几何拓扑

Let $S_g$ be a closed orientable surface of genus $g\geq 2$. A collection $\Omega = \{ \gamma_1, \dots, \gamma_s\}$ of pairwise non-homotopic simple closed curves on $S_g$ such that $\gamma_i$ and $\gamma_j$ are in minimal position, is…

We use the square peg problem for smooth curves to prove a generalized table Theorem for real valued functions on Riemannian surfaces with odd Euler characteristic. We then use this result to prove the table conjecture for even functions on…

We demonstrate that the contact cosmetic surgery conjecture holds true for all non-trivial Legendrian knots, with the possible exception of Lagrangian slice knots. We also discuss the contact cosmetic surgeries on Legendrian unknots and…

We study a new bordification of the decorated Teichm\"uller space for a multiply punctured surface F by a space of filtered screens on the surface that arises from a natural elaboration of earlier work of McShane-Penner. We identify…

We show that for a rational homology 3-sphere $Y$ equipped with a self-conjugate spin$^c$-structure $\mathfrak s$, the $\operatorname{Pin}(2)$-equivariant monopole Floer homology of $(Y,\mathfrak s)$, as defined by Lin, is isomorphic to the…

We give an explicit algorithm for calculating the Kauffman bracket of a link diagram from a Goeritz matrix for that link. Further, we show how the Jones polynomial can be recovered from a Goeritz matrix when the corresponding checkerboard…

We determine which closed orientable $3$-manifolds $M$ admit a self-homeomorphism restricting to a pseudo-Anosov map on an incompressible subsurface $\Sigma$, which we call a pseudo-Anosov surface. When $M$ is irreducible, we show that the…

We show that for a winding number zero satellite operator $P$ on the knot concordance group, if the axis of $P$ has nontrivial self-pairing under the Blanchfield form of the pattern, then the image of the iteration $P^n$ generates an…

This paper concerns the relationships between continued fractions and the geometry of the Stern-Brocot diagram. Each rational number can be expressed as a continued fraction $[a_0; a_1, \ldots, a_n]$ whose terms $a_i$ are integers and are…

The paper is devoted to the well-known problem of smooth structures on moment-angle manifolds. Each real or complex moment-angle manifold has an equivariant smooth structure given by an intersection of quadrics corresponding to a geometric…

We give an explicit construction of a family of closed arithmetic hyperbolic 5-manifolds, tessellated by $117 964 800 = 512 \cdot 16 \cdot 14400$ copies of a Coxeter simplicial prism. We proceed to study various properties of these…

In this article, we propose a new approach for describing and understanding knots and links in a 3-manifold through the use of an embedded non-orientable surface. Specifically, we define a plat-like representation based on this…

In all dimensions $n \ge 5$, we prove the existence of closed orientable hyperbolic manifolds that do not admit any $\text{spin}^c$ structure, and in fact we show that there are infinitely many commensurability classes of such manifolds.…

The Kauffman bracket skein algebra of a surface is a generalization of the Jones polynomial invariant for links and plays a principal role in the Witten-Reshetikhin- Turaev topological quantum field theory. However, the multiplicative…

Cappell-Shaneson homotopy 4-spheres (CS spheres) are potential counterexamples of the smooth 4-dimensional Poincar\'e conjecture. Akbulut proved that infinite CS spheres are diffeomorphic to the standard 4-sphere by Kirby calculus. Kim and…

The fine curve graph was introduced to study homeomorphism group of surfaces. In this paper we study the topology of the Gromov boundary of this graph for closed surfaces with higher genus. We first prove a bounded geodesic image theorem…

We give an explicit bijective correspondence between between nonzero pairs of complex numbers, which we regard as spinors or spin vectors, and horospheres in 3-dimensional hyperbolic space decorated with certain spinorial directions. This…

We note an adjunction inequality in $k\overline{\mathbb{CP}^2}$ for the $s$-version of the $Sq^1$-refinement of Rasmussen's $s$-invariant. This does not hold for general spatial refinements of $s$-invariants.

In this paper, we analyze the homology of the Yang-Baxter Operators $R_{(m)}$ yielding the HOMFLYPT polynomial, reducing the computation of the $n$-th homology of $R_{(m)}$ for arbitrary $m$ to the computation of $n+1$ initial conditions.…

An $n$-mean (also called a ''topological social choice rule'') on a topological space $X$ is a continuous function $p:X^n\to X$ satisfying $p(x,\dots, x)=x$ for every $x\in X$ and $p(x_1,\dots, x_n)=p(x_{\sigma(1)},\dots x_{\sigma(n)})$ for…