arXiv research

Study nearly parallel G2-structures with torus symmetry using multi-moment maps.

Classifies Legendrian and transverse torus knots in $S^3$.

New Garside structures found for torus knot groups and related braid groups.

Study rigidifies torus bundles under first Betti number constraints.

The present paper is an addendum to "Spherical structures on torus knots and links", arXiv:1008.0312, and concerns more general case of torus knot and link cone-manifolds.

We compute the Bott-Morse Floer cohomology of the Clifford torus in $\CP^n$ with all possible spin-structures. Each spin structure is known to determine an orientation of the moduli space of holomorphic discs, and we analyze the change of orientation according to the change of spin structure of the Clifford torus. Also…

The paper classifies all tight contact structures on a solid torus.

We study weak versus strong symplectic fillability of some tight contact structures on torus bundles over the circle. In particular, we prove that almost all of these tight contact structures are weakly, but not strongly symplectically fillable. For the 3-torus this theorem was established by Eliashberg.

Study theta-curves on torus in 3-sphere, classifying them.

Researchers determine quantum filtration structure of torus links.

Efficient algorithms for WRT invariants of torus bundles using algebraic structures.

Exotic hypercomplex structures on a torus are proven to not exist.

In this paper, we study strong symplectic fillability and Stein fillability of some tight contact structures on negative parabolic and negative hyperbolic torus bundles over the circle. For the universally tight contact structure with twisting $π$ in $S^1$-direction on a negative parabolic torus bundle, we completely d…

SU(n)-structures derived from Kähler manifolds with torus actions.

Study the geometry of torus link character varieties, finding unexpected relations.

Classifies knots in a special 3D space.

We prove the existence of essential loops in the space of contact structures on torus bundles over the circle.

Study of rod packings in 3-torus using 3-manifold geometry.

Global invariant for path structures and differential equations defined on torus.

We use Menke's JSJ-type decomposition theorem for symplectic fillings to reduce the classification of strong and exact symplectic fillings of virtually overtwisted torus bundles to the same problem for tight lens spaces. For virtually overtwisted structures on elliptic or parabolic torus bundles, this gives a complete …

Study describes Morse flows on a torus with up to six singular points.

We classify positive transversal torus knots in tight contact structures up to transversal isotopy.

In this note, we first classify all topological torus knots lying on the Heegaard torus in lens spaces, and then we study Legendrian representatives of these knots. We classify oriented positive Legendrian torus knots in the universally tight contact structures on the lens spaces up to contactomorphism.

The group action which defines the moduli problem for the deformation space of flat affine structures on the two-torus is the action of the affine group $\Aff(2)$ on $\bbR^2$. Since this action has non-compact stabiliser $\GL(2,\bbR)$, the underlying locally homogeneous geometry is highly non-Riemannian. In this articl…

Each closed oriented 3-manifold $M$ is naturally associated with a set of integers $D(M)$, the degrees of all self-maps on $M$. $D(M)$ is determined for each torus bundle and torus semi-bundle $M$. The structure of torus semi-bundle is studied in detail. The paper is a part of a project to determine $D(M)$ for all 3-ma…

Study reveals $\mathfrak{sl}_2$ action on link homology of specific torus links.

Combinatorial definition of bordered Floer theory for torus-boundary manifolds.

A near-symplectic structure on a 4-manifold is a closed 2-form that is symplectic away from the 1-dimensional submanifold along which it vanishes and that satisfies a certain transversality condition along this vanishing locus. We investigate near-symplectic 4-manifolds equipped with singular Lagrangian torus fibration…

We present classification results for exceptional Legendrian realisations of torus knots. These are the first results of that kind for non-trivial topological knot types. Enumeration results of Ding-Li-Zhang concerning tight contact structures on certain Seifert fibred manifolds with boundary allow us to place upper bo…

We show that the Kauffman bracket skein module of a cylinder over the torus embeds as a subalgebra of the noncommutative torus. Using this we derive nice formulas for the Jones-Wenzl idempotents and analyze the structure of the Kauffman bracket skein module of the unknot as a module over the Kauffman bracket skein modu…

Generalized complex structures on certain torus bundles are explored.

We prove that an iterated torus knot type fails the uniform thickness property (UTP) if and only if all of its iterations are positive cablings, which is precisely when an iterated torus knot type supports the standard contact structure. We also show that all iterated torus knots that fail the UTP support cabling knot …

The paper proves a pseudo-Kähler structure on a torus's projective space.

A solvmanifold is a compact differentiable manifold M on which a connected solvable Lie group G acts transitively. As the main result, we will see, applying a result of Arapura and Nori on solvable Kaehler groups and some of the author's previous results, that a compact solvmanifold admits a Kaehler structure if and on…

In this paper we give a construction of Lagrangian torus fibration for Fermat type quintic \cy hypersurfaces via the method of gradient flow. We also compute the monodromy of the expected special Lagrangian torus fibration and discuss structures of singular fibers.

Non-exact Poisson structures found on toric varieties.

Derives adjoint polynomials of torus knots in explicit form.

Consider a parallel plane foliation on real finite-dimensional linear vector space. It induces a foliation on the torus obtained by factorization of the space by the integer lattice (let us denote the latter foliation by F). Let g be arbitrary metric on the torus. It induces a complex structure on each leaf of F such t…

We compute bordered Floer homology CFDD of (2,2n)-torus link complement, and discuss assorted examples and type-DD structure homotopy equivalence.

We classify the Legendrian torus knots in S^1\times S^2 with its standard tight contact structure up to Legendrian isotopy.

The paper studies quaternionic structures on GKM graphs and their relation to torus actions on quaternionic projective spaces.

Complete classification of rod complements in 3-torus using topology.

Study on quandle coloring quivers for (p, 2)-torus knots and links.

Study non-Kähler metrics on complex nilmanifolds, proving torus structure under certain conditions.

This paper is a continuation of our paper math.AG/0205321 where we have built a combinatorial model for the torus fibrations of Calabi-Yau toric hypersurfaces. This part addresses the connection between the model torus fibration and the complex and Kähler geometry of the hypersurfaces.

The present paper considers two infinite families of cone-manifolds endowed with spherical metric. The singular strata is either the torus knot ${\rm t}(2n+1, 2)$ or the torus link ${\rm t}(2n, 2)$. Domains of existence for a spherical metric are found in terms of cone angles and volume formulæ are presented.

The family of negative torus links $T_{p,q}$ over a fixed number of strands $p$ admits a stable limit in reduced Khovanov homology as $q$ grows to infinity. In this paper, we endow this stable space with a bi-graded commutative algebra structure. We describe these algebras explicitly for $p=2,3,4$. As an application, w…

In 2006 Masuda and Suh asked if two compact non-singular toric varieties having isomorphic cohomology rings are homeomorphic. In the first part of this paper we discuss this question for topological generalizations of toric varieties, so-called torus manifolds. For example we show that there are homotopy equivalent tor…