几何拓扑

We classify fillable contact structures on all negative-definite star-shaped plumbings. Along the way, we show that such Seifert fibred spaces admit a unique negative maximal twisting number, and compute it explicitly using the Alexander…

We study scale-invariant geometric quantities associated with embedded closed curves in Euclidean three-space, with an emphasis on their behavior under optimization within a fixed knot type. Given a Euclidean-invariant and scale-covariant…

It is not known whether the realisation part of the $s$-cobordism theorem holds for smooth 4-manifolds, nor whether every pair of smoothly $h$-cobordant 4-manifolds is also smoothly $s$-cobordant. We provide some new conditions under which…

We introduce the notions of $\mathbb{K}$-framings, based $\mathbb{K}$-framings and relative $\mathbb{K}$-framings of a compact connected oriented surface $\Sigma$ for any commutative ring $\mathbb{K}$ with unit, and a map which maps a based…

In a paper of Mathews, an isomorphism is constructed between two-component complex spinors and horospheres in H^3 carrying `spin decorations'. A recent arXiv preprint of Mathews and Varsha arXiv:2412.06572 extends this result to the case of…

We initiate the study of Reidemeister hardness of Legendrian unknot front projections. Using normal rulings, we obstruct several infinite families of hard unknot diagrams from being drawn with max-tb unknot fronts, along with 1.7 million of…

In this paper we show that the $\theta$ invariant generalizes the Rozansky-Overbay invariant.

We study a family of scale-invariant $p$-densities of knot types in $R^3$, defined as the ratio of length to an $L^p$-type spread of pairwise distances along a curve. The first point of the paper is that the unconstrained theory has a…

Every stable 4-sphere is identified with the double branched covering space of a trivial surface-knot space. As a result of Wall, it is known that any two orthogonal bases of every stable 4-sphere are transformed into each other by an…

Charts are oriented labeled graphs in a disk. Any simple surface braid (2-dimensional braid) can be described by using a chart. Also, a chart represents an oriented closed surface embedded in 4-space. In this paper, we investigate embedded…

This purpose of this paper is to prove the following result: let phi be a strictly convex, smooth, convex body in the Euclidean plane, if the intersection of n translates of phi has a non-empty interior, and all of the translates contribute…

Submersions with definite folds are submersions on manifolds with boundary whose restrictions to the boundary are definite fold maps. In this paper, we study the properties from the viewpoint of differential topology of manifolds with…

We develop a reinforcement learning pipeline for simplifying knot diagrams. A trained agent learns move proposals and a value heuristic for navigating Reidemeister moves. The pipeline applies to arbitrary knots and links; we test it on…

We show that every non-trivial strongly quasipositive link is smoothly concordant to infinitely many pairwise non-isotopic strongly quasipositive links. In contrast to our result, Baker conjectured that smoothly concordant strongly…

We establish a direct connection between the analytic data of weakly isolated mixed singularities and the topology of their associated links. More precisely, we prove that the existence of essential tori, topological information, in the…

We compute the $k$-colored $\mathfrak{sl}(N)$ homology of the torus knot $T(2,2m+1)$, and we show that it stabilizes as $m\to\infty$ to the integral homology of the free loop space of the complex Grassmannian $\mathrm{Gr}(k,N)$. In…

The fine curve graph of a surface is a graph whose vertices are essential simple closed curves and whose edges connect disjoint curves. Following a rich history of hyperbolicity of various graphs associated to surfaces, the fine curve graph…

We show that the Charney--Davis strict hyperbolization procedure can preserve stable tangent bundles, answering a question of Charney and Davis. The key input is the construction of many hyperbolizing pieces, obtained using separability…

Since Thurston pioneered the connection between circle packing (abbr. CP) and three-dimensional geometric topology, the characterization of CPs and hyperbolic polyhedra has become increasingly profound. Some milestones have been achieved,…

The purpose of this expository article is to give a down-to-hearth introduction to the notion of an arithmetic group and arithmetic manifold. To achieve this we have decided to bring two geometrical questions relating the growth of systole…