arXiv research

We show that the skein vector space of the 3-torus is finitely generated. We show that it is generated by 9 elements: the empty set, some simple closed curves representing the non null elements of the first homology group with coefficients in \Z_2, and a link consisting of two parallel copies of one of the previous non…

Following the Euclidean results of Varopoulos and Pankka--Rajala, we provide a necessary topological condition for a sub-Riemannian 3-manifold $M$ to admit a nonconstant quasiregular mapping from the sub-Riemannian Heisenberg group $\mathbb{H}$. As an application, we show that a link complement $S^3\backslash L$ has a …

We provide an algorithm to determine the Heegaard genus of simple 3-manifolds with non-empty boundary. More generally, we supply an algorithm to determine (up to ambient isotopy) all the Heegaard splittings of any given genus for the manifold. As a consequence, the tunnel number of a hyperbolic link is algorithmically …

Researchers computed Kauffman bracket skein modules of specific Seifert manifolds.

Khovanov introduced a cohomology theory for oriented classical links whose graded Euler characteristic is the Jones polynomial. Since Khovanov's theory is functorial for link cobordisms between classical links, we obtain an invariant of a surface-knot, called the {\it Khovanov-Jacobsson number}, by considering the surf…

New examples of hyperbolic links with generalized torsion elements found.

We give the rectangle condition for strong irreducibility of Heegaard splittings of $3$-manifolds with non-empty boundary. We apply this to a generalized Heegaard splitting of a $2$-fold covering of $S^3$ branched along a link. The condition implies that any thin meridional level surface in the link complement is incom…

Study compact PL 4-manifolds with special handle decompositions.

Paper links set derivatives to its orthogonal projections.

The n-th hull of a union of curves in R^3 is the set of points with the property: Any plane passing through the point intersects the curves at least 2n times. The hull number u(L) of a link L is defined as the minimum number of non-empty hulls a representative of L can have. We show that the hull numbers of torus links…

Take transverse immersions f from a disjoint unin of the three 4-spheres $S^4_1$, $S^4_2$, and $S^4_3$ into $S^6$ with the following properties: (1) The restriction of $f$ to $S^4_i$ is an embedding, (2) The intersection of $f(S^4_i)$ and $f(S^4_j)$ is not empty and connected, (3)The intersection among $f(S^4_1)$, $f(S…

We prove the Turaev-Viro invariants volume conjecture for a "universal" class of cusped hyperbolic 3-manifolds that produces all 3-manifolds with empty or toroidal boundary by Dehn filling. This leads to two-sided bounds on the volume of any hyperbolic 3-manifold with empty or toroidal boundary in terms of the growth r…

Invariants from surface Khovanov-Jacobsson classes help detect knots and slices.

We generalise work of Young-Eun Choi to the setting of ideal triangulations with vertex links of arbitrary genus, showing that the set of all (possibly incomplete) hyperbolic cone-manifold structures realised by positively oriented hyperbolic ideal tetrahedra on a given topological ideal triangulation and with prescrib…

In this paper we give explicit, handle-by-handle constructions of concave symplectic fillings of all closed, oriented contact 3-manifolds. These constructions combine recent results of Giroux relating contact structures and open book decompositions of 3-manifolds, earlier results of the author on attaching 4-dimensiona…

The paper defines and analyzes a volume invariant for 3-manifolds.

This is a review article on the Bennequin-Birman-Menasco machinery for studying embedded incompressible surfaces in 3-space via their `braid foliations'. Two cases are investigated: case (1) The surface has non-empty boundary; the boundary is a knot or link which is represented as a closed braid, Case (2) The surface i…

Study infinitesimal characters on semi-groups to prove interior properties and apply to Teichmüller spaces.

Turaev's shadow formula calculates the SU(2)-Reshetikhin-Turaev-Witten invariants using shadows, and its expression is somehow similar to a Euler characteristic. We give a short proof of this formula using skein theory. The formula applies to pairs (M,G) where M is a closed oriented 3-manifold and GcM is a (possibly em…

We describe a natural decomposition of a normal complex surface singularity $(X,0)$ into its "thick" and "thin" parts. The former is essentially metrically conical, while the latter shrinks rapidly in thickness as it approaches the origin. The thin part is empty if and only if the singularity is metrically conical; the…

Researchers compute the skein module of a solid torus using braids.

New model uses Half-Full/Half-Empty approach for better portfolio selection.

New invariant for 4-manifolds with framed links, stronger than existing invariants.

Study uses twisted Alexander polynomials to link fibered classes in 3-manifolds.

