhombus but can be cobordant to unions of finitely many such rhombi.

Findings

Not all integral curves are orientably cobordant to a single unit rhombus.

Every integral curve is orientably cobordant to a union of finitely many unit rhombi.

Disproves Glazyrin and Pak's conjecture for oriented domes.

Abstract

An integral curve is a closed piecewise linear curve comprised of unit intervals. A dome is a polyhedral surface whose faces are equilateral triangles and whose boundary is an integral curve. Glazyrin and Pak showed that not every integral curve can be domed by analyzing the case of unit rhombi, and conjectured that every integral curve is cobordant to a unit rhombus. We show that this is false for oriented domes, but that every integral curve is orientably cobordant to the union of finitely many unit rhombi.