arXiv research

Polyhedra volume conjecture supports Stoker conjecture weakly.

Proves volume conjecture for twist knots using complex analysis.

Proof confirms volume conjecture for a specific knot.

Proves volume conjectures for figure-eight knot surgeries.

The paper proves ACC for local volumes under boundedness conditions.

We propose to generalize the volume conjecture to knotted trivalent graphs and we prove the conjecture for all augmented knotted trivalent graphs. As a corollary we find that for any link L there is a link containing L for which the volume conjecture holds.

The paper calculates intertwiners for a torus and proves a conjecture about their limits.

Paper proves Gromov's conjecture on manifolds with certain group properties.

The volume conjecture for surface diffeomorphisms connects volumes to polynomial evaluations.

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

This is an introduction to the Volume Conjecture and its generalizations for nonexperts. The Volume Conjecture states that a certain limit of the colored Jones polynomial of a knot would give the volume of its complement. If we deform the parameter of the colored Jones polynomial we also conjecture that it would also g…

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

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

The paper proves macroscopic versions of conjectures about scalar curvature and volume bounds.

The Mahler volume of a centrally symmetric convex body K is defined as M(K)= (Vol K)(Vol K^dual). Mahler conjectured that this volume is minimized when K is a cube. We introduce the bottleneck conjecture, which stipulates that a certain convex body K^diamond subset K X K^dual has least volume when K is an ellipsoid. If…

The volume conjecture states that for a hyperbolic knot K in the three-sphere S^3 the asymptotic growth of the colored Jones polynomial of K is governed by the hyperbolic volume of the knot complement S^3\K. The conjecture relates two topological invariants, one combinatorial and one geometric, in a very nonobvious, no…

Study shows colored Jones invariants limit to link volumes.

The paper explores volume product and slicing conjectures using convex body deformations.

We establish the volume conjecture for (m,2)-cables of the figure 8 knot, when m is odd. For (m,2)-cables of general knots where m is even, we show that the limit in the volume conjecture depends on the parity of the color (of the Kashaev invariant). There are many cases when the volume conjecture for cables of the fig…

Study proves volume conjecture for specific 3-manifolds.

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

We prove, in the case of hyperbolic 3-space, a couple of conjectures raised by J. J. Seidel in "On the volume of a hyperbolic simplex", Stud. Sci. Math. Hung. 21, 243-249, 1986. These conjectures concern expressing the volume of an ideal hyperbolic tetrahedron as a monotonic function of algebraic maps. More precisely, …

For any closed smooth Riemannian manifold H. Weyl has defined a sequence of numbers called today intrinsic volumes. They include volume, Euler characteristic, and integral of the scalar curvature. We conjecture that absolute values of all intrinsic volumes are bounded by a constant depending only on the dimension of th…

The Chen-Yang volume conjecture is verified for knots in handlebodies with specific boundary components.

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…

Constructs infinite families of hyperbolic knots satisfying a volume conjecture.

The paper confirms a conjecture linking link bipyramid volume and Mahler measure.

The volume conjecture, formulated recently by H. Murakami and J. Murakami, is proved for the case of torus knots.

Proves volume conjecture for double twist knots using complexified tetrahedrons.

Researchers create links in 3-sphere satisfying volume conjecture.

In this note, I will discuss a possible relation between the Mahler measure of the colored Jones polynomial and the volume conjecture. In particular, I will study the colored Jones polynomial of the figure-eight knot on the unit circle. I will also propose a method to prove the volume conjecture for satellites of the f…

The generalized volume conjecture relates asymptotic behavior of the colored Jones polynomials to objects naturally defined on an algebraic curve, the zero locus of the A-polynomial $A(x,y)$. Another "family version" of the volume conjecture depends on a quantization parameter, usually denoted $q$ or $\hbar$; this quan…

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

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 conjecture that for every dimension n not equal 3 there exists a noncompact hyperbolic n-manifold whose volume is smaller than the volume of any compact hyperbolic n-manifold. For dimensions n at most 4 and n=6 this conjecture follows from the known results. In this paper we show that the conjecture is true for arit…

In this paper, we show that the derivative of the genus-1 Virasoro conjecture for Gromov-Witten invariants along the direction of quantum volume element holds for all smooth projective varieties. This result provides new evidence for the Virasoro conjecture.

We prove the volume conjecture for an infinite family of links called Whitehead chains that generalizes both the Whitehead link and the Borromean rings.

In this article, we give a rough, and so not complete yet, proof of Kashaev's conjecture, that is, the volume conjecture for hyperbolic knots, where the hyperbolicity equations associated to knot diagrams appear as the stationary phase equations for Kashaev's invariants.

In this report I discuss the relations between systoles and volumes of hyperbolic manifolds and a conjecture of Lehmer about the Mahler measure of non-cyclotomic polynomials.

Study proves Volume Conjecture for Reshetikhin-Turaev invariants.

Improved bounds on volume-det inequality for links with many twists.

The volume conjecture is proven for twist knots after Dehn filling.

We propose a version of the volume conjecture that would relate a certain limit of the colored Jones polynomials of a knot to the volume function defined by a representation of the fundamental group of the knot complement to the special linear group of degree two over complex numbers. We also confirm the conjecture for…

Counterexamples found for volume entropy conjecture in hyperbolic 3-manifolds.

Researchers compute and predict knot volumes using colored Jones polynomials.

Loosely speaking, the Volume Conjecture states that the limit of the n-th colored Jones polynomial of a hyperbolic knot, evaluated at the primitive complex n-th root of unity is a sequence of complex numbers that grows exponentially. Moreover, the exponential growth rate is proportional to the hyperbolic volume of the …

Proves a conjecture about the maximum tet-volume of triangulations of a 2-sphere.

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