arXiv research

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

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

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

Study classifies metrics on a twice-punctured sphere, proving Delaunay metrics are complete.

Connected graph for twice-punctured torus curves.

New algebra for twice-punctured torus curves.

Hyperbolic links in handlebodies can be composed, unlike in 3-sphere.

New proof of SL(n) skein algebra for twice punctured sphere, showing it's a polynomial algebra.

We describe a scheme for constructing generating sets for Kronheimer and Mrowka's singular instanton knot homology for the case of knots in lens spaces. The scheme involves Heegaard-splitting a lens space containing a knot into two solid tori. One solid torus contains a portion of the knot consisting of an unknotted ar…

In this note we construct a family of immersions with constant mean curvature of the twice-punctured Riemann sphere into R^3 from the Bessel equation.

We develop an algebraic representation for (1,1)-knots using the mapping class group of the twice punctured torus MCG(T,2). We prove that every (1,1)-knot in a lens space L(p,q) can be represented by the composition of an element of a certain rank two free subgroup of MCG(T,2) with a standard element only depending on …

We define the extremal length of elements of the fundamental group of the twice punctured complex plane and give upper and lower bounds for this invariant. The bounds differ by a multiplicative constant. The main motivation comes from $3$-braid invariants and their application.

A new method converts knot Floer homology to immersed curves.

We give a new proof that the completion of the Weil-Petersson metric on Teichmüller space is Gromov-hyperbolic if the surface is a five-times punctured sphere or a twice-punctured torus. Our methods make use of the synthetic geometry of the Weil-Petersson metric.

Augmented alternating links are links obtained by adding trivial components that bound twice-punctured disks to non-split reduced non-2-braid prime alternating projections. These links are known to be hyperbolic. Here, we extend to show that generalized augmented alternating links, which allow for new trivial component…

New homomorphism proven using immersed curves on disks.

The Maskit embedding M of a surface Σis the space of geometrically finite groups on the boundary of quasifuchsian space for which the `top' end is homeomorphic to Σ, while the `bottom' end consists of two triply punctured spheres, the remains of Σwhen two fixed disjoint curves have been pinched. As such representations…

We present two different representations of (1,1)-knots and study some connections between them. The first representation is algebraic: every (1,1)-knot is represented by an element of the pure mapping class group of the twice punctured torus. The second representation is parametric: every (1,1)-knot can be represented…

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

In this paper we compute the automorphism groups $\operatorname{Aut}(\mathbf{P}_n(Σ))$ and $\operatorname{Aut}(\mathbf{B}_n(Σ))$ of braid groups $\mathbf{P}_n(Σ)$ and $\mathbf{B}_n(Σ)$ on every orientable surface $Σ$, which are isomorphic to group extensions of the extended mapping class group $\mathcal{M}^*_n(Σ)$ by t…

We prove that curve complexes of surfaces are finitely rigid: for every orientable surface S of finite topological type, we identify a finite subcomplex X of the curve complex C(S) such that every locally injective simplicial map from X into C(S) is the restriction of an element of Aut(C(S)), unique up to the (finite) …

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

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 show that every strongly-cyclic branched covering of a (1,1)-knot is a Dunwoody manifold. This result, together with the converse statement previously obtained by Grasselli and Mulazzani, proves that the class of Dunwoody manifolds coincides with the class of strongly-cyclic branched coverings of (1,1)-knots. As a c…

Let Mod(S) be the extended mapping class group of a surface S. For S the twice-punctured torus, we show that there exists an isomorphism of finite index subgroups of Mod(S) which is not the restriction of an inner automorphism. For S a torus with at least three punctures, we show that every injection of a finite index …

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.

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.

Study tiling spaces over irrational tori using diffeological classification.

New minimal tori found in curved spaces.

Let F a closed connected orientable surface bounding a genus g handlebody H. In this paper we find a finite set of generators for the subgroup E(2,g) of the pure mapping class group of the twice punctured torus PMCG(2,g), consisting of the isotopy classes of homeomorphisms of F which admit an extension to H keeping a p…

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.