The initial investigation into the charged scalar cloud surrounding Kerr-Newman black holes aimed to meticulously map the flow of energy, both into and out of these enigmatic entities. Black holes, regions of spacetime where gravity is so strong that nothing, not even light, can escape, are not merely passive voids. They are dynamic participants in the cosmic drama, influencing their surroundings in ways that continue to astonish scientists. The Kerr-Newman black hole, a theoretical generalization that incorporates both spin and electric charge, represents a more complete astrophysical scenario than the simpler Schwarzschild or Kerr black holes. Understanding the energy balance around these objects is paramount, as it directly relates to phenomena like Hawking radiation and the stability of matter in their vicinity, offering tantalizing glimpses into the quantum nature of gravity and the ultimate fate of information. This erratum, therefore, is not just a footnote; it’s a pivotal moment in clarifying the delicate equilibrium that governs these cosmic structures, ensuring that future theoretical models are built upon the most accurate foundations possible, a testament to the relentless pursuit of scientific integrity and accuracy in understanding the universe’s most extreme environments.

The concept of a “charged scalar cloud” itself is a fascinating theoretical construct. It suggests that under specific conditions, a field of particles carrying an electric charge and possessing scalar properties—meaning they don’t have a preferred direction—can condense around a black hole, forming a dynamic halo. This cloud is not static; it is in a constant state of flux, absorbing and emitting energy. The balance of these energy flows is crucial for determining the stability of the cloud and its long-term influence on the black hole. The original paper sought to quantify these fluxes, aiming to understand whether the net energy flow leads to growth, decay, or a stable equilibrium of the scalar cloud. This erratum’s significance lies in its ability to bring greater precision to these fundamental energetic calculations, thereby refining our understanding of how these complex astrophysical systems maintain their delicate dynamical states and interact with the broader cosmic environment, offering crucial insights into the interplay of fundamental fields within the extreme gravitational regimes.

The erratum specifically addresses a critical aspect of the flux balance calculation: the precise contribution and interaction of charged scalar fields with the spacetime geometry and electromagnetic fields of the Kerr-Newman black hole. Theoretical physicists rely on sophisticated mathematical frameworks, often involving general relativity and quantum field theory, to model these extreme environments. Errors, even seemingly small ones, in these intricate calculations can propagate and lead to misleading conclusions about the behavior of the system. The correction likely involves a refinement of a specific equation, an adjustment in a numerical simulation, or a clarification of a subtle theoretical assumption, but its impact is far-reaching, ensuring that subsequent theoretical explorations and observational interpretations are grounded in a more robust and accurate understanding of the underlying physics governing these colossal cosmic entities, thereby advancing our quest to decipher the fundamental laws of the universe.

The implications of this refined understanding are vast. For instance, the stability of a charged scalar cloud could have direct consequences for the long-term evolution of black holes and their accretion disks. A stable cloud might contribute to the observed properties of astrophysical black holes, while an unstable one could shed light on processes of energy dissipation and particle creation near the event horizon. The dynamics of energy transfer in these regions are also crucial for understanding phenomena like quasars and active galactic nuclei, which are powered by supermassive black holes at the centers of galaxies. This correction, by providing a more accurate picture of these interactions, allows scientists to build more reliable models of these energetic cosmic engines, leading to a deeper appreciation of the forces that shape galaxies and the universe on grand scales.

Furthermore, this work touches upon the very nature of information paradox in black holes. While not directly resolving it, a precise understanding of what can and cannot escape from a black hole, and how energy is exchanged, is fundamental to tackling this profound theoretical challenge. The Kerr-Newman black hole, with its added complexity of charge and spin, offers a richer playground for exploring these paradoxes. The erratum’s contribution to accurately modeling these energy fluxes could provide crucial stepping stones for theoretical physicists grappling with the question of whether information is truly lost when it falls into a black hole or if it is somehow preserved, a question that probes the very foundations of quantum mechanics and general relativity.

The refinement of theoretical models is an ongoing process, and each correction, like the one concerning the Kerr-Newman black hole’s charged scalar cloud, represents a vital step forward. These refinements are not mere academic exercises; they are essential for interpreting incoming data from advanced telescopes and detectors, such as the Event Horizon Telescope, which has provided unprecedentedly detailed images of black hole shadows. Accurate theoretical predictions are crucial for confirming observations and identifying new phenomena. This erratum, therefore, enhances our ability to not only predict but also to understand the cosmic spectacles we are beginning to witness, solidifying the link between abstract mathematical constructs and concrete astrophysical realities.

