Improved algorithm for modular links provides upper volume bounds.
problem Understanding the geometry of modular links and Lorenz links.
method Bunch algorithm to study modular links and provide upper volume bounds.
result First upper volume bound independent of word exponents and quadratic in braid index.
Computed linking number of modular knots and Lorenz links.
problem Computing the linking number of modular knots in a specific space.
method Using correspondence between modular links and Lorenz links, and intersection number involving Conway topographs.
result Computed linking number of modular knots and compared to a previous formula.
New link types derived from old links.
problem Understanding the structure of modular links.
method Analyzing the complements of arithmetic modular links and comparing them to augmented chainlinks.
result Arithmetic modular links are homeomorphic to augmented chainlinks.
Lower bound found for volumes of modular link complements.
problem Finding a lower bound for the volumes of modular link complements.
method Analyzing the volumes of link complements associated with geodesics in the modular surface.
result First linear lower volume bound in terms of exponents of code words.
Study linking numbers in hyperbolic 3-folds, linking to Siegel modular forms.
problem Linking numbers between geodesics in arithmetic hyperbolic 3-folds.
method Analyzing integral of Kudla--Millson theta series over Seifert surfaces.
result Series converges to genus 2 Siegel modular forms of weight 2.
Linking numbers of modular knots derived from geometric and algebraic properties.
problem Understanding linking numbers between modular knots and the trefoil.
method Geometric and algebraic properties of the modular group and its action on the hyperbolic plane.
result Derived several formulae for linking numbers with arithmetical, combinatorial, topological and group theoretical flavors.
Formula for arborescent link tails using theta functions.
problem Understanding tails of colored Jones polynomials for arborescent links.
method Explicit formula using theta and false theta functions.
result Numerical evidence for modularity of tails for alternating knots.
Modular knots follow Chebotarev law from surgeries on hyperbolic fibered links.
problem Understanding knots that behave like prime numbers.
method Analyzing planetary links and surgeries on hyperbolic fibered links in S3. result Modular knots around torus knots Ka,b follow the Chebotarev law. Modularity-aware GAE and VGAE improve community detection and link prediction.
problem Improving community detection with GAE and VGAE in the absence of node features.
method Introducing a modularity-aware message passing scheme and regularizer to GAE and VGAE encoders.
result Jointly addressing community detection and link prediction with high accuracy is possible.
New invariants derived from a modular category for links and 3-manifolds.
problem Developing new invariants for links and 3-manifolds.
method Extracted invariants from a modular category with two simple objects.
result Invariants for 3-manifolds are related by a specific equation.
Geodesics on modular surface yield arithmetic 3-manifolds.
problem Understanding arithmetic properties of modular surfaces.
method Constructing geodesics and analyzing their lifts.
result Complements of canonical lifts are arithmetic 3-manifolds.
Study q-series for 3-manifolds with line defects, proving homomorphism and conjecturing holomorphic modularity.
problem Understanding BPS q-series for 3-manifolds with line defects. method Proving homomorphism from skein module to space of q-series, conjecturing holomorphic modularity. result Holomorphic quantum modularity of q-series suggests new approach to Langlands duality. A formula for Rademacher symbols in triangle groups is provided.
problem No specific problem stated; focuses on a mathematical formula.
method Presentation of an explicit formula for Rademacher symbols.
result Generalizes Ghys' proof of modular knot linking numbers.
Quantum invariants of 3-manifolds and links reviewed, with connections to other topological invariants.
problem Quantum invariants of 3-manifolds and links.
method Review of recent developments and connections to other invariants.
result Rich features of quantum invariants like quantum modularity and Verma module structures.
We consider an asymptotic expansion of Kashaev's invariant or the colored Jones function for the torus link T(2,2m). We shall give q-series identity related to these invariants, and show that the invariant is regarded as a limit of q being N-th root of unity of the Eichler integral of the modular form of weight 3/2.
Study reveals connection between torus links and logarithmic VOAs.
problem Understanding the relationship between torus links and logarithmic VOAs.
method Proposed a geometric method to compute the singlet character of (s,t)-log VOA. result The singlet character of (s,t)-log VOA at the root of unity coincides with the Kashaev invariant and exhibits quantum modularity. Holomorphic quantum modular forms linked to knot volumes.
problem Understanding algebraic properties of quantum modular forms.
method Analyzing descendant state integrals for specific knots.
result Illustrated algebraic properties for the (-2,3,7)-pretzel knot.