Invariants for 3-manifolds with embedded links using Hopf algebras.

New concept of k-holes in simple drawings and convex drawings.

A representation for compact 3-manifolds with non-empty non-spherical boundary via 4-colored graphs (i.e., 4-regular graphs endowed with a proper edge-coloration with four colors) has been recently introduced by two of the authors, and an initial classification of such manifolds has been obtained up to 8 vertices of th…

The study of Seifert linking forms for punctured n-manifolds in (2n-1)-space.

Let k be a subring of the field of rational functions in α, s which contains α^{1}, α^{-1}, s^{1}, s^{-1}, . Let M be a compact oriented 3-manifold, and let K(M) denote the Kauffman skein module of M over k. Then K(M) is the free k-module generated by isotopy classes of framed links in M modulo the Kauffman skein relat…

Study classifies Lie group representations with non-empty boundary orbit space.

We study the Sasaki cone of a CR structure of Sasaki type on a given closed manifold. We introduce an energy functional over the cone, and use its critical points to single out the strongly extremal Reeb vectors fields. Should one such vector field be a member of the extremal set, the scalar curvature of a Sasaki extre…

Let M be a compact connected orientable 3-manifold, with non-empty boundary that contains no 2-spheres. We investigate the existence of two properly embedded disjoint surfaces S_1 and S_2 such that M - (S_1 \cup S_2) is connected. We show that there exist two such surfaces if and only if M is neither a Z_2 homology sol…

We construct {\it quantum hyperbolic invariants} (QHI) for triples $(W,L,ρ)$, where $W$ is a compact closed oriented 3-manifold, $ρ$ is a flat principal bundle over $W$ with structural group $PSL(2,\mc)$, and $L$ is a non-empty link in $W$. These invariants are based on the Faddeev-Kashaev's {\it quantum dilogarithms},…

New spectral invariants recover Calabi invariant for surface dynamics.

The paper connects 3D manifold invariants to hyperbolic cone metrics and discrete Fourier transforms.

We provide combinatorial realizations, according to the usual objects/moves scheme, of the following three topological categories: (1) pairs (M,v) where M is a 3-manifold (up to diffeomorphism) and v is a (non-singular vector) field, up to homotopy; here possibly the boundary of M is non-empty and v may be tangent to t…

Study axisymmetric $σ_k$-Nirenberg problem on spheres.

In this paper we introduce a new invariant of virtual knots and links that is non-trivial for infinitely many virtuals, but is trivial on classical knots and links. The invariant is initially be expressed in terms of a relative of the bracket polynomial and then extracted from this polynomial in terms of its exponents,…

Proves finiteness of Kauffman bracket skein modules for 3-manifolds.

Study empty polar varieties' impact on singular function-germs.

In the recent paper \cite{DGNP} we have proved that the only stable $C^2$ minimal surfaces in the first Heisenberg group $\Hn$ which are graphs over some plane and have empty characteristic locus must be vertical planes. This result represents a sub-Riemannian version of the celebrated theorem of Bernstein. In this pap…

Turaev's shadow can be seen locally as the Stein factorization of a stable map. In this paper, we define the notion of stable map complexity for a compact orientable 3-manifold bounded by (possibly empty) tori counting, with some weights, the minimal number of singular fibers of codimension 2 of stable maps into the re…

Let n>2 and let M be an orientable complete finite volume hyperbolic n-manifold with (possibly empty) geodesic boundary having Riemannian volume vol(M) and simplicial volume ||M||. A celebrated result by Gromov and Thurston states that if M has empty boundary then the ratio between vol(M) and ||M|| is equal to v_n, whe…

Consider a quantum particle trapped between a curved layer of constant width built over a complete, non-compact, $\mathcal C^2$ smooth surface embedded in $\mathbb{R}^3$. We assume that the surface is asymptotically flat in the sense that the second fundamental form vanishes at infinity, and that the surface is not tot…

Given a rational homology 3-sphere M with the first integral homology of rank b and a link L inside M, colored by odd numbers, we construct a unified invariant I_{M,L} belonging to a modification of the Habiro ring where b is inverted. Our unified invariant dominates the whole set of the SO(3) Witten-Reshetikhin-Turaev…

The book defines an invariant for framed links and 3-manifolds using graph configurations.

The paper explores existence and non-existence of constant scalar curvature and extremal Sasaki metrics.

In homogenous space Sol we study compact surfaces with constant mean curvature and with non-empty boundary. We ask how the geometry of the boundary curve imposes restrictions over all possible configurations that the surface can adopt. We obtain a flux formula and we establish results that assert that, under some restr…