几何拓扑

For a negative definite plumbed three-manifold, we give an integral representation of the appropriate average of the GPPV invariants of Gukov--Pei--Putrov--Vafa, which implies that this average admits a resurgent asymptotic expansion, the…

We are presenting proofs of fundamental results related to homotopy idempotents, proofs that are sufficiently simple so that even the author can understand them. The first one is that homotopy idempotents in the category of pointed…

Let $S$ be a boundaryless infinite-type surface with finitely many ends and consider an end-periodic homeomorphism $f$ of S. The end-periodicity of $f$ ensures that $M_f$, its associated mapping torus, has a compactification as a…

We consider 3-manifolds given as Heegaard splittings $M=H^-\cup_\Sigma H^+$ with the aim to describe the hyperbolic metric of $M$ under topological conditions on the splitting guaranteeing that the manifold is hyperbolic. In particular,…

For a hyperbolic fully augmented link in $S^3$, its \emph{FAL volume density} is the ratio of its volume to the number of augmentations. We show that the set of FAL volume densities is dense in $[2\voct, 10\vtet)$, but discrete in…

The space of monic squarefree polynomials has a stratification according to the multiplicities of the critical points, called the equicritical stratification. Tracking the positions of roots and critical points, there is a map from the…

We prove that some relative character varieties of the fundamental group of a punctured sphere into the Hermitian Lie groups $\mathrm{SU}(p,q)$ admit compact connected components. The representations in these components have several…

We define and study the contact cut graph which is an analogue of Hatcher and Thurston's cut graph for contact geometry, inspired by contact Heegaard splittings. We show how oriented paths in the contact cut graph correspond to Lefschetz…

We give an alternate proof of the left-orderability of the mapping class group of a connected oriented infinite-type surface with a non-empty boundary. Our main strategy involves the inductive construction of a countable stable Alexander…

We introduce and study the notion of filling links in 3-manifolds: a link L is filling in M if for any 1-spine G of M which is disjoint from L, $\pi_1(G)$ injects into $\pi_1(M\smallsetminus L)$. A weaker "k-filling" version concerns…

Riemann surfaces are among the simplest and most basic geometric objects. They appear as key players in many branches of physics, mathematics, and other sciences. Despite their widespread significance, how to compute distances between pairs…

In 2014 Andrey Perfiliev introduced the so-called electric invariant for non-oriented knots. This invariant was motivated by using Kirchhoff's laws for the dual graph of the knot diagram. Later, in 2020, Anastasiya Galkina generalised this…

The $k$th module of a surface-knot of a genus $g$ in the 4-sphere is the $k$th integral homology module of the infinite cyclic covering of the surface-knot complement. The reduced first module is the quotient module of the first module by…

We obtain a localized version of the configuration space integral for the Casson knot invariant, where the standard symmetric Gauss form is replaced with a locally supported form. An interesting technical difference between the arguments…

We prove that certain specific sum of enhanced states produce torsion of order two in the Khovanov homology.

Kleiner-Leeb and Quint showed that convex subsets in higher-rank symmetric spaces are very rigid compared to rank 1 symmetric spaces. Motivated by this, we consider convex subsets in products of proper CAT(0) spaces $X_1\times X_2$ and show…

We consider the curve graph in the cases where it is not a Farey graph, and show that its Gromov boundary is linearly connected. For a fixed center point c and radius r, we define the sphere of radius r to be the induced subgraph on the set…

Here we discuss $r-$shake slice knots, and their relation to corks, we then prove that $0$-shake slice knots are slice.

We give a nice description for a Zariski open subset of the ${\rm SL}(3,\mathbb{C})$-character variety of the Whitehead link.

This note is a survey on the topology of hyperplane arrangements. We mainly focus on the relationship between topology and the real structure, such as adjacent relations of chambers and stratifications related to real structures.