几何拓扑
The limiting character, introduced by Tillmann, has been studied recently in the context of Culler-Shalen theory. We extend the methods of the author's previous work to show that certain families of essential twice-punctured tori are…
We improve the bound on the number of tetrahedra in the veering triangulation of a fully-punctured pseudo-Anosov mapping torus in terms of the normalized dilatation. When the mapping torus has only one boundary component, we can improve the…
We describe the geometry of the character variety of representations of the knot group $\Gamma_{m,n}=\langle x,y| x^n=y^m\rangle$ into the group $\mathrm{SU}(3)$, by stratifying the character variety into strata correspoding to totally…
We provide bi-Lipschitz invariants for finitely determined map germs $f: (\mathbb{K}^n,0) \to (\mathbb{K}^p, 0)$, where $\mathbb{K} = \mathbb{R}$ or $ \mathbb{C}$. The aim of the paper is to provide partial answers to the following…
We provide a bound on the maximum degree of the Jones polynomial of any positive link with second Jones coefficient equal to $\pm 1$. This builds on the result of our previous work, in which we found such a bound for positive fibered links.
This paper contains linear systems of equations which can distinguish knots without knot invariants. Let $M_n$ be the topological moduli space of all n-component string links and such that a fixed projection into the plane is an immersion.…
Given a non-compact semisimple real Lie group $G$ and an Anosov subgroup $\Gamma$, we utilize the correspondence between $\mathbb R$-valued additive characters on Levi subgroups $L$ of $G$ and $\mathbb R$-affine homogeneous line bundles…
The character variety $\Xi$ of a finitely generated group $\Gamma$ in $\mathrm{PSL}_2(\mathbb{R})$ has many compactifications. We construct a continuous surjection from the real spectrum compactification $\Xi^{\mathrm{RSp}}$ to the oriented…
We complete the work of Lanneau and M\"oller to show that there are no primitive Teichm\"uller curves in Prym(2,2).
A point of a Veech surface is periodic if it has a finite orbit under the surface's affine automorphism group. We show that the periodic points of Prym eigenforms in the minimal strata of translation surfaces in genera 2, 3 and 4 are the…
The nonorientable four-ball genus of a knot $K$ in $S^3$ is the minimal first Betti number of nonorientable surfaces in $B^4$ bounded by $K$. By amalgamating ideas from involutive knot Floer homology and unoriented knot Floer homology, we…
We describe a method by which the number of intersections one ellipse makes inside the plane of another can be determined. The method is based on applying a transformation that reverts one ellipse to the unit circle, and examining the…
We expand a result of Birman and Series (1985) by proving that the set of geodesics whose self-intersection angles are bounded from below has Hausdorff dimension zero. In addition, we show that the set of geodesics that do not bound a…
Brain network topology, derived from functional magnetic resonance imaging (fMRI), holds promise for improving Alzheimer's disease (AD) diagnosis. Current methods primarily focus on lower-order topological features, often overlooking the…
Given an elliptic fibration $\pi : M \to S^2$ with singular locus $\Delta \subseteq S^2$, let $\operatorname{Br}(\pi) < \operatorname{Mod}(S^2,\Delta)$ be the subgroup of the spherical braid group consisting of those braids that lift to a…
Following a recent work of Kastenholz, we show existence of infinitely many parallelizable closed oriented 4-manifolds, none of which admit an open book decomposition. This implies there is no analogue of the Giroux correspondence for Engel…
We show that if $Y$ is a toroidal closed graph manifold rational homology $3$-sphere with $|H_1(Y;\mathbb{Z})| \le 5$, then there exists an irreducible representation $\fund{Y} \to SU(2)$, using topological methods and avoiding the use of…
Gilmer and Masbaum use Witten-Reshetikhin-Turaev (WRT) invariants to define a map from the Kauffman bracket skein module to a set of complex-valued functions defined on roots of unity in order to provide a lower bound for its dimension. We…
Every geodesic current on a hyperbolic surface has an associated dual space. If the current is a lamination, this dual embeds isometrically into a real tree. We show that, in general, the dual space is a Gromov hyperbolic metric tree-graded…
We prove that if a conjugation quandle is Hopfian, then its underlying group is also Hopfian. We also show that the converse does not hold by providing an example. This highlights a distinction between conjugation quandles and their…