The research also delves into the fundamental interactions between gravity, electromagnetism, and quantum fields. The Kerr-Newman black hole is a perfect laboratory for studying these interactions in their most extreme manifestations. The presence of charge and spin introduces electromagnetic fields that interact with the charged scalar cloud, while the immense gravitational field warps spacetime. Understanding how these forces interplay and how energy is conserved or dissipated in this complex environment is key to developing a unified theory of everything, a long-sought-after goal in physics. This erratum, by clarifying the energy flux balance, provides a more precise data point in the immense puzzle of unifying the fundamental forces of nature.

The concept of a “flux balance” implies a crucial equilibrium. If incoming energy consistently exceeds outgoing energy, the scalar cloud would grow, potentially altering the black hole’s properties. Conversely, if energy is consistently lost, the cloud would dissipate. Understanding the precise conditions under which these systems achieve a stable balance is critical for predicting their long-term behavior and their impact on their cosmic surroundings. The erratum’s correction likely pinpoints a specific reason why the previous calculations might have predicted an incorrect balance, allowing for a more accurate determination of the stability regime for these charged scalar clouds, leading to a more robust understanding of their persistence and influence in the universe.

The allure of black holes lies not only in their immense gravitational pull but also in the exotic physics that governs their vicinity. Charged scalar clouds represent one such exotic phenomenon, pushing the boundaries of our theoretical understanding. The fact that such a correction has been published underscores the rigor and self-correcting nature of the scientific process. It is a testament to the dedication of researchers to ensure that the foundations of our knowledge are as sound as possible, even when dealing with the most abstract and challenging aspects of theoretical physics, fostering a culture of continuous improvement and deep intellectual inquiry.

The European Physical Journal C, as a leading publication in particle physics, astrophysics, and cosmology, serves as a vital platform for disseminating these critical updates. The erratum signals to the entire research community that a nuanced re-evaluation has taken place, prompting a reassessment of related theoretical work and potentially inspiring new research directions. This collaborative and transparent approach to scientific progress is what drives our understanding of the universe forward, ensuring that discoveries are built upon a solid and evolving bedrock of knowledge, thereby accelerating the pace of scientific discovery.

This refined understanding of Kerr-Newman black holes and their charged scalar clouds has implications that extend beyond pure theory. It could influence our search for alternative theories of gravity or new fundamental particles. By precisely modeling the behavior of these cosmic objects, scientists can better distinguish between predictions made by established theories and those made by speculative ones, guiding future experimental and observational efforts and refining our cosmic roadmap.

The scientific community’s response to this erratum is likely to be one of careful examination and integration. Researchers will be keen to understand the specifics of the correction and how it modifies existing theoretical frameworks. This process of verification and assimilation is crucial for the robustness of scientific knowledge, ensuring that conclusions are not based on flawed premises and that progress is built on verifiable facts and accurate calculations, thus strengthening the foundations of our cosmic understanding.

In essence, this erratum is a powerful reminder that science is a dynamic and evolving discipline. It is a process of constant questioning, rigorous testing, and meticulous refinement. The correction to the study of Kerr-Newman black holes’ charged scalar cloud is a shining example of this, reinforcing our commitment to accuracy and deepening our appreciation for the complex and awe-inspiring universe we inhabit, pushing the boundaries of human knowledge further into the unknown, and inspiring future generations of scientists to continue this grand endeavor.

Subject of Research: The behavior and energy flux balance of charged scalar clouds surrounding Kerr-Newman black holes.

Article Title: Revisiting Kerr–Newman black hole’s charged scalar cloud: flux balance.

Article References:

Senjaya, D. Erratum to: Revisiting Kerr–Newman black hole’s charged scalar cloud: flux balance.
Eur. Phys. J. C 86, 38 (2026). https://doi.org/10.1140/epjc/s10052-025-15274-8

Image Credits: AI Generated

DOI: https://doi.org/10.1140/epjc/s10052-025-15274-8

Keywords: Kerr-Newman black holes, charged scalar clouds, flux balance, general relativity, quantum field theory, theoretical astrophysics, spacetime dynamics, energy conservation, gravitational interactions, cosmic phenomena.

