arXiv research

Essential tori in certain 3-manifolds are missed by ideal points in character varieties.

The study limits the number of 2-holed tori in knot exteriors.

New method detects essential tori in mixed singularity links.

Study shows how certain knots and tori are detected by ideal points in character varieties.

We give an algorithm to find vertical essential tori in small Seifert fiber spaces with infinite fundamental groups. This implies that there are algorithms to decide whether a 3-manifold is a Seifert fiber space.

A formula is given for the Seiberg-Witten invariants of a 4-manifold that is cut along certain kinds of 3-dimensional tori. The formula involves a Seiberg-Witten invariant for each of the resulting pieces.

Study essentiality and simplicial volume of manifolds fibered over spheres.

Study on cylindrical handlebody-knots with symmetry and rigidity properties.

Essential dimension of a family of complex manifolds is the dimension of the image of its base in the Kuranishi space of the fiber. We prove that any family of hyperkähler manifolds over a compact simply connected base has essential dimension not greater than $1$. A similar result about families of complex tori is also…

Study shows constraints on slopes for knot manifolds with specific tori.

New families of twisted torus knots found with essential surfaces.

The fundamental group of $M = \sharp_n (S^2\times S^1)$ is $F_n$, the free group with $n$ generators. There is a 1-1 correspondence between the equivalence classes of $\mathbb{Z}$-- splittings of $F_n$ and homotopy classes of embedded essential tori in $M$. We define and prove a local notion of minimal intersection of …

We study the symplectic topology of some finite algebraic quotients of the An Milnor fibre which are diffeomorphic to the rational homology balls that appear in Fintushel and Stern's rational blowdown construction. We prove that these affine surfaces have no closed exact Lagrangian submanifolds by using the already ava…

In this paper, we will compute the dimension of the space of spun and ordinary normal surfaces in an ideal triangulation of the interior of a compact 3-manifold with incompressible tori or Klein bottle components. Spun normal surfaces have been described in unpublished work of Thurston. We also define a boundary map fr…

Let $M$ be a simple 3-manifold with a toral boundary component. It is known that if two Dehn fillings on $M$ along the boundary produce a reducible manifold and a toroidal manifold, then the distance between the filling slopes is at most three. This paper gives a remarkably short proof of this result.

Haken showed that the Heegaard splittings of reducible 3-manifolds are reducible, that is, a reducing 2-sphere can be found which intersects the Heegaard surface in a single simple closed curve. When the genus of the "interesting" surface increases from zero, more complicated phenomena occur. Kobayashi showed that if a…

The paper studies properties of stated SL(n)-skein algebras and their centers.

The paper generalizes index theory for periodic manifolds and proves equivalence of signatures for tori.

In a recent work "Arc-presentation of links: Monotonic simplification" Ivan Dynnikov showed that each rectangular diagram of the unknot, composite link, or split link can be monotonically simplified into a trivial, composite, or split diagram, respectively. The following natural question arises: Is it always possible t…

Let M be a compact, connected, orientable, irreducible 3-manifold and T' an incompressible torus boundary component of M such that the pair (M,T') is not cabled. By a result of C. Gordon, if S and T are incompressible punctured tori in M with boundary on T' and boundary slopes at distance d, then d is at most 8, and th…

This paper explores twisted Lagrangian tori in C^2 and their Hamiltonian stationarity.

Smooth 2-tori in R^4 can be approximated by polyhedral Lagrangian or isotropic tori.

In this paper we study exceptional Dehn fillings on hyperbolic knot manifolds which contain an essential once-punctured torus. Let $M$ be such a knot manifold and let $β$ be the boundary slope of such an essential once-punctured torus. We prove that if Dehn filling $M$ with slope $α$ produces a Seifert fibred manifold,…

The paper classifies minimal immersions of flat 3- and 4-tori in spheres by their first eigenfunctions.

Characterizes conformal classes of tori using differential geometry.

Study finds non-isotopic transverse tori in Engel manifolds.

We study square-tiled tori, that is, tori obtained from a finite collection of unit squares by parallel side identifications. Square-tiled tori can be parametrized in a natural way that allows to count the number of square-tiled tori tiled by a given number of square tiles. There is a natural $\mathrm{SL}(2,\mathbf{Z})…

New findings on isospectral tori and harmonic maps between flat tori.

Constructs flows of tori in sphere perturbations for Morse homology.

We prove that the conformal immersions of complex two tori into $S^3$ which locally minimize their conformal volume in their conformal class all satisfy some elliptic PDE. We prove that they are either minimal tori, CMC flat tori, elliptic conformally constrained minimal tori or critical point of the area under some fi…

Ellipsoids host infinitely many minimal tori, bifurcating from a 2-torus orbit.

Study of critical tori for mean curvature energies in Killing submersions.

We consider proper-biharmonic flat tori with constant mean curvature (CMC) in spheres and find necessary and sufficient conditions for certain rectangular tori and square tori to admit full CMC proper-biharmonic immersions in $\mathbb{S}^n$, as well as the explicit expressions of some of these immersions.

The Clifford torus is a torus in a three-dimensional sphere. Homogeneous tori are simple generalization of the Clifford torus which still in a three-dimensional sphere. There is a way to construct tori in a three-dimensional sphere using the Hopf fibration. In this paper, all Hamiltonian stationary Lagrangian tori whic…

Isothermic tori with one planar curvature line found and characterized.

Study tiling spaces over irrational tori using diffeological classification.

For all positive integers n we construct a 1-parameter family of conformal tori of revolution in the 3-sphere with n bulges. These tori arise by Darboux transformations of constant mean curvature tori but do not have constant mean curvature in the 3-sphere.

New minimal tori found in curved spaces.

Given a smooth positive function $f$ defined on the unit circle satisfying a simple condition, we obtain a Poincaré-type inequality for an arbitrary function $u$ whose weighted average with respect to $f$ is zero. The proof uses Fenchel's theorem about the total curvature of closed space curves in an essential way. Nex…

Classifies mapping tori of specific groups, generalizing known results.

Otsuki tori form a countable family of immersed minimal two-dimensional tori in the unitary three-dimensional sphere. According to El Soufi-Ilias theorem, the metrics on the Otsuki tori are extremal for some unknown eigenvalues of the Laplace-Beltrami operator. Despite the fact that the Otsuki tori are defined in quite…

Paper explains dynamics of homeomorphisms to mapping tori geometry.

Smooth tori in S^4 are topologically unknotted.

In \cite{BSV}, Borisov, Salamon and Viaclovsky constructed non-standard orthogonal complex structures on flat tori $T^{2n}_{\mathbb R}$ for any $n\geq 3$. We will call these examples BSV-tori. In this note, we show that on a flat $6$-torus, all the orthogonal complex structures are either the complex tori or the BSV-to…

Counts minimal tori in Riemannian manifolds with 6 or more dimensions.

The paper finds non-contractible loops of Legendrian tori from knot families.

Engel manifolds show transverse tori can be made to have various formal invariants.

New non-Kähler examples of generalized Kähler manifolds constructed via mapping tori.