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Sajal Mukherjee

Publications and source records attributed to Sajal Mukherjee.

At least 19 recordsLinked to original sources
arXiv2026

Tidally-enhanced resonances in extreme-mass-ratio inspirals: A tertiary path to chaos

Extreme-mass-ratio inspirals (EMRIs) provide a unique laboratory for probing strong-field gravity and complex relativistic dynamics. We study a tidally deformed EMRI composed of a stellar-mass secondary orbiting a supermassive (non-)rotating black hole embedded in an external, adiabatically varying tidal environment. These systems provide a restricted, yet astrophysically motivated, realization of the relativistic three-body problem, expected to appear in active galactic nuclei, with a clear hierarchy of masses and radiation-reaction timescales. The external tidal deformation breaks the axisymmetry of the Kerr spacetime, rendering the geodesic dynamics non-integrable and giving rise to chaotic motion. The resulting signature of non-integrability is characterized through Poincar\'e maps and rotation curves constructed from the ratios of the fundamental frequencies of bound radial, polar, and azimuthal motion. We identify two prominent plateaus whose widths increase with the tidal field amplitude, signaling a transition from weak to strong chaos. We then demonstrate the sensitivity of the chaotic dynamics to the orientation of the orbit relative to the tidal field. We further analyze the proper-time evolution of the action-angle variables, showing that the angle combinations associated with the dominant commensurabilities are phase locked, thereby allowing the associated tidal contributions to induce secular changes in the constants of motion, whereas off-plateau angle combinations circulate. These results clarify the dynamical significance of the prominent plateaus and provide a novel phase-space characterization of tidal resonances in EMRIs. Finally, we discuss the potential role of radiation-reaction effects in driving EMRIs through tidal island crossings and the implications for gravitational-wave inference with future detectors.

gr-qc↗
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 $\mathcal{W}_\alpha$ be a given gradient vector field on $A_{\alpha}(=\cap_{i \in \alpha} A_i)$ for each $\alpha \in \mathcal{N}(X)$. Then, we use the mere information of the gradient trajectories in $A_{\alpha}$ for each $\alpha \in \mathcal{N}(X)$ 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

Cancellation of a critical pair in discrete Morse theory and its effect on (co)boundary operators

Discrete Morse theory helps us compute the homology groups of simplicial complexes in an efficient manner. A "good" gradient vector field reduces the number of critical simplices, simplifying the homology calculations by reducing them to the computation of homology groups of a simpler chain complex. This homology computation hinges on an efficient enumeration of gradient trajectories. The technique of cancelling pairs of critical simplices reduces the number of critical simplices, though it also perturbs the gradient trajectories. In this article, in a purely combinatorial manner, we derive an explicit formula for computing the modified boundary operators after cancelling a critical pair, in terms of the original boundary operators. The same formula can be obtained through a sequence of elementary row operations on the original boundary operators. Thus, it eliminates the need of enumeration of the new gradient trajectories. We also obtain a similar result for coboundary operators.

math.CO↗
arXiv2026

A recursive construction of an acyclic matching on the independence complex of a graph with a simplicial vertex

We provide a recursive construction of an acyclic matching (also known as a gradient vector field, an equivalent notion to a discrete Morse function) on the independence complex of a graph with a simplicial vertex using given acyclic matchings on the independence complexes of specific subgraphs. As an application, we determine the homotopy type of the independence complexes of the family of chordal graphs and of a class of graphs generalising the comparability graphs of grid posets in an algorithmic and combinatorial manner via discrete Morse theory, some of which were previously obtained by sophisticated homotopy theoretic techniques. Even when the homotopy type is not easily determinable, our construction may be applied to obtain a pre-processing framework for efficient homology computation.

math.CO↗
arXiv2026

$2$-colourability of the maximum ranked elements of a combinatorially sphere-like ranked poset

We obtain a higher dimensional analogue of a classical theorem which states that a polygonally cellulated $2$-sphere in $\mathbb{R}^3$, such that each vertex has even degree, is $2$-face-colourable. In order to formulate our result, we introduce the notion of combinatorially sphere-like ranked posets, which are ranked posets that generalise combinatorial spheres. We prove that, in a combinatorially sphere-like ranked poset $S$ of rank $k$, if each element of rank $(k-2)$ is covered by an even number of elements, then the maximum ranked elements of $S$ admit a proper $2$-colouring, i.e., any two adjacent maximum ranked elements have different colours.