Delving deeper into the theoretical underpinnings of black bounces, the research explores the profound implications of what happens when matter or energy approaches such exotic gravitational entities. Unlike the well-understood phenomenon of tidal forces near a classical black hole, where an object is irrevocably stretched and compressed into oblivion, the concept of a black bounce suggests a more nuanced interaction. Imagine an object approaching a black bounce. Instead of an inevitable plunge into a singularity, the object might experience extreme tidal forces – the difference in gravitational pull across its extended form – but this stretching and compression might not lead to destruction. Instead, it could be a precursor to a “bounce,” a point where the inward collapse is arrested, and the object is, in a sense, pushed outward or redirected. This hypothetical scenario fundamentally alters our understanding of gravitational interactions at these extreme densities and curvatures of spacetime, moving beyond the singularity paradigm that has long dominated black hole physics. The mathematical frameworks employed to describe these phenomena are incredibly complex, often involving advanced concepts from quantum gravity and modified theories of gravity.

The concept of tidal stretching and compression, even in this black bounce context, remains a critical aspect. Tidal forces are a direct consequence of the non-uniform gravitational field. For an object falling towards any massive body, the part of the object closer to the body feels a stronger gravitational pull than the part further away. This differential pull results in stretching along the direction of the pull and compression perpendicular to it. Near a black hole, these forces become infinitely strong at the singularity. However, in the black bounce scenario, the point of maximum tidal effect might not be a destructive singularity but rather an inflection point where the gravitational path dramatically changes. The paper, in its original and corrected forms, likely uses tensor calculus and differential geometry to model these spacetime distortions, grappling with equations that describe how the curvature of spacetime dictates the paths of objects and the very nature of gravity.

The erratum itself, while a technical detail, underscores the rigorous scientific process. Science is a self-correcting mechanism, and even the most cutting-edge theoretical work is subject to scrutiny and refinement. The initial publication might have contained a minor error in its formulation or presentation, leading to the publisher’s correction. However, this correction doesn’t diminish the significance of the research; rather, it highlights the careful attention to detail required when exploring such speculative frontiers. The fact that a publisher felt the need to issue this specific correction points to the complexity of the mathematical models and the sensitivity of the results. It’s akin to fine-tuning a complex instrument to capture the faintest cosmic signals; even a minute adjustment can be crucial for accurate interpretation, especially when dealing with concepts that push the boundaries of our current physical understanding.

The theoretical framework of black bounces emerges from attempts to resolve some of the most perplexing paradoxes associated with classical black holes, particularly the information loss paradox. According to general relativity, anything that falls into a black hole is lost forever, taking its information with it. This violates a fundamental principle of quantum mechanics, which states that information can never be truly destroyed. Black bounces offer a potential avenue for resolving this paradox. If, instead of a singularity, there’s a bounce, then the matter and energy that fell in might, in principle, be able to escape, carrying their information with them. This elegantly sidesteps the information loss problem by proposing a mechanism for the egress of material and, crucially, the information it contains, from what otherwise appears to be a cosmic trap.

Furthermore, the idea of black bounces opens up tantalizing possibilities for cosmology. Some theoretical models suggest that these bounces could be remnants of the Big Bang itself. If the universe began not with a singularity but with a bounce from a previous contracting phase, then the inflationary epoch, which explains the rapid expansion of the early universe, could be a consequence of this cosmic rebound. This radical idea connects the microscopic realm of quantum gravity with the macroscopic evolution of the entire cosmos, suggesting that the explosive birth of our universe might be a repeating or cyclical phenomenon, a breathtaking concept to contemplate.

The mathematical descriptions of black bounces often involve modifications to Einstein’s theory of general relativity, incorporating quantum effects at extremely high energy densities. These modifications can introduce new fields or alter the fundamental equations governing gravity, allowing for the possibility of non-singular gravitational collapses. Techniques from quantum field theory in curved spacetime, string theory, or loop quantum gravity might be employed to construct these theoretical models. The resulting equations are incredibly difficult to solve, often requiring sophisticated numerical simulations to explore their behavior and predict observable consequences, if any.

The implications for observational astronomy are equally profound, even if currently indirect. While directly observing a black bounce is likely beyond our present technological capabilities, understanding their theoretical properties could help us interpret existing astronomical data in new ways. Anomalies in the cosmic microwave background radiation, gravitational wave signals, or the dynamics of galactic centers might, in the future, be explained by the presence of these exotic objects. Physicists are constantly searching for deviations from the predictions of general relativity, and black bounces, if they exist, would represent a significant departure, potentially offering clues to the fundamental nature of gravity and the universe.

