arXiv research

We show that any non-minimal bridge decomposition of a torus knot is stabilized and that $n$-bridge decompositions of a torus knot are unique for any integer $n$. This implies that a knot in a bridge position is a torus knot if and only if there exists a torus containing the knot such that it intersects the bridge sphe…

Characterizes Legendrian knots in lens spaces.

In the present paper, we will show that for any integer n>0 there are infinitely many twisted torus knots with n-string essential tangle decompositions.

Study shows knots surgered elliptic surfaces admit handle decompositions without 1- and 3-handles.

Classifies symplectic fillings of specific torus bundles.

Torus decomposition shows foliation detected slopes for glued knot manifolds.

Hyperbolic links in thickened torus decompose into angled tetrahedra.

In this paper, we determine the canonical polyhedral decomposition of every hyperbolic once-punctured torus bundle over the circle. In fact, we show that the only ideal polyhedral decomposition that is straight in the hyperbolic structure and that is invariant under a certain involution is the ideal triangulation defin…

Study Dolbeault cohomology on complex manifolds with torus action.

In this paper we show that the convergence of complete Kahler-Einstein hypersurfaces in complex torus in the sense of Cheeger-Gromov will canonically degenerate the underlying manifolds into "pair of pants" decomposition. We also construct minimal Lagrangian tori that represent the vanishing cycles of the degeneration.

Characterizes separable subgroups in 3-manifold groups.

Study contact geometry of symplectic divisors, invariant under specific transformations.

Optimal Euclidean structure minimizes energy in weighted toroidal graphs.

Abstract: Proves relative versions of group splitting results.

We consider knots whose diagrams have a high amount of twisting of multiple strands. By encircling twists on multiple strands with unknotted curves, we obtain a link called a generalized augmented link. Dehn filling this link gives the original knot. We classify those generalized augmented links that are Seifert fibere…

We show that associating the Euclidean cell decomposition due to Cooper and Long to each point of the moduli space of framed strictly convex real projective structures of finite volume on the once-punctured torus gives this moduli space a natural cell decomposition. The proof makes use of coordinates due to Fock and Go…

We study a certain class of embedded two-foams that arise from gluing discs into ribbon torus knots along nonintersecting torus meridians. We exhibit several equivalent diagrammatic formalisms for these objects and identify several of their invariants, including a unique prime decomposition.

We describe a new approach to the canonical decompositions of 3-manifolds along tori and annuli due to Jaco-Shalen and Johannson (with ideas from Waldhausen) - the so-called JSJ-decomposition theorem. This approach gives an accessible proof of the decomposition theorem; in particular it does not use the annulus-torus t…

Let $M$ be a closed, oriented and smooth manifold of dimension $d$. Let $ŁM$ be the space of smooth loops in $M$. Chas and Sullivan introduced loop product, a product of degree $-d$ on the homology of $LM$. In this paper we show how for three manifolds the ``nontriviality'' of the loop product relates to the ``hyperbol…

This paper provides a new method to construct $b$-symplectic toric manifolds from toric manifolds.

New 3D shapes can't be split into torus pieces.

We determine the set of all genus g bridge numbers of many iterated torus knots, listing these numbers in a sequence called the bridge spectrum. In addition, we prove a structural lemma about the decomposition of a strongly irreducible bridge surface induced by cutting along a collection of essential surfaces.

Discretizes Hodge-Dirac operators on a torus.

We show that the interior of the convex core of a quasifuchsian punctured-torus group admits an ideal decomposition (usually an infinite triangulation) which is canonical in two different senses: in a combinatorial sense via the pleating invariants, and in a geometric sense via an Epstein-Penner convex hull constructio…

L. Paoluzzi constructed a family of compact orientable three-dimensional hyperbolic manifolds with totally geodesic boundary, which were, by construction, closely related to the three-dimensional torus. This paper gives their complete classification up to isometry, and also their isometry groups. The key tool is the so…

Study shows $(p,q)$-cable knots cannot undergo certain types of surgery.

New CR structures found for once-punctured torus bundles.

Researchers calculate dimensions of skein modules for 2-torus mapping tori.

Study satellite knots and their quandles related to incompressible tori.

Study fully augmented links in thickened torus, generalizing $S^3$ results.

Study of knot polynomials for twist satellites, generalizing cabling.

Proves LeBrun-Salamon Conjecture for low-dimensional contact manifolds.

The study examines Morse diagrams and their behavior under Murasugi sums, leading to contact structure classifications.

We propose a method to compute complex volume of 2-bridge link complements. Our construction sheds light on a relationship between cluster variables with coefficients and canonical decompositions of link complements.

Alternative proof of Dynnikov's three-page index for torus links.

There is no 5,7-triangulation of the torus, that is, no triangulation with exactly two exceptional vertices, of degree 5 and 7. Similarly, there is no 3,5-quadrangulation. The vertices of a 2,4-hexangulation of the torus cannot be bicolored. Similar statements hold for 4,8-triangulations and 2,6-quadrangulations. We pr…

Let $M$ be a 3-manifold with torus boundary components $T_1$ and $T_2$. Let $φ\colon T_1 \to T_2$ be a homeomorphism, $M_φ$ the manifold obtained from $M$ by gluing $T_1$ to $T_2$ via the map $φ$, and $T$ the image of $T_1$ in $M_φ$. We show that if $φ$ is "sufficiently complicated" then any incompressible or strongly …

Study of panhandle polynomials of torus links with geometric applications.

We present a complete classification of elements in the mapping class group of the torus which have a representative that can be written as a product of two orientation reversing involutions. Our interest in such decompositions is motivated by features of the monodromy maps of real fibrations. We employ the property th…

Compact pseudo-Hermitian spaces have rigid holomorphic isometries.

Let X be a subcomplex of the standard CW-decomposition of the n-dimensional torus. We exhibit an explicit optimal motion planning algorithm for X. This construction is used to calculate the topological complexity of complements of general position arrangements and Eilenberg-Mac Lane spaces associated to right-angled Ar…

Decomposes singular Kähler spaces with trivial first Chern class into simpler components.

Develops a diagrammatic method for symplectic filling classifications.

Cannon, Floyd, and Parry have studied subdivisions of the 2-sphere extensively, especially those corresponding to 3-manifolds, in an attempt to prove Cannon's conjecture. There has been a recent interest in generalizing some of their tools, such as extremal length, to higher dimensions. We define finite subdivision rul…

The paper explores triangulations of spheres and projective spaces, focusing on Hopf triangulations and equilibrium structures.

We consider the representation space of a compact surface, that is the space of morphisms from the fundamental group to SU(2) up to conjugation. We show that the trace functions associated to multicurves on the surface are linearly independent as functions on the representation space. The proof relies on the Fourier de…

To each once-punctured-torus bundle, $T_φ$, over the circle with pseudo-Anosov monodromy $φ$, there are associated two tessellations of the complex plane: one, $Δ(φ)$, is (the projection from $\infty$ of) the triangulation of a horosphere at $\infty$ induced by the canonical decomposition into ideal tetrahedra, and the…

Study connects plumbing baskets to 3-sphere contact structures.