Upper bounds for volume spectrum depend on volume, dimension, and a conformal invariant.
problem Bounding the volume spectrum of Riemannian manifolds.
method Proves upper bounds that depend on volume, dimension, and a conformal invariant.
result Upper bounds for the volume spectrum are established.
Study shows colored Jones invariants limit to link volumes.
problem Volume conjecture for colored Jones invariants.
method Deformation of hyperbolic structure for link complements.
result Limits of colored Jones invariants related to link volumes.
Formulae for volume and Chern-Simons invariant of hyperbolic knot orbifolds.
problem Computing volume and Chern-Simons invariant for hyperbolic knot orbifolds.
method Extending Neumann's methods to derive explicit formulae.
result Explicit formulae for volume and Chern-Simons invariant.
Paper proves stability of volume conjecture under cabling.
problem Stability of volume conjecture under cabling.
method Analyzes Turaev-Viro invariants and simplicial volume.
result Stability of Chen-Yang conjecture under (2n+1,2)-cabling. Formula connects Turaev-Viro invariants to colored Jones polynomials and hyperbolic volumes.
problem Relating Turaev-Viro invariants to hyperbolic volumes of link complements.
method Using colored Jones polynomials and asymptotic behavior of Turaev-Viro invariants.
result Asymptotics of Turaev-Viro invariants determine hyperbolic volumes of specific link complements.
Computes circular area and spherical volume invariants via integrals.
problem Computing circular area and spherical volume invariants for curves and surfaces.
method Using the Divergence Theorem, express area and volume integrals as line and surface integrals against kernels, then compute analytically on triangulated meshes.
result Simple algorithm for computing spherical volume invariant for triangulated surfaces without discretizing ambient space.
The study of equal-volume polygons in 3D space and their affine invariants.
problem Estimating projective invariants of planar curves.
method Developing a theory of discrete affine invariants from equal-volume polygons.
result Equal-volume polygons can be used to estimate projective invariants of a planar curve.
New invariants defined for volume-preserving flows on 3-manifolds.
problem Defining invariants for volume-preserving flows.
method Extending wrapping number and trunk to define invariants of links and flows.
result Wrappingness and trunkenness are not functions of helicity.
The volume conjecture is extended for surface diffeomorphisms with quantum invariants.
problem Extending the volume conjecture for quantum invariants of surface diffeomorphisms.
method Relating asymptotics of quantum invariants to hyperbolic cone structures on mapping tori.
result The conjecture is proven for a specific case of the once-punctured torus bundle.
Investigates simplicial volume over finite fields and compares it with other coefficients.
problem Examines simplicial volume over Fp coefficients. method Analyzes simplicial volume and gradient invariants over Fp coefficients, comparing with other coefficient rings. result Compares simplicial volumes and Betti numbers over different coefficient rings.
We give an efficient simplicial formula for the volume and Chern-Simons invariant of a boundary-parabolic PSL(2,C)-representation of a tame 3-manifold. If the representation is the geometric representation of a hyperbolic 3-manifold, our formula computes the volume and Chern-Simons invariant directly from an ideal tria…
A new polynomial invariant for links in a thickened torus exhibits volume conjecture behavior.
problem Defining and characterizing a new polynomial invariant for links in a thickened torus.
method Defining a new invariant JnT, proving properties, and providing constructions. result The invariant JnT exhibits volume conjecture behavior, providing the first example of this in a virtual link. Study Turaev-Viro invariants of Seifert fibered 3-manifolds, proving volume conjecture.
problem Understanding the asymptotic behavior of Turaev-Viro invariants for Seifert fibered 3-manifolds.
method Analysis of large r asymptotic behavior of Turaev-Viro invariants. result Proved the volume conjecture for Seifert fibered 3-manifolds with empty and non-empty boundaries.
The paper defines and analyzes a volume invariant for 3-manifolds.
problem Defining and analyzing a topological invariant for 3-manifolds.
method Definition and analysis of topological volume, refinements, bounds determination, classification of manifolds.
result Asymptotically tight upper and lower bounds for topological volume, classification of non-hyperbolic 3-manifolds.
Let Γ be a lattice in a connected semisimple Lie group G with trivial center and no compact factors. We introduce a volume invariant for representations of Γ into G, which generalizes the volume invariant for representations of uniform lattices introduced by Goldman. Then, we show that the maximality of this vo…
The paper explores the connection between Poisson-Lie structures and invariant volume forms in Hamiltonian dynamics.
problem Understanding the relationship between Poisson-Lie structures and invariant volume forms in Hamiltonian systems.
method Analyzing the existence and preservation of invariant volume forms under Hamiltonian vector fields on Poisson-Lie groups.
result A unimodular Poisson-Lie structure ensures the preservation of a multiple of any left-invariant volume, and the existence of a preserving volume form implies unimodularity.