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↗
arXiv2025

Combinatorial degree version of a generalized $\mathbb{Z}_p$-Tucker's lemma with a combinatorial proof

Combinatorial analogues of classical Borsuk-Ulam-type theorems (e.g., Tucker's lemma, $\mathbb{Z}_p$-Tucker's lemma, etc.) have numerous important applications in combinatorics. In this paper, we formulate a combinatorial degree version of a generalized $\mathbb{Z}_p$-Tucker's lemma. Our proof is purely combinatorial in the sense that it does not involve homology, cohomology or any other notions from continuous topology. In order to prove the aforementioned degree theorem, as a main technical tool, we prove a Hopf trace-type formula, which is also purely combinatorial and involves no homology. This combinatorial Hopf trace formula is of independent interest.

math.CO↗
arXiv2025

Gravitational waves from parabolic encounters: A study of linear and nonlinear memory

The memory effect is known to introduce a permanent displacement in the gravitational wave (GW) detectors after the passage of a GW signal. While the $\textit{linear memory}$ adheres to the source properties, the $\textit{non-linear memory}$ is a secondary effect sourced by the GW itself. In the present work, we discuss GW signals with both these kinds of memory effects, while focusing on the parabolic limit of an encounter. This special case is theoretically intriguing and emerges as a limiting situation for both eccentric and hyperbolic events. However, in this paper, we argue that a simple extrapolation of memory calculations for eccentric or hyperbolic cases to the parabolic case may lead to incorrect estimations. Therefore, we treat the parabola as a special case and use an intrinsic parameterization, with which we calculate gravitational wave signals and their energy spectrum via an effective field theory formalism. Unlike the hyperbolic case, which is known to have linear memory, we notice that parabolic encounters bring out new features in the zero frequency limit (ZFL). The exactly parabolic case is studied here primarily as an idealized separatrix between bound and unbound motion, and our analysis highlights some of the key challenges and salient aspects of GW memory in this regime.

gr-qc↗
arXiv2025

Impact of a third body on binary neutron star tidal interactions

For waveform modelling of compact binary coalescence, it is conventionally assumed that the binary is in isolation. In this work, we break that assumption and introduce a third body at a distance. The primary goal is to understand how the distant third body would affect the binary dynamics. However, in the present work, we treat the three-body problem perturbatively and study tidal interaction in the binary due to the third body's presence. We introduce appropriate modifications to the equations governing the orbital motions and the evolution equations of the binary component's quadrupole moment. Further, we obtain the radiated energy and accumulated dephasing for the binary. We show that for b-EMRI, the effect is weak in the tidal sector, while for systems such as b-IMRIs, it would be most relevant to study these effects.

gr-qc↗
arXiv2025

Pure Gauss-Bonnet NUT Black Hole Solution: II

In the present article, we have obtained an exact analytical solution of six-dimensional pure Gauss-Bonnet gravity in the presence of both NUT and Maxwell charges. The topology of the horizon is chosen to be the product of two 2-spheres. Upon evaluating the solution, we study the spacetime properties, such as event horizon and singularity, and obtain the ranges of parameter space where the solution is valid. We discuss how the presence of Maxwell charges may impact the solution's asymptotic expansion and what distinctive effects it will bring to the geometry. The thermodynamic properties of the solution are also discussed, emphasizing the interplay between NUT and Maxwell charges.

gr-qc↗
arXiv2025

Signatures of gravitational wave memory in the radiative process of entangled quantum probes

In this article, we examine entangled quantum probes in geodesic trajectories in a flat background with a gravitational wave (GW) burst. In particular, these quantum probes are prepared initially either in the symmetric or anti-symmetric Bell's states, and we study the radiative process as the GW burst passes. We split a generic GW burst into two profiles with and without memory. GW burst with (without) memory profiles have different (similar) asymptotic strains between early and late times. We observe that for eternal switching, there is a finite change in the collective atomic transition rate due to the memory part of the GW burst, while the contribution from the without memory counterpart vanishes. We also consider finite Gaussian switching and observe characteristic differences in the radiative process between the GW backgrounds with and without memory. Notably, if the Gaussian switching is peaked much later compared to the passing of GW, only the memory part contributes to the radiative process. Thus, although examined in a simplified set-up, our findings suggest the potential to distinguish bursts with and without GW memory based on the radiative process of entangled detectors.