The sheer audacity of the black bounce concept is what makes it so captivating. It challenges a cornerstone of modern physics – the singularity. For decades, the singularity has been the ultimate endpoint of gravitational collapse, a point of infinite density and curvature where the laws of physics break down. Black bounces propose a way around this seemingly insurmountable barrier, offering a more gentle and perhaps cyclical view of cosmic evolution. This isn’t just about theoretical physics; it’s about redefining our understanding of the most extreme environments in the universe and our place within it. The universe, it seems, may be far more dynamic and inventive than we previously imagined.

The initial paper, by focusing on tidal stretching and compression within these black bounce backgrounds, likely explored how matter would be affected as it approaches and potentially “bounces” off these objects. This would involve calculating the geodesic paths of particles and light, and how their shapes would be distorted by the extreme spacetime curvature. The analysis would scrutinize the gradients in the gravitational field, quantifying the stretching and squeezing forces that would act upon any infalling object. Understanding these tidal effects is crucial for distinguishing black bounces from classical black holes, as the ultimate fate of an object near the former would be drastically different from its fate near the latter.

The work by Crispim, Silva, Alencar, and their colleagues, even with its publisher’s correction, contributes to a growing body of theoretical research exploring scenarios beyond the standard cosmic model. These investigations are vital for pushing the boundaries of our knowledge and for developing a more complete picture of the universe, from its earliest moments to its most extreme gravitational phenomena. The rigor of publishing in a peer-reviewed journal like The European Physical Journal C ensures that these complex theoretical ideas are subjected to critical evaluation, leading to a more robust understanding of the cosmos.

This engagement with theoretical physics, particularly concerning black bounces, is not merely an academic exercise. It represents humanity’s enduring drive to comprehend the fundamental laws that govern reality. The very concept of a “black bounce” suggests a universe that is not simply ending in black holes, but potentially evolving, transforming, and perhaps even repeating. This cyclical or transitional nature of cosmic events challenges our linear perception of time and existence, prompting us to consider a universe that is far more alive and dynamic than previously conceived by many.

The DOI provided, https://doi.org/10.1140/epjc/s10052-025-14985-2, serves as a permanent digital identifier for this specific publication. In the realm of scientific literature, DOIs are essential for ensuring that research papers can be reliably located and accessed by the global scientific community. For this particular corrected article, the DOI will point to the most up-to-date version, incorporating any necessary amendments. This system is crucial for maintaining the integrity of scientific records and for facilitating smooth communication and collaboration among researchers worldwide.

The subject matter of this research, black bounces, is at the cutting edge of theoretical astrophysics and cosmology. It represents an attempt to unify general relativity with quantum mechanics in regimes of extreme gravity where our current understanding falters. The exploration of tidal forces within these exotic backgrounds is a critical step in characterizing their physical properties and potential observability, even if such observations are currently of a theoretical nature and await future advancements in detection capabilities.

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Subject of Research: Theoretical Astrophysics and Cosmology, exploring the nature of gravitational objects beyond classical black holes, specifically “black bounces.”

Article Title: Tidal stretching and compression in black bounce backgrounds

Article References: Crispim, T.M., de S. Silva, M.V., Alencar, G. et al. Publisher Erratum: Tidal stretching and compression in black bounce backgrounds. Eur. Phys. J. C 85, 1248 (2025). https://doi.org/10.1140/epjc/s10052-025-14985-2

DOI: 10.1140/epjc/s10052-025-14985-2

Keywords: Black bounces, tidal forces, general relativity, quantum gravity, cosmology, singularity resolution.

In a groundbreaking development that has sent ripples of excitement through the theoretical physics and mathematics communities, a team of researchers has unveiled a novel computational framework that promises to illuminate the intricate geometries of “bipartite links.” This sophisticated new approach, detailed in a recent publication in the European Physical Journal C, introduces a “Khovanov–Rozansky cycle calculus,” a powerful tool that allows for a deeper and more precise understanding of these abstract mathematical structures. The implications of this work are far-reaching, potentially impacting fields as diverse as quantum field theory, condensed matter physics, and even the very fabric of spacetime. For decades, mathematicians and physicists have grappled with the enigmatic nature of links, which are essentially closed loops embedded in three-dimensional space. While simple links, like a single untangled circle, are easily visualized, the study of more complex arrangements, particularly those with interwoven strands, has presented formidable challenges. The introduction of Khovanov homology several years ago offered a significant breakthrough by assigning algebraic invariants to these links, transforming the study from a purely geometric endeavor to one with a rich algebraic underpinning. This new cycle calculus builds directly upon that foundation, refining the computational machinery and unlocking new avenues for exploration.

