几何拓扑

We introduce a new numerical invariant for special, reduced, alternating diagrams of oriented knots and links, defined in terms of the Laplacian matrix of the associated Tait graph. For a special alternating diagram, the Laplacian encodes…

The 3-dimensional clasp number $cl(K)$ of a knot $K$ is the minimum number of clasp singularities of clasp disk, a singular immersed disk bounding $K$ whose singular set consists of only clasp singularities. We give a classification of…

Starting with a pseudo-Anosov flow $\varphi$ on a closed hyperbolic $3$-manifold $M$ and an embedded surface $S \subset M$ that is (almost) transverse to $\varphi$, we relate the hyperbolic geometry of $M$ (e.g. volume, circumference, short…

In this paper, we introduce the 0-smoothing invariant $\mathcal{F}$ of virtual knotoids constructed from local modification at classical crossings, which take values in a free $\mathbb Z$-module generated by non-oriented flat virtual…

Khovanov homology and singular instanton Floer homology are both functorial with respect to link cobordisms. Although the two homology groups are related by a spectral sequence, direct correspondence between the cobordism maps has not been…

In 2019 P. Patak and M. Tancer obtained the following higher-dimensional generalization of the Heawood inequality on embeddings of graphs into surfaces. We present a short well-structured proof accessible to non-specialists in the field.…

We prove uniform linear bounds on the volume variation under drilling and filling operations on finite volume hyperbolic 3-manifolds.

We study Milnor fibers and symplectic fillings of links of sandwiched singularities, with the goal of contrasting their algebro-geometric deformation theory and symplectic topology. In the algebro-geometric setting, smoothings of sandwiched…

A square-tiled surface (STS) is a (finite, possibly branched) cover of the standard square-torus with possible branching over exactly 1 point. Alternately, STSs can be viewed as finitely many axis-parallel squares with sides glued in…

We show that any ruled surface $X$ with $\chi(X) < 0$ admits infinitely many inequivalent Lefschetz pencils of fixed genus and number of base points. Our proof proceeds by building infinitely many inequivalent Lefschetz fibrations on a…

For each link type $K$ in the 3-sphere, we show that there is a polynomial $p_K$ such that any two diagrams of $K$ with $c_1$ and $c_2$ crossings differ by at most $p_K(c_1) + p_K(c_2)$ Reidemeister moves. As a consequence, the problem of…

The fine curve complex of a surface is a simplicial complex whose vertices are essential simple closed curves and whose $k$-simplices are collections of $k+1$ disjoint curves. We prove that the fine curve complex is homotopy equivalent to…

Suppose $Y$ is a compact, connected, oriented 3-manifold possibly with boundary, such that $\pi_1(Y)$ is infinite. Let $\operatorname{Diff}_\partial(I\times Y)$ denote the group of self-diffeomorphisms of $I\times Y$ that are equal to the…

We give explicit matrix presentations of the Blanchfield pairing and certain twisted Blanchfield pairings of the $(m,n)$-torus knot $T(m,n)$. Our method uses a taut identity realizing a genus-two Heegaard splitting of the manifold…

We construct compact hyperbolic 3-manifolds with totally geodesic boundary, such that the closed 3-pseudomanifolds obtained by coning off the boundary components are negatively curved and contain locally convex subspaces whose fundamental…

If a rational homology 3-sphere $M$ bounds a rational homology 4-ball $W$, then the kernel of the inclusion-induced homomorphism $H_1(M;\mathbb{Z})\to H_1(W;\mathbb{Z})$ is a Lagrangian for the $\mathbb{Q}/\mathbb{Z}$-valued torsion linking…

In this article, we construct infinitely many (small Seifert fibred, hyperbolic and toroidal) rational homology $3$-spheres that admit co-orientable taut foliations, but none with vanishing Euler class. In the context of the $L$-space…

For a non-split multi-crossing diagram $D$ of a link $L$ we show that $\alpha(L)-2 \leq c_2(D) + \sum_{n> 2}(2n-4)c_n(D)$ holds. Here $\alpha(L)$ is the arc index and $c_n(D)$ is the number of $n$-crossings of $D$. This generalizes and…

We review the main achievements regarding the interactions between gem theory (which makes use of edge-colored graphs to represent PL-manifolds of arbitrary dimension) and both the classical representation of PL 4-manifolds via Kirby…

The Ma-Qiu index of a group is the minimum number of normal generators of the commutator subgroup. We show that the Ma-Qiu index gives a lower bound of the presentation distance of two groups, the minimum number of relator replacements to…