arXiv subjects

Alexandra Kjuchukova

Publications and source records attributed to Alexandra Kjuchukova.

At least 19 recordsLinked to original sources

arXiv2026

Cobordism groups of dihedral branched covers

For every integer $n \geq 1$, we compute the cobordism groups of dihedral $n$-fold branched covers of $S^3$ with oriented and non-oriented branching sets. We show that the groups are cyclic, generated by the $n$-fold connected dihedral covers of $(2,n)$-torus links. The isomorphism types of the groups are detected using explicit cobordism invariants defined in terms of the Seifert forms on the branching sets, generalizing Cappell-Shaneson characteristic knots associated to dihedral covers.

math.GT↗

arXiv2026

Extending Quotients of Knot Groups over Surfaces in $B^4$

Let $K\subseteq S^3$ be a knot with exterior $E_K$, and denote by $ρ\colon π_1(E_K)\twoheadrightarrow G$ a quotient of its group. We give a sharp obstruction to the existence of a connected, oriented, smooth surface $F\subseteq B^4$ with $\partial F = K$ over whose exterior $ρ$ extends surjectively. Equivalently, we determine whether the cover of $S^3$ branched over $K$ and induced by $ρ$ bounds a connected cover of $B^4$ branched along such a surface. When $G$ is a dihedral group, we show the obstruction can be computed by evaluating the Seifert form of $K$ on a single curve, a so-called characteristic knot associated to $ρ$. When the dihedral obstruction vanishes, we construct the surface $F$ explicitly.

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arXiv2025

Branched Covers of Open Manifolds

For $m=2$ and $m=3$ we prove that any connected, oriented, open manifold $M^m$ admits a simple branched covering map over $\mathbb{R}^m$. When $M$ has $k$ ends and $k$ is finite, the degree of the cover can be taken to be $mk$. Regardless of the number of ends, $M$ admits a branched covering map of countably infinite degree over $\mathbb{R}^m$. We also investigate which compact manifolds are universal bases, that is, are branch covered by all compact manifolds in the same dimension.

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arXiv2025

Rep-Tiles

An $n$-dimensional rep-tile is a compact, connected submanifold of $\mathbb{R}^n$ with non-empty interior which can be decomposed into pairwise isometric rescaled copies of itself whose interiors are disjoint. We show that every smooth compact $n$-dimensional submanifold of $\mathbb{R}^n$ with connected boundary is topologically isotopic to a polycube that tiles the $n$-cube, and hence is topologically isotopic to a rep-tile. It follows that there is a rep-tile in the homotopy type of any finite CW complex. In addition to classifying rep-tiles in all dimensions up to isotopy, we also give new explicit constructions of rep-tiles, namely examples in the homotopy type of any finite bouquet of spheres.

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arXiv2025

The Plain Sphere Number of a Link

Let $L$ be a link in $S^3$. We consider the class of meridional presentations for $π_1(S^3\backslash L)$ in which the relations are witnessed by embedded two-spheres which can be represented simultaneously in a fixed diagram of $L$, analogously to decomposition spheres studied by Cromwell, Menasco and others. Wirtinger relations are witnessed by such spheres and the Wirtinger presentation is a special case of the ones we study. We prove that the smallest number of generators of $π_1(S^3\backslash L)$ over all such presentations equals the bridge number of $L$.

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arXiv2024

Adding a suitable unknot to any link equates bridge number and meridional rank

Given any link $L\subseteq S^3$, we show that it is possible to embed an unknot $U$ in its complement so that the link $L\cup U$ satisfies the Meridional Rank Conjecture (MRC). The bridge numbers in our construction fit into the equality $β(L\cup U)=2β(L)-1=\text{rank}(π_1(S^3\backslash (L\cup U)))$. In addition, we prove the MRC for new infinite families of links and distinguish them from previously settled cases through an application of bridge distance.

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arXiv2023

Brunnian exotic surface links in the 4-ball

This paper investigates the exotic phenomena exhibited by links of disconnected surfaces with boundary that are properly embedded in the 4-ball. Our main results provide two different constructions of exotic pairs of surface links that are Brunnian, meaning that all proper sublinks of the surface are trivial. We then modify these core constructions to vary the number of components in the exotic links, the genera of the components, and the number of components that must be removed before the surfaces become unlinked. Our arguments extend two tools from 3-dimensional knot theory into the 4-dimensional setting: satellite operations, especially Bing doubling, and covering links in branched covers.

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arXiv2023

Linking in Cyclic Branched Covers and Satellite (non)-Homomorphisms

Let $K\subset S^3$ be a knot and $η, γ\subset S^3\backslash K$ be simple closed curves. Denote by $Σ_q(K)$ the $q$-fold cyclic branched cover of $K$. We give an explicit formula for computing the linking numbers between lifts of $η$ and $γ$ to $Σ_q(K)$. As an application, we evaluate, in a variety of cases, an obstruction to satellite operations inducing homomorphisms on smooth concordance.

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arXiv2022

Coxeter quotients of knot groups through 16 crossings

We find explicit maximal rank Coxeter quotients for the knot groups of 595,515 out of the 1,701,936 knots through 16 crossings. We thus calculate the bridge numbers and verify Cappell and Shaneson's Meridional Rank Conjecture for these knots. In addition, we provide a computational tool for establishing the conjecture for knots beyond 16 crossings whose meridional ranks can be detected via finite Coxeter quotients.

