Papers

A similarity pair is the dynamical system in $\mathbb{C}$ generated by two maps $f:z \to sz-1$ and $g:z \to sz+1$ for $|s|<1$. Associated to the dynamical system is an attractor $\Lambda$. The Barnsley–Harrington Mandelbrot set $\mathcal{M}$ is the set of $s\in \mathbb{D}$ for which $\Lambda$ is connected. Let $K$ denote the filled set of a connected $\Lambda$. For $s\in \partial \mathcal{M}$ we show that the (partially defined) action of the semigroup on $\partial K$ is topologically conjugate to a (discontinuous) piecewise linear action of constant slope. Conditional on a conjecture (satisfied for `most' $s\in \partial \mathcal{M}$) we give a necessary and sufficient condition in terms of the dynamics on $\partial K$ for $K$ to contain cut points, and we describe the set of all such cut points in terms of infinite walks in a directed graph $\textrm{IG}$ obtained by an explicit recursive algorithm. The structure of the `dynamical cut point set' for a 2-dimensional family of piecewise linear actions (containing those coming from $s\in \partial \mathcal{M}$) recovers and generalizes the Douady–Hubbard–Thurston quadratic minor lamination for the abstract Mandelbrot set.

For every $g>0$, we construct a closed Haken 3-manifold $M_g$, whose only connected, incompressible surface is a non-separating surface of genus $g$ and discuss the torsion behavior of the Kauffman bracket skein module $\S(M_g,\Z[A^{\pm 1}])$. We also use properties of shared surgeries between knots to estimate (and often compute) the dimension of the Kauffman bracket skein module over $\Q(A)$, for families of hyperbolic 3-manifolds obtained by surgery on double twist knots. We have similar computations for some surgeries on several knot complements in the SnapPy census of cusped hyperbolic manifolds with triangulations up to 9 tetrahedra.

The main result of this article shows that the quiver mutation class of a malleable real Morsification uniquely determines the integral monodromy module of a plane curve singularity. In particular, we show that the quiver mutation class of a malleable divide determines the complex topological type of an irreducible plane curve singularity. This establishes the algebraic-to-topological implication of a conjecture of S.~Fomin, P.~Pylyavskyy, E.~Shustin and D.~Thurston in the malleable irreducible case. The result is proven by developing representation-theoretic techniques based on an equivariant Euler pairing in the derived category of continuous finite-dimensional dg modules over a differential bigraded Ginzburg algebra. A key step uses these techniques to show that the quiver mutation class of a plabic fence uniquely recovers the torsion part of the Alexander module of the associated smooth link.

We study the quasisymmetric classification of limit sets of Kleinian groups and obtain a characterization of those geometrically finite limit sets that are quasisymmetrically universal. This allows us to obtain a quasi-isometric classification of finitely generated Kleinian groups. We also obtain a classification of virtually topologically rigid geometrically finite Kleinian groups.

For every $n\ge 8$, we construct explicit families of generating pairs of $GL(n,\mathbb Z)$ and $SL(n,\mathbb Z)$ representing countably infinitely many Nielsen equivalence classes. These pairs arise as the homology images of explicit torsion generating pairs in $Out(F_n)$ and its index-two subgroup $SOut(F_n)$, yielding countably infinitely many Nielsen classes of generating pairs in both outer groups. Within each constructed outer family, all pairs become Nielsen equivalent after one stabilization, obtained by adjoining a trivial coordinate. We derive a relation-module obstruction to Nielsen equivalence of once-stabilized generating tuples. Evans's non-cancellation examples show that this obstruction is nontrivial and can distinguish Nielsen classes of once-stabilized generating tuples. We also record a quotient relation-module refinement for possible future applications.

We develop computational tools for constructing the minimal resolution graphs of suspension singularities. In particular, we provide a Magma program that implements Nemethi's algorithm that can be used to construct the minimal resolution graph of any suspension singularity. The program outputs the resulting resolution graph as structured graph data, encoding the vertices, edges, and decorations of the graph. We further develop a SageMath program that takes this data as input and produces graphical representations of the resolution graphs, facilitating their visualization and analysis. For three families of suspension singularities, we prove that the corresponding minimal resolution graphs are periodic, with periods equal to the orders of the corresponding monodromy factorizations. We also provide Magma programs for generating compactified resolution graphs of suspension singularities and checking graph isomorphisms over a specified range of parameters.

Let $s782$ be the 2-cusped hyperbolic 3-manifold in the SnapPy census. Its spherical CR uniformization was established in \cite{JWX2023} using the Ford domain of the complex hyperbolic triangle group $\Delta_{4,4,\infty;\infty}$. By comparing the combinatorial structures of the Ford domain of $\Delta_{4,4,\infty;\infty}$ and the Dirichlet domain of $\Delta_{4,4,n;\infty}$, we prove that for each $n \geq 5$, the Dehn filling of $s782$ along the slope $(n-1)\mathcal{m}_1 + \mathcal{l}_1$ on its second cusp admits a spherical CR uniformization, where $(\mathcal{m}_1, \mathcal{l}_1)$ denotes the meridian-longitude system of a cusp in SnapPy notation.