Study reveals failure of uniqueness in dynamical invariants for 3D volume-preserving diffeomorphisms.
problem Uniqueness of dynamical invariants for 3D volume-preserving diffeomorphisms.
method Examined failure of uniqueness on integral homology spheres and arbitrary three-manifolds using local and global invariants.
result Failure of uniqueness is severe, with continuous and non-constant invariants appearing in C1-open sets of nonvanishing exact fields of fixed helicity. Geodesically equivalent Finsler metrics share invariant volume forms and first integrals.
problem Understanding shared properties of geodesically equivalent Finsler metrics.
method Computing first integrals as coefficients of a characteristic polynomial.
result Geodesically invariant functions are first integrals of geodesically equivalent Finsler metrics.
The paper defines unimodularity for coisotropic Poisson spaces and discusses invariant volume forms.
problem Understanding unimodularity and invariant volume forms for Hamiltonian dynamics on coisotropic Poisson spaces.
method Introducing multiplicative unimodularity and discussing its properties for coisotropic Poisson homogeneous spaces.
result Existence of invariant volume forms for explicit Hamiltonian systems on coisotropic Poisson spaces.
We give a volume formula of hyperbolic knot complements using twisted Alexander invariants.
New invariant from links to polyhedra volumes.
problem Computing hyperbolic volumes of link complements.
method Geometric, topological, and combinatorial methods to decompose link complements into ideal polyhedra.
result A new geometric link invariant, the right-angled volume, is a lower bound for hyperbolic volume.
Defines a new version of Turaev-Viro invariants for 3-manifolds with boundaries.
problem Computing the volume of hyperbolic polyhedral 3-manifolds.
method Introduces a relative version of Turaev-Viro invariants for ideally triangulated compact 3-manifolds with boundaries and a coloring on edges.
result Proves the Volume Conjecture for these invariants, suggesting a method to solve the conjecture for hyperbolic 3-manifolds with totally geodesic boundary.
Generalizes DW invariants for cusped 3-manifolds, distinguishing some pairs.
problem Distinguishing cusped 3-manifolds with same volumes and invariants.
method Introducing and analyzing generalized Dijkgraaf-Witten invariants.
result Generalized DW invariants can distinguish some pairs of cusped hyperbolic 3-manifolds.
We investigate the conjectural relations between the Reshetikhin-Turaev-Witten quantum SU(2) invariants and the volume of hyperbolic 3-manifolds. Given a finite set of sufficiently large positive integers, say J, we construct examples of closed hyperbolic 3-manifolds with the same invariants at all levels in J and diff…
Study proves Volume Conjecture for Reshetikhin-Turaev invariants.
problem Volume Conjecture for Reshetikhin-Turaev invariants.
method Hyperbolic cone metrics and discrete Fourier transforms.
result Proves Volume Conjecture for most figure-8 knot configurations.
We show that given n>0, there exists a hyperbolic knot K with trivial Alexander polynomial, trivial finite type invariants of order <=n, and such that the volume of the complement of K is larger than n. This contrasts with the known statement that the volume of the complement of a hyperbolic alternating knot is bounded…
Renormalized volume invariant for knots in 3-sphere computed.
problem Computing renormalized volume for knot embeddings in 3-sphere.
method Renormalizing volume associated to singular Yamabe metric.
result Renormalized volume is a global conformal invariant for knots in 3-sphere.
Study counts hyperbolic 4-manifolds with vanishing Seiberg-Witten invariants up to volume v.
problem Counting hyperbolic 4-manifolds with specific topological properties.
method Used volume bounds and commensurability to estimate the number of such manifolds.
result The number of hyperbolic 4-manifolds with vanishing Seiberg-Witten invariants up to volume v is asymptotically bounded by vcv. New formula calculates volumes and Chern-Simons invariants for closed 3-manifolds.
problem Computing volumes and Chern-Simons invariants for non-parabolic representations.
method Introducing deformed Ptolemy varieties to extend Zickert's formula.
result Volume and Chern-Simons invariants computed for closed 3-manifolds.
We study the variation of a smooth volume form along extremals of a variational problem with nonholonomic constraints and an action-like Lagrangian. We introduce a new invariant describing the interaction of the volume with the dynamics and we study its basic properties. We then show how this invariant, together with c…
This paper discuss an intrinsic relation among congruent relations \cite{CLPZ}, cyclotomic expansion and Volume Conjecture for SU(n) invariants. Motivated by the congruent relations for SU(n) invariants obtained in our previous work \cite{CLPZ}, we study certain limits of the SU(n) invariants at various roots of …
We calculate the asymptotic behavior of hyperbolic volume and Chern-Simons invariant.
problem Calculating the asymptotic behavior of hyperbolic volume and Chern-Simons invariant.
method Renormalization of the Chern-Simons invariant using asymptotics along an equidistance foliation.
result The leading coefficient introduces a complex-valued quantity consisting of mean curvature and torsion 2-form.