The paper studies knots in modular flows using self-covers.
problem Understanding topological properties of modular knots.
method Constructing templates for Anosov flows and studying closed geodesics.
result Explicit construction of an infinite family of links with the trefoil.
Recent work extends Turaev's modular categories to non-semisimple settings.
problem Building TQFTs beyond semisimplicity.
method Generalized modular categories.
result Success in extending Turaev's construction to non-semisimple settings.
Contemporary sensorimotor learning approaches typically start with an existing complex agent (e.g., a robotic arm), which they learn to control. In contrast, this paper investigates a modular co-evolution strategy: a collection of primitive agents learns to dynamically self-assemble into composite bodies while also lea…
We study the asymptotic behaviour of the quantum representations of the modular group in the large level limit. We prove that each element of the modular group acts as a Fourier integral operator. This provides a link between the classical and quantum Chern-Simons theories for the torus. From this result we deduce the …
Proves representability of complex semigroup systems.
problem Representability of systems of proportionally modular numerical semigroups.
method Canonical equivariant resolution of weighted homogeneous surface singularities.
result Every system of proportionally modular numerical semigroups is representable.
Improved community detection and link prediction with GAE and VGAE.
problem Jointly improving community detection and link prediction with graph autoencoders.
method Introducing a community-preserving message passing scheme and doping encoders with Louvain-based prior communities.
result Empirical effectiveness of Modularity-Aware GAE and VGAE on various real-world graphs.
New techniques prove quantum modularity for various functions.
problem Proving quantum modularity of false theta functions and related series.
method Developed techniques including Poisson summation formula and modular series framework.
result Unified approach to proving quantum modularity for various functions.
Study counts geodesics on modular surface, linking to necklace counting.
problem Counting geodesics on modular surface with specific winding numbers.
method Asymptotic expansion, generating function analysis, correspondence to necklace counting.
result Obtained asymptotic growth rate of m low-lying geodesics in terms of word length.
Modular categories are a well-known source of quantum 3-manifold invariants. In this paper we study structures on modular categories which allow to define refinements of quantum 3-manifold invariants involving cohomology classes or generalized spin and complex spin structures. A crucial role in our construction is play…
The paper proves a quantum modularity conjecture for 3-manifolds.
problem Quantum invariants of 3-manifolds at roots of unity.
method Formulates and proves a strong version of the conjecture for geometric 3-manifolds.
result The conjecture holds for Brieskorn homology spheres and some other examples.
A very simple expression is conjectured for arbitrary colored Jones and HOMFLY polynomials of a rich (g+1)-parametric family of Pretzel knots and links. The answer for the Jones and HOMFLY polynomials is fully and explicitly expressed through the Racah matrix of U_q(SU_N), and looks related to a modular transformatio…
Cone structures in quantum field theory linked to information geometry.
problem Understanding geometric structures in quantum field theory.
method Analyzing invariant cones under modular automorphism groups and their relation to Wishart laws.
result Explicit connection between CAH cones and Wishart laws.
The Yokonuma-Hecke algebras are quotients of the modular framed braid group and they support Markov traces. In this paper, which is sequel to Juyumaya and Lambropoulou (2007), we explore further the structures of the p-adic framed braids and the p-adic Yokonuma-Hecke algebras constructed in Juyumaya and Lambropoulo…
É.Ghys proved that the linking numbers of modular knots and the "missing" trefoil K2,3 in S3 coincide with the values of a highly ubiquitous function called the Rademacher symbol for SL2Z. In this paper, we replace SL2Z=Γ2,3 by the triangle group Γp,q for any coprime …
This paper provides both a detailed study of color-dependence of link homologies, as realized in physics as certain spaces of BPS states, and a broad study of the behavior of BPS states in general. We consider how the spectrum of BPS states varies as continuous parameters of a theory are perturbed. This question can be…
Holomorphic functions from knot complements link to quantum modular forms.
problem Analyzing holomorphic functions from knot complements.
method Matrix-valued holomorphic functions, cocycles, and quantum modularity.
result Identifies a matrix-valued holomorphic quantum modular form.
