arXiv subjects

Arundhati Rakshit

Publications and source records attributed to Arundhati Rakshit.

3 recordsLinked to original sources

arXiv2026

A combinatorial nerve theorem for effective homology computation

The celebrated (homological) nerve theorem makes use of spectral sequences to determine the homology of a simplicial complex. However, this theorem cannot effectively compute the homology in every circumstance. In this paper, we develop an effective version of the nerve theorem, yielding a new and powerful tool for homology computation. The essence of our theorem can be formulated in the following manner. Suppose, $X$ is a simplicial complex with covering subcomplexes $A_1, \dots ,A_k$, that is, $X= \cup_{i=1}^k A_i$ and $\mathcal{N}(X)$ is the nerve of $X$ with respect to its covering. Let $\mathcal{W}_α$ be a given gradient vector field on $A_α(=\cap_{i \in α} A_i)$ for each $α\in \mathcal{N}(X)$. Then, we use the mere information of the gradient trajectories in $A_α$ for each $α\in \mathcal{N}(X)$ to explicitly compute the homology groups of $X$. Furthermore, we point out here, that these gradient vector fields do not need to be coherent, that is, they do not need to coincide on the intersections, which gives us ample flexibility to apply our theorem. Moreover, we can further simplify the computation of the homology groups using a gradient vector field on the nerve of $X$. Our approach is purely combinatorial, in the sense that it does not involve any notions of geometric realisation, continuity or homotopy, which makes it more amenable to computation and coding.

math.CO↗

arXiv2026

The number of Pfaffian orientations on punctured polygonally cellulated surfaces

In this paper, we introduce the notion of Pfaffian orientations on (punctured) polygonally cellulated orientable surfaces, and provide an expression for the number of such orientations. This generalizes the notion of Pfaffian orientations on planar graph, where a planar graph is seen as a punctured $2$-sphere, embedded in $\mathbb{R}^3$. So, as a direct corollary of our main theorem, we derive the number of Pfaffian orientations on a planar graph.

math.CO↗

arXiv2026

An effective Mayer-Vietoris Theorem for discrete Morse homology

The Mayer-Vietoris theorem is known for its wide applications, especially in determining homology. In fact, this theorem provides us with a long exact sequence, where the underlying homology groups fit in. However, this theorem does not provide an explicit way to compute homology. In this paper we prove an ``effective" version of the Mayer-Vietoris theorem using discrete Morse theory. Suppose, we have a Mayer-Vietoris type setup, i.e., let $X$ be a simplicial complex and $A$ and $B$ be two subcomplexes of $X$, such that $A \cup B=X$. Moreover, let $\mathcal{W}_A$, $\mathcal{W}_{B}$ and $\mathcal{W}_{A \cap B}$ be gradient vector fields on $A$, $B$ and $A \cap B$ respectively (which need not be ``coherent", i.e., they do not need to coincide on their intersection). Then, the main theorem of our paper provides an explicit way to compute the homology groups of $X$, using the combinatorial information regarding the trajectories of the aforementioned gradient vector fields, we do not even need to know the individual homology groups $H_{*}(A)$, $H_{*}(B)$ and $H_{*}(A \cap B)$. In principle, the homology of $X$ can always be computed explicitly using our theorem irrespective of the choice of the gradient vector fields. Further, if we choose the subcomplexes $A$ and $B$ wisely so that each of $A$, $B$ and $A \cap B$ admits an efficient gradient vector field, then the computation of the homology groups is considerably reduced.

math.CO↗