Study Turaev-Viro invariants of 3-manifolds with toroidal boundary.
problem Volume conjecture for Seifert fibered and graph 3-manifolds.
method Large r asymptotic behavior of Turaev-Viro invariants under gluing operation. result Volume conjecture proven for Seifert fibered and graph 3-manifolds.
The study identifies Hermitian metrics preserving the total Monge-Ampere volume.
problem Understanding Hermitian metrics preserving volume invariance.
method Characterizations and comparison principles for complex Monge-Ampere operator.
result Several characterizations of Hermitian metrics satisfying the comparison principle.
The paper introduces a new complex analytic invariant called the pointed harmonic volume and its relation to the Johnson homomorphism.
problem Exploring new complex analytic invariants related to the complex structure of Riemann surfaces.
method Defining and computing the pointed harmonic volume as a natural extension of Chen's iterated integrals.
result Established a relationship between the harmonic volume and the first extended Johnson homomorphism.
The paper proves an asymptotic additivity of Turaev-Viro invariants for a family of 3-manifolds.
problem Preserving the Turaev-Viro invariant volume conjecture under gluings of toroidal boundary components.
method Using a construction of hyperbolic cusped 3-manifolds by Agol, the authors show that the asymptotics of Turaev-Viro invariants are additive under certain gluings of elementary pieces.
result The Turaev-Viro invariant volume conjecture is preserved under specific gluings of 3-manifolds.
We construct a new invariant-the trunkenness-for volume-perserving vector fields on S^3 up to volume-preserving diffeomorphism. We prove that the trunkenness is independent from the helicity and that it is the limit of a knot invariant (called the trunk) computed on long pieces of orbits.
We prove that any regular integral invariant of volume-preserving transformations is equivalent to the helicity. Specifically, given a functional I defined on exact divergence-free vector fields of class C1 on a compact 3-manifold that is associated with a well-behaved integral kernel, we prove that $\mat…
Optimal volume limit found for Kähler manifolds with positive Ricci curvature.
problem Bounding the volume of Kähler manifolds with positive Ricci curvature.
method Using δ-invariants and Newton--Okounkov bodies.
result Derive the optimal volume upper bound and new characterization of the complex projective space.
Proves volume conjectures for figure-eight knot surgeries.
problem Volume conjectures for hyperbolic 3-manifolds.
method Ohtsuki's method applied to figure-eight knot surgeries.
result Proves Asymptotic Expansion and Volume Conjectures for figure-eight knot surgeries.
The volume density of a hyperbolic link is defined as the ratio of hyperbolic volume to crossing number. We study its properties and a closely-related invariant called the determinant density. It is known that the sets of volume densities and determinant densities of links are dense in the interval [0,v_{oct}]. We cons…
Study diverging sequences of unit volume metrics with bounded curvature on homogeneous spaces.
problem Understanding 1-parameter families of invariant metrics with bounded curvature.
method Analyzing diverging sequences in the space of G-invariant, unit volume metrics on compact homogeneous spaces. result Prove structure results for diverging sequences with bounded curvature.
Proof confirms volume conjecture for a specific knot.
problem Verifying the volume conjecture for a specific knot.
method Using a generalized topological quantum field theory and a tetrahedral decomposition.
result Volume conjecture holds for the 73 knot in S3. Formula calculates volume of two-bridge knots.
problem Calculating the volume of two-bridge knots.
method Derived from Hopf formula and Fox derivatives.
result Closed formula for the volume of two-bridge knots.
Researchers compute quantum invariant for four-puncture sphere, verifying volume conjecture.
problem Verifying the Bonahon-Wong-Yang volume conjecture for a specific case.
method Representation theory of the Checkov-Fock algebra to compute quantum invariant.
result Verification of the volume conjecture for four-puncture sphere bundles with technical conditions.
New invariant measures knot geometry, improving volume-volume inequality.
problem Understanding hyperbolic knots through geometry.
method Introducing natural slope based on cusp geometry, using machine learning for detection.
result Improved inequality linking knot signature, volume, and injectivity radius.
New invariant found in spaces with curvature below, related to volume.
problem Geometric invariant of spaces with curvature below.
method Relations between new invariant and normalized volume, rigidity for maximal case.
result New invariant related to volume and rigidity for maximal case.
We study how the genus, the simplicial volume and the L2-Alexander invariant of W. Li and W. Zhang can detect individual knots among all others. In particular, we use various techniques coming from hyperbolic geometry and topology to prove that the L2-Alexander invariant contains strictly more information than th…