几何拓扑

This paper introduces relative versions of the inner automorphism group and the transvection group associated with surjective quandle homomorphisms.By using the relative inner automorphism group, we define a notion of \emph{connectedness}…

The Birman-Craggs-Johnson homomorphism is a homomorphism $\sigma \colon \mathcal{I}_g \to \mathbb{B}_3'$ from the Torelli group to a certain $\mathbb{Z}/2\mathbb{Z}$-vector space of Boolean polynomials. In 1983, Johnson computed…

For knot surgery in $S^3$, Heegaard Floer homology provides an obstruction due to Hom--Karakurt--Lidman. We extend this obstruction to all integer homology spheres $Y$, for both positive and negative 1/m surgeries. This is used to test…

In this paper, we show that if the link of an isolated complex surface singularity is either a $Sol^3$-manifold or an $\widetilde{SL}(2;\mathbb{R})$-manifold with its canonical contact structure, then it admits infinitely many strong…

Let $\Gamma$ be a complex reflection group acting on the complex affine or hyperbolic space $X$ with the set of reflecting hyperplanes $\mathcal{H}$. We define an augmented rack $(G, \mathcal{K}, p)$ associated to the orbifold fundamental…

Given a pseudo-Anosov flow $\phi$ on a closed atoroidal $3$--manifold $M$ and a closed surface $S$ almost transverse to $\phi$, we give a homological characterization of when $S$ can be completed to an almost transverse depth one lamination…

Let $\Gamma$ be a finite simplicial graph of girth at least five. In this short note, we give a proof that if $M$ is a finite volume hyperbolic $3$--manifold, then the right-angled Artin group $A(\Gamma)$ cannot contain $\pi_1(M)$ as a…

We present a list of 3028 obstructions to knotless embedding. We survey recent work in this area including: 1) A bibliography of graphs proven to be intrinsically knotted without relying on computers; 2) An updated listing of obstructions…

We introduce a generalization of the Jones polynomial for classical and virtual knots and links using colorings by a permutation $\sigma:X\to X$ of a finite set $X$. For $X=\{1\}$ and for classical knots, the invariant is equivalent to the…

Alexander's conjecture states that for every two finite triangulations of the same topological space, if they have a common subdivision, then they have a common stellar subdivision. We generalize the recent result of Adiprasito and Pak, who…

The mapping class group of a $3$-dimensional handlebody of genus $g$, denoted by $\mathcal{M}(V_g)$, is a fundamental object of study in geometric topology. Building upon the initial generators introduced by Suzuki and their explicit…

We describe a dense subset of the Gromov boundary of the grand arc graph of an infinite-type surface as a space of geodesic laminations, analogous to Klarreich's description of the Gromov boundary of the curve complex. After showing that…

We introduce a chain map from quandle homology to relative group homology, and construct several quandle cocycles through the chain map. We also relate this chain map to triangulations of Seifert (hyper)surfaces of 1- and 2-dimensional…

The marked Schottky space records, up to conjugacy, all actions of a free group of fixed rank as a Schottky group on hyperbolic space of fixed dimension. In dimension three it is the classical Schottky space covering the moduli space of…

A diffeomorphism of a $4$-manifold is said to be exotic if it is continuously isotopic to the identity but not smoothly isotopic to the identity. Ruberman constructed the first examples of exotic diffeomorphisms on simply-connected closed…

We derive explicit formulas for the HOMFLY polynomials of the torus links $T(3,n)$ using braid groups and the skein relation. We first treat the case of $T(2,n)$ and then derive a five-term linear recurrence for an auxiliary sequence…

On the Kirby list, Akbulut poses the question of whether there exists a homology 3-sphere $Y$, other than $S^3$, with the following property: Any knot $K$, representing $0\in\pi_{1}(Y),$ which is slice in some contractible 4-manifold $X$…

We examine how the Rasmussen invariant, satellite operations, and null-homologous twists can be used to establish infinite order of knots in the smooth concordance group. As an application, we show that the Conway knot has infinite…

In this paper, we upgrade the instanton TQFT from ordinary categories to a functor $CI$ from an $\infty$-cobordism category $\mathrm{BI}$ for instantons to an $\infty$-derived category $\mathsf{D}$ of $2$-periodic chain complexes and sums…

We prove the surgery exact triangle for monopole (Seiberg--Witten) Floer homology over integer coefficients, extending the work of Kronheimer--Mrowka--Ozsv\'{a}th--Szab\'{o} over $\mathbb{Z}/2$, Lin--Ruberman--Saveliev over $\mathbb{Q}$,…