Papers by Alexandra Kjuchukova
We prove the meridional rank conjecture for twisted links and arborescent links associated to bip... more 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.
arXiv: Geometric Topology, 2017
Consider a dihedral cover $f: Y\to X$ with $X$ and $Y$ four-manifolds and $f$ branched along an o... more Consider a dihedral cover $f: Y\to X$ with $X$ and $Y$ four-manifolds and $f$ branched along an oriented surface embedded in $X$ with isolated cone singularities. We prove that only a slice knot can arise as the unique singularity on an irregular dihedral cover $f: Y\to S^4$ if $Y$ is homotopy equivalent to $\mathbb{CP}^2$ and construct an explicit infinite family of such covers with $Y$ diffeomorphic to $\mathbb{CP}^2$. An obstruction to a knot being homotopically ribbon arises in this setting, and we describe a class of potential counter-examples to the Slice-Ribbon Conjecture. Our tools include lifting a trisection of a singularly embedded surface in a four-manifold $X$ to obtain a trisection of the corresponding irregular dihedral branched cover of $X$, when such a cover exists. We also develop a combinatorial procedure to compute, using a formula by the second author, the contribution to the signature of the covering manifold which results from the presence of a singularity on ...
arXiv: Geometric Topology, 2018
Kjuchukova's $\Xi_p$ invariant gives a ribbon obstruction for Fox $p$-colored knots. The inva... more Kjuchukova's $\Xi_p$ 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 $\Xi_p$ from a colored knot diagram, and compute a couple of examples.
arXiv: Geometric Topology, 2019
We show that any 4-manifold admitting a $(g;k,0,0)$-trisection is an irregular 3-fold cover of th... more We show that any 4-manifold admitting a $(g;k,0,0)$-trisection is an irregular 3-fold cover of the 4-sphere, branched along an embedded surface with two singularities. A 4-manifold admits such a trisection if and only if it has a handle decomposition with no 1-handles and no 3-handles; it is conjectured that all simply-connected 4-manifolds have this property.
Discret. Comput. Geom., 2021
Let M be a connected, closed, oriented three-manifold and K, L two rationally null-homologous ori... more 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 three-fold cover of $$S^3$$ S 3 branched along a knot $$\alpha \subset S^3$$ α ⊂ 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 $$\alpha $$ α can be derived from dihedral covers of $$\alpha $$ 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.
arXiv: Geometric Topology, 2020
We prove the meridional rank conjecture for arborescent links associated to plane trees with the ... more 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.
This paper investigates the exotic phenomena exhibited by links of disconnected surfaces with bou... more 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.
Mathematische Annalen
We prove the meridional rank conjecture for twisted links and arborescent links associated to bip... more 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.
International Mathematics Research Notices
Let $X^4$ and $Y^4$ be smooth manifolds and $f: X\to Y$ a branched cover with branching set $B$. ... more 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 $\sigma (X)$ 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 $\Xi (K)$ to $\sigma (X)$ resulting from the non-smooth point. We extend the above results to the case where $Y$ is a topological four-manifold and $B$ is locally flat, away from the possible singularity. Owing to the presence of points on $B$ which are not locally flat, $X$ in this setting is a stratified pseudomanifold, and we use the intersection homology signature of $X$, $\sigma _{IH}(X)$. For any knot $K$ whose determinant is not $\pm 1$, a homotopy ribbon obstruction is derived from $\Xi (K)$, providing a new technique to potentially detect slice knots that are n...
Communications in Analysis and Geometry
We define the Wirtinger number of a link, an invariant closely related to the meridional rank. Th... more We define the Wirtinger number of a link, an invariant closely related to the meridional rank. The Wirtinger number is the minimum number of generators of the fundamental group of the link complement over all meridional presentations in which every relation is an iterated Wirtinger relation arising in a diagram. We prove that the Wirtinger number of a link equals its bridge number. This equality can be viewed as establishing a weak version of Cappell and Shaneson's Meridional Rank Conjecture, and suggests a new approach to this conjecture. Our result also leads to a combinatorial technique for obtaining strong upper bounds on bridge numbers. This technique has so far allowed us to add the bridge numbers of approximately 50,000 prime knots of up to 14 crossings to the knot table. As another application, we use the Wirtinger number to show there exists a universal constant C with the property that the hyperbolic volume of a prime alternating link L is bounded below by C times the bridge number of L.
Proceedings of the American Mathematical Society
We prove that the expected value of the ratio between the smooth four-genus and the Seifert genus... more 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.
Advances in Mathematics
Given a closed oriented PL four-manifold X and a closed surface B embedded in X with isolated con... more Given a closed oriented PL four-manifold X and a closed surface B embedded in X with isolated cone singularities, we give a formula for the signature of an irregular dihedral cover of X branched along B. For X simply-connected, we deduce a necessary condition on the intersection form of a simply-connected irregular dihedral branched cover of (X, B). When the singularities on B are two-bridge slice, we prove that the necessary condition on the intersection form of the cover is sharp. For X a simply-connected PL four-manifold with non-zero second Betti number, we construct infinite families of simply-connected PL manifolds which are irregular dihedral branched coverings of X. Given two four-manifolds X and Y whose intersection forms are odd, we obtain a necessary and sufficient condition for Y to be homeomorphic to an irregular dihedral p-fold cover of X, branched over a surface with a two-bridge slice singularity.
Discrete Mathematics
We define and compare several natural ways to compute the bridge number of a knot diagram. We stu... more We define and compare several natural ways to compute the bridge number of a knot diagram. We study bridge numbers of crossing number minimizing diagrams, as well as the behavior of diagrammatic bridge numbers under the connected sum operation. For each notion of diagrammatic bridge number considered, we find crossing number minimizing knot diagrams which fail to minimize bridge number. Furthermore, we construct a family of minimal crossing diagrams for which the difference between diagrammatic bridge number and the actual bridge number of the knot grows to infinity.
Algebraic & Geometric Topology
Let K ⊂ S 3 be a Fox p-colored knot and assume K bounds a locally flat surface S ⊂ B 4 over which... more Let K ⊂ S 3 be a Fox p-colored knot and assume K bounds a locally flat surface S ⊂ B 4 over which the given p-coloring extends. This coloring of S induces a dihedral branched cover X → 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 Km 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.
Mathematical Proceedings of the Cambridge Philosophical Society
We define a metric filtration of the Gordian graph by an infinite family of 1-dense subgraphs. Th... more We define a metric filtration of the Gordian graph by an infinite family of 1-dense subgraphs. The nth subgraph of this family is generated by all knots whose fundamental groups surject to a symmetric group with parameter at least n, where all meridians are mapped to transpositions. Incidentally, we verify the Meridional Rank Conjecture for a family of knots with unknotting number one yet arbitrarily high bridge number.
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Papers by Alexandra Kjuchukova