# A Coupled Alpha Complex

arXiv: https://arxiv.org/abs/2105.08113 · HTML: https://mlsanity.com/paper/2105.08113/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 17 May 2021.

## Built on by (most cited first, 1 of 1)

- [Fast Topological Signal Identification and Persistent Cohomological Cycle Matching](https://mlsanity.com/paper/2209.15446/) (2022)

## Abstract (arXiv)

The alpha complex is a subset of the Delaunay triangulation and is often used in computational geometry and topology. One of the main drawbacks of using the alpha complex is that it is non-monotone, in the sense that if cal Xsubsetcal X' it is not necessarily (and generically not) the case that the corresponding alpha complexes satisfy cal Ar(cal X)subsetcal Ar(cal X'). The lack of monotonicity may introduce significant computational costs when using the alpha complex, and in some cases even render it unusable. In this work we present a new construction based on the alpha complex, that is homotopy equivalent to the alpha complex while maintaining monotonicity. We provide the formal definitions and algorithms required to construct this complex, and to compute its homology. In addition, we analyze the size of this complex in order to argue that it is not significantly more costly to use than the standard alpha complex.