几何拓扑

We study whether quandle colorings can detect causality of events for links realized as skies in a $(2+1)$-dimensional globally hyperbolic spacetime $X$. Building off the Allen--Swenberg paper in which their $2$-sky link was conjectured to…

Let $\Sigma_g$ be a closed oriented surface of genus $g$, in this paper we discuss how to define coloring invariants and its generalizations for links in $\Sigma_g\times S^1$.

The ropelength of a knot or link is the minimal number of inches of 1-inch-thick rope that it takes to tie it. The relationship of this measurement to knot and link invariants has been studied by various authors. We give the first results…

We study some generalisations to mixed braid groups of the Fadell-Neuwirth short exact sequence and the possible splitting of this sequence. In certain cases, we determine conditions under which the projection from the mixed braid group…

The character variety $\mathscr{X}(S,G)$ associated to an oriented compact surface $S$ with boundary and a real reductive Lie group $G$ admits a Poisson structure and is foliated by symplectic leaves. When $G$ is a matrix group, any closed…

We point out that the claim in Theorem 1.1 of Yashiro's paper ``Pseudo-cycles of surface-knots'' is not true, giving a counter example.

We prove that any compact, orientable 3-manifold with empty or toral boundary is profinitely almost rigid among all compact, orientable 3-manifolds. In other words, the profinite completion of its fundamental group determines its…

Let $K_m$ be the result of applying $m$ full twists to $n$ parallel strands in a knot $K$. We prove that extremal knot Floer homologies of $K_m$ stabilize as $m$ goes to infinity.

The work of Mann and Rafi gives a classification surfaces $\Sigma$ when $\textrm{Map}(\Sigma)$ is globally CB, locally CB, and CB generated under the technical assumption of tameness. In this article, we restrict our study to the pure…

We show that the knot quandle of the $3$-, $4$-, or $5$-twist-spun trefoil is isomorphic to a quandle related to the $16$-, $24$-, or $600$-cell respectively. We further show that the cardinality of the knot quandle of the $m$-twist-spun…

We describe the lower algebraic $K$-theory of the integral group ring of both the pure and full braid groups of the real projective plane $\mathbb{R}P^2$ with $3$ strings, as well as that of the integral group ring of the mapping class…

Let $S_g$ be the closed orientable surface of genus $g \geq 2$, and let $\mathrm{Mod}(S_g)$ be the mapping class group of $S_g$. Let $A_n$ denote the alternating group on $n$ letters. We derive necessary and sufficient conditions under…

Work of Kalelkar, Schleimer, and Segerman shows that, with some exceptions, the set of essential ideal triangulations of an orientable cusped hyperbolic 3-manifold is connected via 2-3 and 3-2 moves. It is natural to ask if the subgraph…

A periodic link, is link in $S^3$ with action of $\mathbb{Z}_p$ by rotation with $2\pi/p$ around a fixed unknot $U$. The equivariant Khovanov homology of periodic links has been studied in \cite{BP17}. We prove that the equivariant Khovanov…

This monograph is on convex real projective structures on strongly tame n-orbifolds with some appropriate conditions on ends.

We show that if there exists a knot in $S^3$ that admits purely cosmetic surgeries, then there exists a hyperbolic one with this property.

The study of rod complements is motivated by rod packing structures in crystallography. We view them as complements of links comprised of Euclidean geodesics in the 3-torus. Recent work of the second author classifies when such rod…

We study the density of the Burau representation from the perspective of a non-semisimple TQFT at a fourth root of unity. This gives a TQFT construction of Squier's Hermitian form on the Burau representation with possibly mixed signature.…

Wajnryb proved that the mapping class group of a closed oriented surface is generated by two elements. We proved that the mapping class group is generated by two pseudo-Anosov elements. In particular, if the genus is greater than or equal…

We prove that two virtual knots have equivalent intersection graphs if and only if they have the same writhe polynomial.