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arXiv2021

Slicing knots in definite 4-manifolds

We study the $\mathbb{CP}^2$-slicing number of knots, i.e. the smallest $m\geq 0$ such that a knot $K\subseteq S^3$ bounds a properly embedded, null-homologous disk in a punctured connected sum $(\#^m\mathbb{CP}^2)^{\times}$. We give a lower bound on the smooth $\mathbb{CP}^2$-slicing number of a knot in terms of its double branched cover, and we find knots with arbitrarily large but finite smooth $\mathbb{CP}^2$-slicing number. We also give an upper bound on the topological $\mathbb{CP}^2$-slicing number in terms of the Seifert form and find knots for which the smooth and topological $\mathbb{CP}^2$-slicing numbers are both finite, nonzero, and distinct.

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arXiv2021

Average four-genus of two-bridge knots

We prove that the expected value of the ratio between the smooth four-genus and the Seifert genus of two-bridge knots tends to zero as the crossing number tends to infinity.

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arXiv2020

Coxeter groups and meridional rank of links

We prove the meridional rank conjecture for twisted links and arborescent links associated to bipartite trees with even weights. These links are substantial generalizations of pretzels and two-bridge links, respectively. Lower bounds on meridional rank are obtained via Coxeter quotients of the groups of link complements. Matching upper bounds on bridge number are found using the Wirtinger numbers of link diagrams, a combinatorial tool developed by the authors.

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arXiv2020

A note on three-fold branched covers of $S^4$

We show that any 4-manifold admitting a $(g;k_1,k_2,0)$-trisection is an irregular 3-fold cover of the 4-sphere whose branching set is a surface in $S^4$, smoothly embedded except for one singular point which is the cone on a link. A 4-manifold admits such a trisection if and only if it has a handle decomposition with no 1-handles; it is conjectured that all simply-connected 4-manifolds have this property.

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arXiv2020

Linking Numbers in Three-Manifolds

Let $M$ be a connected, closed, oriented three-manifold and $K$, $L$ two rationally null-homologous oriented simple closed curves in $M$. We give an explicit algorithm for computing the linking number between $K$ and $L$ in terms of a presentation of $M$ as an irregular dihedral $3$-fold cover of $S^3$ branched along a knot $α\subset S^3$. Since every closed, oriented three-manifold admits such a presentation, our results apply to all (well-defined) linking numbers in all three-manifolds. Furthermore, ribbon obstructions for a knot $α$ can be derived from dihedral covers of $α$. The linking numbers we compute are necessary for evaluating one such obstruction. This work is a step toward testing potential counter-examples to the Slice-Ribbon Conjecture, among other applications.

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arXiv2020

Signatures of topological branched covers

Let $X^4$ and $Y^4$ be smooth manifolds and $f: X\to Y$ a branched cover with branching set $B$. Classically, if $B$ is smoothly embedded in $Y$, the signature $σ(X)$ can be computed from data about $Y$, $B$ and the local degrees of $f$. When $f$ is an irregular dihedral cover and $B\subset Y$ smoothly embedded away from a cone singularity whose link is $K$, the second author gave a formula for the contribution $Ξ(K)$ to $σ(X)$ resulting from the non-smooth point. We extend the above results to the case where $Y$ is a {\it topological} four-manifold and $B$ is locally flat, away from the possible singularity. Owing to the presence of non-locally-flat points on $B$, $X$ in this setting is a stratified pseudomanifold, and we use the Intersection Homology signature of $X$, $σ_{IH}(X)$. For any knot $K$ whose determinant is not $\pm 1$, a homotopy ribbon obstruction is derived from $Ξ(K)$, providing a new technique to potentially detect slice knots that are not ribbon.

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arXiv2020

The Dihedral Genus of a Knot

Let $K\subset S^3$ be a Fox $p$-colored knot and assume $K$ bounds a locally flat surface $S\subset B^4$ over which the given $p$-coloring extends. This coloring of $S$ induces a dihedral branched cover $X\to S^4$. Its branching set is a closed surface embedded in $S^4$ locally flatly away from one singularity whose link is $K$. When $S$ is homotopy ribbon and $X$ a definite four-manifold, a condition relating the signature of $X$ and the Murasugi signature of $K$ guarantees that $S$ in fact realizes the four-genus of $K$. We exhibit an infinite family of knots $K_m$ with this property, each with a {Fox 3-}colored surface of minimal genus $m$. As a consequence, we classify the signatures of manifolds $X$ which arise as dihedral covers of $S^4$ in the above sense.

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arXiv2020

Computing Ribbon Obstructions for Colored Knots

Kjuchukova's $Ξ_p$ invariant gives a ribbon obstruction for Fox $p$-colored knots. The invariant is derived from dihedral branched covers of 4-manifolds, and is needed to calculate the signatures of these covers, when singularities on the branching sets are present. In this note, we give an algorithm for evaluating $Ξ_p$ from a colored knot diagram, and compute a couple of examples.

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arXiv2020

The bridge number of arborescent links with many twigs

We prove the meridional rank conjecture for arborescent links associated to plane trees with the following property: all branching points carry a straight branch to at least three leaves. The proof involves an upper bound on the bridge number in terms of the maximal number of link components of the underlying tree, valid for all arborescent links.

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