arXiv research

Affirmative proof that rank 3 3-manifolds have filling links.

Study filling links in 3-manifolds to understand their topological properties.

Positive braids have endless filling possibilities.

Study finds many Lagrangian fillings for certain Legendrian links.

Study Stein and Milnor fillings of links from surface singularities.

Study of Legendrian links using Floer theory and cluster varieties.

The paper proves there are many Lagrangian fillings for Legendrian links of affine type.

Computes A-polynomials of knots from Whitehead sister link fillings.

The paper connects Legendrian links to cluster theory and exact Lagrangian fillings.

The paper proves infinitely many strong symplectic fillings for cusp singularity links.

Study finds symplectic fillings' properties for specific contact covers.

New examples of Legendrian links with infinitely many fillings.

We prove that any knot or link in any 3-manifold can be nicely decomposed (splitted) by a filling Dehn sphere. This has interesting consequences in the study of branched coverings over knots and links. We give an algorithm for computing Johansson diagrams of filling Dehn surfaces out from coverings of 3-manifolds branc…

Study shows arithmetic properties of specific hyperbolic Dehn fillings.

New links in 3-manifolds have large systole.

The study finds many Lagrangian fillings for Legendrian links of specific types.

In a recent paper of Akhmedov, Etnyre, Mark and Smith, it was shown that there exist infinitely many contact Seifert fibered 3-manifolds each of which admits infinitely many exotic (homeomorphic but pairwise non-diffeomorphic) simply-connected Stein fillings. Here we extend this result to a larger set of contact Seifer…

When a Dehn filled link manifold contains a geometrically incompressible one-sided surface, it is shown there is a unique boundary incompressible position that the surface can take in the link space. The proof uses a version of the sweep-out technique from two-sided Heegaard splitting theory. When applied to one-sided …

We study symplectic deformation types of minimal symplectic fillings of links of quotient surface singularities. In particular, there are only finitely many symplectic deformation types for each quotient surface singularity.

We investigate great circle links in the three-sphere, the class of links where each component is a great circle. Using the geometry of their complements, we classify such links up to five components. For any two-bridge knot complement, there is a finite cover that is the complement of a link of great circles in $S^3$.…

For a Legendrian $(2,n)$ torus knot or link with maximal Thurston-Bennequin number, Ekholm, Honda, and Kálmán constructed $C_n$ exact Lagrangian fillings, where $C_n$ is the $n$-th Catalan number. We show that these exact Lagrangian fillings are pairwise non-isotopic through exact Lagrangian isotopy. To do that, we com…

We prove the existence of Lagrangian fillings for $D_n$-type Legendrian links.

Study uses Newton polytopes to distinguish Lagrangian fillings of Legendrian submanifolds.

We construct infinitely many Legendrian links in the standard contact $\mathbb{R}^3$ with arbitrarily many topologically distinct Lagrangian fillings. The construction is used to find links in $S^3$ that bound topologically distinct pieces of algebraic curves in $B^4 \subset \mathbb{C}^2$, is applied to find contact 3-…

Profinite rigidity of certain 3-manifolds detected through Dehn fillings.

The study computes trace fields and minimal polynomials for specific knots and links.

No characterizing slopes for multi-component links.

In this article, we construct a genus-$0$ or genus-$1$ positive allowable Lefschetz fibration on any minimal symplectic filling of the link of non-cyclic quotient surface singularities. As a byproduct, we also show that any minimal symplectic filling of the link of quotient surface singularities can be obtained from a …

It has been observed that most manifolds in the Callahan-Hildebrand-Weeks census of cusped hyperbolic $3$-manifolds are obtained by surgery on the minimally twisted 5-chain link. A full classification of the exceptional surgeries on the 5-chain link has recently been completed. In this article, we provide a complete cl…

The paper extends techniques to create non-left-orderable manifolds from links with multiple boundary components.

Algorithm finds minimal volume hyperbolic links in 3-manifolds.

New proof shows unique symplectic fillings for certain surface singularity links.

We consider in this paper the minimally twisted chain link with 5 components in the 3-sphere, and we analyze the Dehn surgeries on it, namely the Dehn fillings on its exterior M5. The 3-manifold M5 is a nicely symmetric hyperbolic one, filling which one gets a wealth of hyperbolic 3-manifolds having 4 or fewer (includi…

Study symplectic fillings of sandwiched singularities.

Let $H\subseteq S^3$ be the two-component Hopf link. After choosing a Legendrian representative of $H$ with respect to the standard tight contact structure on $S^3$ we perform contact $(-1)$-surgery on the link itself. The manifold we get is a lens space together with a tight contact strucure on it, which depends on th…

New Stein fillings found for rational surface singularities.

We prove that for any V>0, there exist a hyperbolic manifold M_V, so that Vol(M_V) < 2.03 and LinVol(M_V) > V. The proof requires study of cosmetic surgery on links (equivalently, fillings of manifolds with boundary tori). There is no bound on the number of components of the link (or boundary components). For statement…

Study on hidden symmetries in Dehn fillings of tetrahedral links.

We investigate the filling area conjecture, optimal systolic inequalities, and the related problem of the nonvanishing of certain linking numbers in 3-manifolds.

The volume conjecture is proven for twist knots after Dehn filling.

The paper classifies hyperbolic and satellite T-links formed by twisting.

We study some properties of decomposable exact Lagrangian cobordisms between Legendrian links in $\mathbb{R}^3$ with the standard contact structure. In particular, for any decomposable exact Lagrangian filling $L$ of a Legendrian link $K$, we may obtain a normal ruling of $K$ associated with $L$. We prove that the asso…

We show that if a simple 3-manifold $M$ has two Dehn fillings at distance $Δ\geq 4$, each of which contains an essential annulus, then $M$ is one of three specific 2-component link exteriors in $S^3$. One of these has such a pair of annular fillings with $Δ= 5$, and the other two have pairs with $Δ= 4$.

The paper finds canonical triangulations for specific 3-manifolds.

Worldsheet skein D-module for Hopf link conormal uniquely determines partition functions.

Suppose $M$ is a hyperbolic 3-manifold which admits two Dehn fillings $M(r_1)$ and $M(r_2)$ such that $M(r_1)$ contains an essential torus and $M(r_2)$ contains an essential annulus. It is known that $Δ= Δ(r_1, r_2) \leq 5$. We will show that if $Δ= 5$ then $M$ is the Whitehead sister link exterior, and if $Δ= 4$ then …

The study of symplectic fillings for rational cuspidal curves.

New Stein fillings found for non-weighted homogeneous singularities.