gr-qc↗
arXiv2024

Topology of matching complexes of complete graphs via discrete Morse theory

Bouc (1992) first studied the topological properties of $M_n$, the matching complex of the complete graph of order $n$, in connection with Brown complexes and Quillen complexes. Björner et al. (1994) showed that $M_n$ is homotopically $(ν_n-1)$-connected, where $ν_n=\lfloor{\frac{n+1}{3}}\rfloor-1$, and conjectured that this connectivity bound is sharp. Shareshian and Wachs (2007) settled the conjecture by inductively showing that the $ν_n$-dimensional homology group of $M_n$ is nontrivial, with Bouc's calculation of $H_1(M_7)$ serving as the pivotal base step. In general, the topology of $M_n$ is not very well-understood, even for a small $n$. In the present article, we look into the topology of $M_n$, and $M_7$ in particular, in the light of discrete Morse theory as developed by Forman (1998). We first construct a gradient vector field on $M_n$ (for $n \ge 5$) that doesn't admit any critical simplices of dimension up to $ν_n-1$, except one unavoidable $0$-simplex, which also leads to the aforementioned $(ν_n-1)$-connectedness of $M_n$ in a purely combinatorial way. However, for an efficient homology computation by discrete Morse theoretic techniques, we are required to work with a gradient vector field that admits a low number of critical simplices, and also allows an efficient enumeration of gradient paths. An optimal gradient vector field is one with the least number of critical simplices, but the problem of finding an optimal gradient vector field, in general, is an NP-hard problem (even for $2$-dimensional complexes). We improve the gradient vector field constructed on $M_7$ in particular to a much more efficient (near-optimal) one, and then with the help of this improved gradient vector field, compute the homology groups of $M_7$ in an efficient and algorithmic manner. We also augment this near-optimal gradient vector field to one that we conjecture to be optimal.

math.CO↗
arXiv2024

Analytical model of precessing binaries using post-Newtonian theory in the extreme mass-ratio limit I: General Formalism

We develop a fully analytical waveform model for precessing binaries with arbitrary spin vectors using post-Newtonian~(PN) theory in the extreme mass-ratio limit and a hierarchical multi-scale analysis. The analytical model incorporates leading PN order spin precession dynamics from spin-orbit, spin-spin, and quadrupole-monopole couplings, and 2PN order dissipative dynamics truncated to first order in the mass ratio $q \ll 1$. Due to the pure analytic nature of the model, the framework developed herein can readily be extended to both higher PN and higher-$q$ order. Although the PN series is asymptotic to this limit, our results can be used to estimate how precession affects the measurability of certain binary parameters, and to inform and compare with other waveform approximants, such as effective-one-body models, hybrid waveforms, and self-force calculations.

gr-qc↗
arXiv2024

Detectability of stochastic gravitational wave background from weakly hyperbolic encounters

We compute the stochastic gravitational wave (GW) background generated by black hole-black hole (BH-BH) hyperbolic encounters with eccentricities close to one and compare them with the respective sensitivity curves of planned GW detectors. We use the Keplerian potential to model the orbits of the encounters and the quadrupole formula to compute the emitted GWs. We take into account hyperbolic encounters that take place in clusters up to redshift $5$ and with BH masses spanning from $5 M_{\odot}$ to $55 M_{\odot}$. We assume the clusters to be virialized and study several cluster models with different mass and virial velocity, and finally obtain an accumulative result, displaying the background as an average. Using the maxima and minima of our accumulative result for each frequency, we provide analytical expressions for both optimistic and pessimistic scenarios. Our results suggest that the background from these encounters is likely to be detected by the third-generation detectors Cosmic explorer and Einstein telescope, while the tail section at lower frequencies intersects with DECIGO, making it a potential target source for both ground- and space-based future GW detectors.

gr-qc↗
arXiv2024

Discrete Morse theory and the topology of matching complexes of complete graphs

We denote the matching complex of the complete graph with $n$ vertices by $M_n$. Bouc first studied the topological properties of $M_n$ in connection with the Quillen complex. Later Björner, Lovász, Vrećica, and Živaljević showed that $M_n$ is homotopically $(ν_n-1)$-connected, where $ν_n=\lfloor{\frac{n+1}{3}}\rfloor-1$, but in general the topology of $M_n$ is not very well-understood even for smaller natural numbers. Forman developed discrete Morse theory, which has various applications in diverse fields of studies. In this article, we develop a discrete Morse theoretic technique to capture deeper structural topological properties of $M_n$. We show that $M_n$ is \emph{geometrically} $(ν_n-1)$-connected, where the notion of geometrical $k$-connectedness as defined in this article, is stronger than that of homotopical $k$-connectedness. Previously, Björner et al. showed that $M_8$ is simply connected, but not 2-connected. The technique developed here helped us determine that $M_8$ is in fact homotopy equivalent to a wedge of 132 spheres of dimension 2.

math.CO↗
arXiv2023

Can extended bodies follow geodesic trajectories?

We provide an extension of the analysis on whether an extended test body can follow a geodesic trajectory given by Mukherjee, S., Lukes-Gerakopoulos, G. and Nayak, R. K. (2022), Extended bodies moving on geodesic trajectories, General Relativity and Gravitation, 54(9), 113, arXiv: 1907.05659. In particular, we consider a test body in a pole-dipole-quadrupole approximation under the Ohashi-Kyrian-Semerák spin supplementary condition moving in the Schwarzschild and Kerr background. Using orbital setups under which a pole-dipole body can follow geodesic motion, we explore under which conditions this can take place also in the pole-dipole-quadrupole approximation, when only the mass quadrupole is taken into account. For our analysis we employ the assumption that the dipole contribution and the quadrupole contribution vanish independently.

gr-qc↗