The essence of this research lies in its ability to connect abstract algebraic concepts with the tangible, albeit geometric, representation of links. Khovanov–Rozansky homology, the theoretical bedrock of this new calculus, provides a graded algebraic structure—akin to a complex numerical fingerprint—that uniquely characterizes a given link. Prior to this development, calculating these fascinating invariants was often a laborious and computationally intensive process, requiring significant expertise and specialized algorithms. The newly developed cycle calculus, however, streamlines this process, offering a more elegant and efficient method for deriving these crucial link invariants. This advancement is akin to discovering a shortcut through a dense mathematical forest, allowing researchers to reach their destination of understanding much faster and with greater clarity. The potential for this refined computational power is immense, enabling scientists to tackle previously intractable problems and explore the properties of links with an unprecedented level of detail.

At its core, the Khovanov–Rozansky cycle calculus introduces a method for associating specific algebraic objects, known as cycles, to the regions and crossings within a bipartite link diagram. Bipartite links, a special class of links characterized by their specific combinatorial structure, possess a particular symmetry that makes them amenable to this new analytical approach. Think of a complex knot as a tangled piece of string; a bipartite link is like a specific type of tangle that can be systematically described by alternating two types of components. This alternating property is crucial, as it allows for a more organized and structured way of assigning the algebraic elements within the calculus. The researchers have devised a way to translate the visual information of the link diagram—the way the strands intertwine and the regions they enclose—into a sequence of algebraic operations. These operations, when performed according to the rules of the cycle calculus, ultimately yield the Khovanov–Rozansky invariant.

The “cycle” in Khovanov–Rozansky cycle calculus refers to specific elements within the underlying algebraic chain complex that represents the link. These cycles, when projected onto certain subcomplexes, reveal deep structural information about the link’s topology. Imagine dissecting a complex origami structure; the cycle calculus allows us to understand the fundamental folds and creases that define the final shape. By studying how these cycles behave and interact within the algebraic framework, researchers can deduce properties of the link that might be obscured from purely visual inspection. This abstraction allows for a level of precision and generality that is often difficult to achieve with purely geometric or combinatorial arguments. The insights gained from this cycle calculus are not merely academic; they offer a new perspective on how to quantify and differentiate between complex topological arrangements.

The elegance of this new calculus lies in its ability to automate and systematize the computation of these vital link invariants. Instead of ad hoc methods, the cycle calculus provides a standardized procedure that can be implemented algorithmically. This opens the door for large-scale computational studies of link invariants, allowing researchers to analyze databases of links and identify patterns and relationships that would be impossible to find manually. The implications for fields that rely on understanding complex structures, such as materials science where knotting can affect material properties, are profound. The ability to predict and analyze the topological characteristics of materials at a fundamental level could lead to the design of novel materials with enhanced functionalities. This signifies a shift from understanding what a link looks like to understanding why it behaves the way it does, based on its underlying algebraic signature.

Beyond computational efficiency, the Khovanov–Rozansky cycle calculus offers a deeper conceptual understanding of the relationship between knot theory and other areas of mathematics and physics. Historically, knot theory has found surprising connections to diverse fields, from statistical mechanics to quantum computation. This new calculus promises to forge even stronger ties, providing a common language and a unified framework for exploring these interdisciplinary links. The researchers are optimistic that their work will serve as a bridge, enabling physicists working on quantum field theories to communicate more effectively with mathematicians specializing in algebraic topology, and vice versa. This cross-pollination of ideas is often where the most significant scientific breakthroughs emerge.

One of the most exciting aspects of this research is its potential to shed light on the structure of quantum field theories themselves. Khovanov homology and its extensions have a known relationship to certain topological quantum field theories (TQFTs). These theories are of immense interest in theoretical physics, particularly in attempts to formulate a unified theory of gravity and understand the fundamental nature of space and time. By providing a more tractable computational tool for analyzing link invariants, the Khovanov–Rozansky cycle calculus could provide concrete ways to test and develop these abstract TQFTs. The ability to connect these mathematical structures to the physical world is a long-standing goal of theoretical physics.

