arXiv research

Generalizes DW invariants for cusped 3-manifolds, distinguishing some pairs.

New invariant cusp crossing density shows cusp densities of hyperbolic knots and links are dense.

Surveying tools for estimating geometric properties of hyperbolic knots.

The cobordism invariance of the index on closed manifolds is reproved using the calculus of cusp pseudodifferential operators on a manifold with boundary. More generally, on a compact manifold with corners, the existence of a symmetric cusp differential operator of order 1 and of Dirac type near the boundary implies th…

Identifies 3D-index as invariant for cusped hyperbolic 3-manifolds.

This study introduces a spin invariant for Prym-Teichmüller curves in genus 3.

Power series invariant of hyperbolic 3-manifolds matches knot invariants.

Study relates Gaussian curvature signs to cuspidal edge types and geometric invariants.

Jones and HOMFLY polynomials are derived from cluster C*-algebras.

Extended Bridgeman-Kahn identity for hyperbolic manifolds with cusps.

We study the eta invariants of Dirac operators and the regularized determinants of Dirac Laplacians over hyperbolic manifolds with cusps. We follow Werner M"uller and use relative traces to define these spectral invariants. We show the regularity of eta and zeta functions at s=0. The Selberg trace formula and the detai…

The paper explains how the conjugate locus of a point on a surface can gain or lose cusps as the point moves.

Computes cusp cobordism groups for Morse functions on manifolds.

The study analyzes spectral asymmetry and index theory on manifolds with generalized hyperbolic cusps.

New invariant measures knot geometry, improving volume-volume inequality.

Let M be a geometrically finite pinched negatively curved Riemannian manifold with at least one cusp. Inspired by the theory of diophantine approximation of a real (or complex) number by rational ones, we develop a theory of approximation of geodesic lines starting from a given cusp by ones returning to it. We define a…

Finite-type k-surfaces in hyperbolic space have finite genus and cusp-like ends.

Study shows arithmetic properties of specific hyperbolic Dehn fillings.

At a 3/2-cusp of a given plane curve $γ(t)$, both of the Euclidean curvature $κ_g$ and the affine curvature $κ_A$ diverge. In this paper, we show that each of $\sqrt{|s_g|}κ_g$ and $(s_A)^2 κ_A$ (called the Euclidean and affine normalized curvature, respectively) at a 3/2-cusp is a smooth function of the variable $t$, …

In this paper we deepen the analysis of certain classes M_{g,k} of hyperbolic 3-manifolds that were introduced in a previous work by B. Martelli, C. Petronio and the author. Each element of M_{g,k} is an oriented complete finite-volume hyperbolic 3-manifold with compact connected geodesic boundary of genus g and k cusp…

We study a computational method of the hyperbolic Reidemeister torsion (also called in the literature the non-abelian Reidemeister torsion) induced by J. Porti for complete hyperbolic three-dimensional manifolds with cusps. The derivative of the twisted Alexander invariant for a hyperbolic knot exterior gives the hyper…

An analytic index is defined for a family of cusp pseudodifferential operators, $P_b,$ on a fibration with fibres which are compact manifolds with boundaries, provided the family is elliptic and has invertible indicial family at the boundary. In fact there is always a perturbation $Q_b$ by a family of cusp operators of…

Algorithm reduces cusps and singular components in maps of surfaces with boundary.

In this paper we will promote the 3D index of an ideal triangulation T of an oriented cusped 3-manifold M (a collection of q-series with integer coefficients, introduced by Dimofte-Gaiotto-Gukov) to a topological invariant of oriented cusped hyperbolic 3-manifolds. To achieve our goal we show that (a) T admits an index…

We consider a compact manifold whose boundary is a locally trivial fiber bundle and an associated pseudodifferential algebra that models fibered cusps at infinity. Using trace-like functionals that generate the 0-dimensional Hochschild cohomology groups, we express the index of a fully elliptic fibered cusp operator as…

We study the adiabatic limit of the eta invariant of the Dirac operator over cofinite quotient of PSL(2,R), which is a noncompact manifold with a nonexact fibred-cusp metric near the ends.

The paper classifies symplectic invariants of specific singularities in integrable Hamiltonian systems.

We study the geometry of the Margulis region associated with an irrational screw translation $g$ acting on the 4-dimensional real hyperbolic space. This is an invariant domain with the parabolic fixed point of $g$ on its boundary which plays the role of an invariant horoball for a translation in dimensions $\leq 3$. Th…

Proves regularity of isomorphisms between hyperbolic 3-manifolds.

Efficient cobordisms show minimal signature values on certain links.

Given a cusped hyperbolic 3-manifold with finite volume, we define two types of complex parameters which capture geometric information about the preimages of geodesic arcs traveling between cusp cross-sections. We prove that these parameters are elements of the invariant trace field of the manifold, providing a connect…

We consider a geometrically finite discrete group of conformal transformations of the sphere. Further we consider distributions which are supported on the limit set and are invariant with conformal weight. We estimate their regularity in terms of the conformal weight, the Hausdorff dimension of the limit set, and the m…

Quantum Teichmüller invariants studied for 3-manifolds.

We define for each g>=2 and k>=0 a set M_{g,k} of orientable hyperbolic 3-manifolds with $k$ toric cusps and a connected totally geodesic boundary of genus g. Manifolds in M_{g,k} have Matveev complexity g+k and Heegaard genus g+1, and their homology, volume, and Turaev-Viro invariants depend only on g and k. In additi…

We study a particular class of open manifolds. In the category of Riemannian manifolds these are complete manifolds with cylindrical ends. We give a natural setting for the conformal geometry on such manifolds including an appropriate notion of the cylindrical Yamabe constant/invariant. This leads to a corresponding ve…

We classify the orientable finite-volume hyperbolic 3-manifolds having non-empty compact totally geodesic boundary and admitting an ideal triangulation with at most four tetrahedra. We also compute the volume of all such manifolds, we describe their canonical Kojima decomposition, and we discuss manifolds having cusps.…

4-manifolds show every flat 3-manifold as cusp sections.

For a closed locally symmetric space M=Γ\G/K and a representation of G we consider the push-forward of the fundamental class in the homology of the linear group and a related invariant in algebraic K-theory. We discuss the nontriviality of this invariant and we generalize the construction to cusped locally symmetric sp…

The paper verifies no cosmetic surgeries on knots and 3-manifolds using hyperbolic geometry.

Characterizes holonomies of convex projective cusps.

The state sums defining the quantum hyperbolic invariants (QHI) of hyperbolic oriented cusped $3$-manifolds can be split in a "symmetrization" factor and a "reduced" state sum. We show that these factors are invariants on their own, that we call "symmetry defects" and "reduced QHI", provided the manifolds are endowed w…

1-loop invariant equals torsion for 2-bridge knots.

Study geodesics entering a fixed cusp neighborhood multiple times.

The study proves exponential mixing for specific Weil-Petersson flows in exceptional moduli spaces.

Computes presentations for specific arithmetic hyperbolic lattices.

Proves methods for creating convex projective 3-manifolds with cusps.

By regular tessellation, we mean any hyperbolic 3-manifold tessellated by ideal Platonic solids such that the symmetry group acts transitively on oriented flags. A regular tessellation has an invariant we call the cusp modulus. For small cusp modulus, we classify all regular tessellations. For large cusp modulus, we pr…

The paper classifies different types of cusps on plane curves.