arXiv research

Upper bounds for volume spectrum depend on volume, dimension, and a conformal invariant.

Study shows colored Jones invariants limit to link volumes.

New invariants defined for volume-preserving flows on 3-manifolds.

The volume conjecture is extended for surface diffeomorphisms with quantum invariants.

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…

Study Turaev-Viro invariants of Seifert fibered 3-manifolds, proving volume conjecture.

A new polynomial invariant for links in a thickened torus exhibits volume conjecture behavior.

The paper defines and analyzes a volume invariant for 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.

The paper defines unimodularity for coisotropic Poisson spaces and discusses invariant volume forms.

Geodesically equivalent Finsler metrics share invariant volume forms and first integrals.

Study reveals failure of uniqueness in dynamical invariants for 3D volume-preserving diffeomorphisms.

We give a volume formula of hyperbolic knot complements using twisted Alexander invariants.

Defines a new version of Turaev-Viro invariants for 3-manifolds with boundaries.

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.

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…

Study counts hyperbolic 4-manifolds with vanishing Seiberg-Witten invariants up to volume $v$.

Renormalized volume invariant for knots in 3-sphere computed.

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.

Study Turaev-Viro invariants of 3-manifolds with toroidal boundary.

We obtain a formula for the Turaev-Viro invariants of a link complement in terms of values of the colored Jones polynomial of the link. As an application we give the first examples for which the volume conjecture of Chen and the third named author\,\cite{Chen-Yang} is verified. Namely, we show that the asymptotics of t…

We show how to compute the circular area invariant of planar curves, and the spherical volume invariant of surfaces, in terms of line and surface integrals, respectively. We use the Divergence Theorem to express the area and volume integrals as line and surface integrals, respectively, against particular kernels; our r…

Equal-volume polygons are obtained from adequate discretizations of curves in 3-space, contained or not in surfaces. In this paper we explore the similarities of these polygons with the affine arc-length parameterized smooth curves to develop a theory of discrete affine invariants. Besides obtaining discrete affine inv…

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.

The paper proves an asymptotic additivity of Turaev-Viro invariants for a family of 3-manifolds.

For primes $p$, we investigate an $\mathbb{F}_p$-version of simplicial volume and compare these invariants with their siblings over other coefficient rings. We will also consider the associated gradient invariants, obtained by stabilisation along finite coverings. Throughout, we will discuss the relation between such s…

We prove that any regular integral invariant of volume-preserving transformations is equivalent to the helicity. Specifically, given a functional $\mathcal I$ defined on exact divergence-free vector fields of class $C^1$ on a compact 3-manifold that is associated with a well-behaved integral kernel, we prove that $\mat…

Proves 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…

We extend the Neumann's methods and give the explicit formulae for the volume and the Chern-Simons invariant for hyperbolic alternating knot orbifolds.

Proof confirms volume conjecture for a specific knot.

Formula calculates volume of two-bridge knots.

Researchers compute quantum invariant for four-puncture sphere, verifying volume conjecture.

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

The goal of this article is to establish estimates involving the Yamabe minimal volume, mixed minimal volume and some topological invariants on compact 4-manifolds. In addition, we provide topological sphere theorems for compact submanifolds of spheres and Euclidean spaces, provided that the full norm of the second fun…

We study how the genus, the simplicial volume and the $L^2$-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 $L^2$-Alexander invariant contains strictly more information than th…

In this paper, we prove that the systolic volume of a closed aspherical 3-manifold is bounded below in terms of complexity. Systolic volume is defined as the optimal constant in a systolic inequality. Babenko showed that the systolic volume is a homotopy invariant. Moreover, Gromov proved that the systolic volume depen…

Study on Berwald-Weyl curvature with projective invariance and vanishing results.

The paper connects quantum $6j$-symbols to tetrahedra volumes via discrete Fourier transforms.

We define invariants for colored oriented spatial graphs by generalizing CM invariants, which were defined via non-integral highest weight representations of $U_q(sl_2)$. We apply the same method to define Yokota's invariants, and we call these invariants Yokota type invariants. Then we propose a volume conjecture of t…

Study calculates quantum hyperbolic invariants for figure-eight knot complement, finding it either 0 or half the volume.

The paper introduces a volume invariant for Hermitian-symplectic metrics and proves its critical points are Kähler.

We introduce a generalization of the Dijkgraaf-Witten invariants for cusped or compact oriented 3-manifolds. We show that the generalized DW invariants distinguish some pairs of cusped hyperbolic 3-manifolds with the same hyperbolic volumes and with the same Turaev-Viro invariants. We also present an example of a pair …

Simplicial volume vanishes for 4-manifolds with open book decompositions.