Consider the exploration of spacetime. In certain theoretical models, the very structure of spacetime might be thought of as being woven from fundamental loops or threads. Understanding the topology of these threads, and how they can be knotted or linked, is crucial for developing a complete picture of the universe at its most fundamental level. The Khovanov–Rozansky cycle calculus offers a novel mathematical language to describe and analyze these potential “spacetime knots.” This could lead to entirely new ways of thinking about phenomena like black holes, wormholes, and the early universe, potentially unlocking secrets that have remained hidden for decades due to the limitations of previous analytical tools. The deep connections being unearthed suggest that the study of abstract knots is not merely an intellectual exercise but a fundamental inquiry into the nature of reality.

The researchers are particularly enthusiastic about the implications for understanding complex systems in physics. Many physical phenomena, from the swirling patterns of fluids to the intricate folding of proteins, can be described using topological concepts. The Khovanov–Rozansky cycle calculus, by providing a powerful way to distinguish and analyze different topological configurations, could offer unprecedented insights into these systems. Imagine being able to predict how a protein will fold based on its underlying topological structure, or how a turbulent fluid will behave based on the entanglement of its flow lines. This level of predictive power would be revolutionary across many scientific disciplines.

Furthermore, the development of the Khovanov–Rozansky cycle calculus for bipartite links represents a significant advance in the field of low-dimensional topology, the study of spaces that are essentially one, two, or three-dimensional. The mathematical tools developed here could have broad applicability within this subfield, leading to new classifications and understandings of topological objects. The elegance of the approach suggests that there may be further simplifications and extensions to discover, pushing the boundaries of what is currently understood about the world of knots and links. It is a testament to the enduring power of abstract mathematics to provide new lenses through which to view the universe.

The implications for quantum computing are also gaining attention. Quantum computers rely on manipulating delicate quantum states, and the robustness of these states against errors is a major challenge. Topological quantum computing is an emerging paradigm that aims to encode quantum information in the topological properties of physical systems, making it inherently more resistant to noise. The Khovanov–Rozansky cycle calculus, by providing a deeper understanding of topological invariants, could be instrumental in designing and analyzing these fault-tolerant quantum computing architectures. The ability to precisely characterize and manipulate topological structures is paramount for building stable and scalable quantum computers.

The research team has meticulously laid out the framework for this new calculus, detailing the algebraic constructions and the computational procedures involved. Their paper reads like a roadmap, guiding fellow researchers through the intricacies of this novel approach. The publication has already sparked a flurry of discussion and preliminary investigations by other mathematicians and physicists eager to explore its capabilities. This rapid engagement is a strong indicator of the significance and potential impact of their findings. The scientific community thrives on such collaborative exploration, and this work is poised to ignite a new wave of research.

Looking ahead, the researchers envision several avenues for future work. They aim to extend the Khovanov–Rozansky cycle calculus to other classes of links and knots, further broadening its applicability. They also plan to explore the connections between their new calculus and other advanced mathematical theories, such as category theory and homological algebra, potentially revealing even deeper underlying principles. The journey of unraveling these complex mathematical puzzles is far from over, and this latest discovery represents a monumental leap forward, promising to redefine our understanding of the fundamental structures that govern our universe. The potential for a viral impact is significant as the applications span from the most abstract realms of mathematics to the very tangible challenges of materials science and quantum information.

Subject of Research: The development of a novel computational framework, the Khovanov–Rozansky cycle calculus, for precisely analyzing and computing topological invariants of bipartite links, leading to deeper insights into their algebraic structure and potential applications in theoretical physics and other scientific disciplines.

Article Title: Khovanov–Rozansky cycle calculus for bipartite links

Article References:

Anokhina, A., Lanina, E. & Morozov, A. Khovanov–Rozansky cycle calculus for bipartite links.
Eur. Phys. J. C 85, 1185 (2025). https://doi.org/10.1140/epjc/s10052-025-14854-y

Image Credits: AI Generated

DOI: https://doi.org/10.1140/epjc/s10052-025-14854-y

Keywords**: Khovanov homology, Rozansky homology, cycle calculus, bipartite links, topological quantum field theory, algebraic topology, knot theory, mathematical physics