This paper proves a conjecture linking quantum modular forms and WRT invariants for specific graphs.
problem Proving a conjecture about quantum modular forms and WRT invariants for unimodular H-graphs.
method Constructed finite sums of rational functions, studied weighted Gauss sums, and combined results to prove the conjecture.
result WRT invariants of H-graphs yield quantum modular forms of depth two and weight one.
Periodic geodesics on the modular surface correspond to periodic orbits of the geodesic flow in its unit tangent bundle PSL2(Z)\PSL2(R). The complement of any finite number of orbits is a hyperbolic 3-manifold, which thus has a well-defined volume. We present strong nu…
Objective: Multimodal measurements of the same phenomena provide complementary information and highlight different perspectives, albeit each with their own limitations. A focus on a single modality may lead to incorrect inferences, which is especially important when a studied phenomenon is a disease. In this paper, we …
Paper constructs new identities linking quantum invariants and modular forms.
problem Connecting quantum invariants with modular forms.
method Bailey pairs, Eichler integrals, and novel relations between sums.
result Infinite families of identities linking q-multisums and Eichler integrals.
New cubic forms linked to η-invariants and mod 2 indices.
problem Anomaly cancellation and modularity in 12-manifolds.
method Combination of Witten classes and affine E8 character. result Relates cubic forms to η-invariants and mod 2 indices.
This paper improves hierarchical community detection efficiency using local structural properties.
problem Efficiency of hierarchical community detection methods in large networks.
method Use of local structural network properties as proxies to improve efficiency.
result Achieves competitive results in modularity with improved efficiency.
Finite specializations of a q-deformed modular group at roots of unity.
problem Understanding the finiteness of specializations of a q-deformed modular group at roots of unity.
method Introduced a q-deformed modular group and studied its specializations at roots of unity.
result For ζn being a primitive nth root of unity, PSLq(2,Z)∣q=ζn is finite if and only if Gq(ζn) is finite. The f-invariant is a higher version of the e-invariant that takes values in the divided congruences between modular forms; it can be formulated as an elliptic genus of manifolds with corners of codimension two. In this thesis, we develop a geometrical interpretation of the f-invariant in terms of index theory, thereby …
We calculate the RT-invariants of all oriented Seifert manifolds directly from surgery presentations. We work in the general framework of an arbitrary modular category as in [V. G. Turaev, Quantum invariants of knots and 3--manifolds, de Gruyter Stud. Math. 18, Walter de Gruyter (1994)], and the invariants are expresse…
We present an invariant of connected and oriented closed 3-manifolds based on a coribbon Weak Hopf Algebra H with a suitable left-integral. Our invariant can be understood as the generalization to Weak Hopf Algebras of the Hennings-Kauffman-Radford evaluation of an unoriented framed link using a dual quantum-trace. Thi…
We develop several applications of the fact that the Yokonuma--Hecke algebra of the general linear group GL is isomorphic to a direct sum of matrix algebras associated to Iwahori--Hecke algebras of type A. This includes a description of the semisimple and modular representation theory of the Yokonuma--Hecke algebras of…
Constructs the moduli stack of elliptic curves as an orbifold.
problem Moduli stack construction for elliptic curves.
method Analytic orbifold construction, non-effective actions.
result Provides a self-contained account for young researchers.
We construct unitary modular categories for a general class of coset conformal field theories based on our previous study of these theories in the algebraic quantum field theory framework using subfactor theory. We also consider the calculations of the corresponding 3-manifold invariants. It is shown that under certain…
Introduces q-transpose for q-deformed modular group matrices.
problem Understanding q-deformed rational numbers and their properties. method Introduces q-transpose and applies it to refine q-deformed modular group actions. result New proof and refinement of Leclere and Morier-Genoud's trace palindromicity theorem.
This chapter provides a self-contained introduction to the use of Bayesian inference to extract large-scale modular structures from network data, based on the stochastic blockmodel (SBM), as well as its degree-corrected and overlapping generalizations. We focus on nonparametric formulations that allow their inference i…