63 papers
This paper establishes a geometric group theory framework for right quasigroups by introducing graph markings and invariants to characterize their Cayley (di)graphs, thereby proving that all racks are realizable by their full Cayley graphs and providing graph-theoretic characterizations for various algebraic structures.
This paper proves that for knots containing specific components like the -torus knot or the figure-eight knot, no compactly supported Hamiltonian diffeomorphism can move their conormal bundles to intersect the zero section cleanly along an unknot, a result established by deriving a unique algebraic constraint on the augmentation variety over the rational numbers using symplectic field theory.
This paper establishes a general framework for visible Lagrangians in Hitchin systems that factor through proper subvarieties of the Hitchin base, computes their fiber-wise Fourier-Mukai transforms to construct mirror dual branes, and provides a detailed new example arising from pillowcase covers that connects to Hausel's toy model.
This paper establishes that the space of infinite circle patterns in the Euclidean plane parameterized by discrete harmonic functions of finite Dirichlet energy forms an infinite-dimensional Hilbert manifold homeomorphic to the Sobolev space of half-differentiable functions, equipped with a Riemannian metric derived from a hyperbolic volume functional that relates to a symplectic form via an analogue of the Hilbert transform, thereby connecting these patterns to the Weil-Petersson class of the universal Teichmüller space.
This paper classifies free graph braid groups using cubical structures and demonstrates that, under specific conditions, the large-scale geometry of graph 2-braid groups is determined by the union of maximal product subcomplexes, yielding new examples of groups that are and are not quasi-isometric to right-angled Artin groups.
This paper investigates higher property T by providing new operator-algebraic characterizations and exploring its relationship with cohomological, rigidity, and geometric phenomena in lattices of semisimple Lie groups below their real rank, culminating in a unifying conjectural framework.
This paper establishes the non-amenability of mapping class groups for infinite-type surfaces and locally finite infinite graphs of higher ranks, provides an example of a non-amenable point stabilizer in a specific hyperbolic Polish group, and identifies a class of amenable mapping class groups associated with trees and rank-one graphs.
This paper establishes a cubic braid-type skein theory for the Links-Gould polynomial, proving its universal computability for oriented links and demonstrating its equivalence to the -polynomial, thereby unifying their shared properties such as specialization to the Alexander polynomial and providing a Seifert genus bound.
This paper presents explicit formulas for the Alexander polynomial of pretzel knots, characterizes those with trivial polynomials, and utilizes these results to construct a new family of knots that are topologically slice but not smoothly slice.
This paper explores invariants of almost embeddings of graphs in the plane by establishing relations among them, connecting these to the homology of the deleted product, constructing examples, and presenting these topological concepts in an accessible manner while highlighting open conjectures.
This paper introduces new finite criteria that enable a practical algorithm to certify projective Anosov subgroups in or , significantly reducing the computational complexity required for verification as demonstrated by a genus 2 surface group example.
This paper establishes that convex-cocompact representations of finitely generated groups into the isometry group of infinite-dimensional hyperbolic space form an open set, enabling the construction of new surface group representations via bending that are distinct from those classified by Monod and Py.
This paper introduces the -adjacency graph to visualize relationships between knots connected by sets of crossing circles and establishes several fundamental properties of this new mathematical structure.
This paper establishes that the -deformed modular group specializes to a finite group at a complex parameter if and only if is a primitive -th root of unity for , in which cases the resulting groups are isomorphic to specific binary polyhedral groups, while the case yields an infinite but "mild" structure with applications to Jones polynomials.
This paper introduces the framework of non-Abelian patchworking, a geometric method for constructing real algebraic surfaces in via the real locus of non-Abelian complex-phase tropical hypersurfaces, which successfully reproduces all isotopy types of surfaces up to degree three and reveals that primitive surfaces can exhibit Euler characteristics distinct from their complex counterparts.
This paper establishes a necessary and sufficient condition for embedding homogeneous quandles associated with a triplet into conjugation quandles, thereby generalizing existing results for Alexander quandles and providing new insights into core, Grassmann, and rotation quandles.
This paper introduces graph-pretzel links as a generalization of classical pretzel links and demonstrates that a specific subfamily associated with the complete graph on four vertices yields an infinite family of distinct ribbon knots that share a trivial Alexander polynomial but are distinguished by their Jones polynomials.
This paper presents a highly symmetric 6-dimensional polytope whose antipodal quotient yields a 24-vertex triangulation of , conjectured to be minimal, while also providing improved 45- and 49-vertex triangulations for .
This paper partially answers Genevois's question regarding when hyperbolic graph braid groups arise as 3-manifold groups by demonstrating that is a 3-manifold group while for is not even quasi-isometric to one.
This paper establishes that two homotopic smooth embeddings of a closed surface into a closed oriented 5-manifold are isotopic if they share a common algebraic dual 3-sphere or if the ambient manifold is simply connected, by introducing a new invariant defined via the manifold's homotopy groups that classifies such isotopy classes.