几何拓扑
Simulations of knotting and unknotting in polymers or other filaments rely on random processes to facilitate topological changes. Here we introduce a method of \textit{topological steering} to determine the optimal pathway by which a…
We give upper and lower bounds on the conformal dimension of the Bowditch boundary of a Coxeter group with defining graph a complete graph and edge labels at least three. The lower bounds are obtained by quasi-isometrically embedding…
Two geometric spaces are in the same topological class if they are related by certain geometric deformations. We propose machine learning methods that automate learning of topological invariance and apply it in the context of knot theory,…
Ivansic proved that there is a link $L$ of five tori in $S^4$ with hyperbolic complement. We describe $L$ explicitly with pictures, study its properties, and discover that $L$ is in many aspects similar to the Borromean rings in $S^3$. In…
Welded knotted objects are a combinatorial extension of knot theory, which can be used as a tool for studying ribbon surfaces in $4$-space. A finite type invariant theory for ribbon knotted surfaces was developped by Kanenobu, Habiro and…
The space of Hitchin representations of the fundamental group of a closed surface $S$ into $\text{SL}_n\mathbb{R}$ embeds naturally in the space of projective oriented geodesic currents on $S$. We find that currents in the boundary have…
In this short note we adapt a proof by Bucher and Neofytidis to prove that the simplicial volume of 4-manifolds admitting an open book decomposition vanishes. In particular this shows that Quinns signature invariant, which detects the…
The simultaneous crossing number is a new knot invariant which is defined for strongly invertible knots having diagrams with two orthogonal transvergent axes of strong inversions. Because the composition of the two inversions gives a cyclic…
We study the large $r$ asymptotic behaviour of the Turaev-Viro invariants of oriented Seifert fibered 3-manifolds at the root $q=e^\frac{2\pi i}{r}$. As an application, we prove the volume conjecture for large families of oriented Seifert…
We present a table of symmetric diagrams for strongly invertible knots up to 10 crossings, point out the similarity of transvergent diagrams for strongly invertible knots with symmetric union diagrams and discuss open questions.
We study the behaviour of the Kauffman bracket skein modules of 3-manifolds under gluing along surfaces. For this purpose we extend the notion of Kauffman bracket skein modules to $3$-manifolds with marking consisting of open intervals and…
We investigate connections between biquandle colorings, quiver enhancements, and several notions of the bridge numbers $b_i(K)$ for virtual links, where $i=1,2$. We show that for any positive integers $m \leq n$, there exists a virtual link…
In this paper we construct two different explicit triangulations of the family of double twist knots $K(p,q)$ using methods of triangulating Dehn fillings, with layered solid tori and their double covers. One construction yields the…
A $G-$family of quandles is an algebraic construction which was proposed by A. Ishii, M. Iwakiri, Y. Jang, K. Oshiro in 2013. The axioms of these algebraic systems were motivated by handlebody-knot theory. In the present work we investigate…
A surface in the Teichm\"uller space, where the systole function admits its maximum, is called a maximal surface. For genus two, a unique maximal surface exists, which is called the Bolza surface, whose systolic geodesics give a…
Multi-virtual knot theory was introduced in $2024$ by the first author. In this paper, we initiate the study of algebraic invariants of multi-virtual links. After determining a generating set of (oriented) multi-virtual Reidemeister moves,…
Guo and Luo introduced generalized circle patterns on surfaces and proved their rigidity. In this paper, we prove the existence of Guo-Luo's generalized circle patterns with prescribed generalized intersection angles on surfaces with cusps,…
We present numerical polyhedron data for the image of a piecewise-linear map from a zero-curvature Klein bottle into Euclidean 3-space such that every point in the domain has a neighborhood which is isometrically embedded. To the author's…
Motivated by work of Kinoshita and Teraska, Lamm introduced the notion of a symmetric union, which can be constructed from a partial knot $J$ by introducing additional crossings to a diagram of $J \# -\!J$ along its axis of symmetry. If…
The existence of quasimorphisms on groups of homeomorphisms of manifolds has been extensively studied under various regularity conditions, such as smooth, volume-preserving, and symplectic. However, in this context, nothing is known about…