We show that for any closed atoroidal 3-manifold $M$, there are only finitely many isotopy classes of links that can arise as singular orbits of pseudo-Anosov flows on $M$.

The odd Vassiliev conjecture asserts that all Jacobi diagrams with an odd number of legs vanish modulo the AS and IHX relations. Equivalently, the conjecture asserts that Vassiliev invariants cannot distinguish a knot from its inverse. The conjecture is elementary for one 1 or 2 loops, and was proven by Moskovich and Ohtsuki for 3 loops. We establish the conjecture for 4 loops. Our proof uses Jacobi diagrams and weight systems with polynomial coefficients, commuting colored legs, and several vanishing criteria.

In the first part of this work, we have, for a not necessarily semisimple modular category, generalized the action of the mapping class groups of surfaces on the spaces of conformal blocks to the so-called derived block spaces. In the second part presented here, we compute this action explicitly in the case of Drinfel'd doubles of finite groups over fields of positive characteristic. To do that, we connect Lyubashenko's approach to mapping class group representations with the theory of representation varieties. In this way, we are able to show that the mapping class group representations on the derived block spaces are in general different from those on the ordinary block spaces.

We prove several bounds on the largest and smallest eigenvalues of the combinatorial Hodge Laplacian $\Delta^H_k$ of a finite simplicial complex $\Sigma.$ As a consequence, we obtain new vanishing criteria for cohomology groups $H^k(\Sigma,\mathbb{R)}$ and confirm a conjecture of O on the dimensional monotonicity of the largest eigenvalue.

We construct examples of fibrewise Anosov flows over the Fenley–Mann–Potrie example of dimension 4. This is a family of non-algebraic Anosov flows with unstable dimension $1+4n$ and stable dimension $2+4n$, for every positive integer $n$.

We prove that the $SL(2,\mathbb{C})$ character varieties of compact oriented surfaces and the generic relative $SL(2,\mathbb{C})$ character varieties of punctured surfaces admit divisorial log terminal (dlt) log Calabi-Yau compactifications. To do this, we establish a general result giving sufficient conditions for a compactification of an affine variety arising from a filtration of its algebra of regular functions to be log Calabi-Yau. We then apply this result to show that the compactifications constructed by Kutteri-Tehrani-Frohman in the compact case and by Tehrani-Frohman in the punctured case are log canonical and log Calabi-Yau.

We prove that the asymptotic cone of any non-abelian free group equipped with the bi-invariant word metric has elements of order 2.

We compute the effect of the inclusion $\mathrm{F}_4 \rightarrow \mathrm{hAut}(\mathbb{O}P^2)$ on rational homotopy groups by determining the rational homotopy class of the isometry action $\mathrm{F}_4 \times \mathbb{O}P^2 \rightarrow \mathbb{O}P^2$ in terms of a homomorphism between the minimal models of source and target. Although special emphasis is put on the Cayley plane, our results generalise to all simply connected rank $1$-symmetric spaces.

We show that the numerical invariants $\nu$ and $|\varepsilon|$ coming from knot Floer homology are knot $n$-trace invariants for any integer $n$, resolving the remaining case in Hayden-Mark-Piccirillo. This extension allows us to construct new families of exotic pairs using Yasui patterns. Moreover, by studying the invariant $\widehat{\nu}(K)=|\varepsilon(K)|(2\nu(K)-1)$, we give a new topological obstruction to a $4$-manifold being a strong symplectic filling of any contact structure on the boundary.

This paper introduces a two-variable polynomial invariant for oriented twisted knots, denoted by $Q_{K}^{z}(s,t)$, refining the $Q$-polynomial of N. Kamada and S. Kamada \cite{NaoSei}. We exhibit an infinite family of twisted knots indistinguishable by the $Q$-polynomial but separated by the $Q^z$-polynomial. To prove invariance, we first determine a generating set of oriented Reidemeister moves for twisted knot diagrams, extending the result of Ali \cite{Dan} for oriented virtual knots; this result is new and of independent interest, as it provides the minimal framework needed to verify invariance of any oriented twisted knot invariant. As further applications, we derive an explicit crossing change formula, obtain lower bounds on the Gordian distance between homotopic twisted knots, examine the existence of cosmetic crossings in a twisted knot diagram, and finally prove that $Q^{z}_{K}(s,t)$ is a Vassiliev invariant of order one.

We construct two Stein structures on the same compact smooth four-manifold whose boundary contact forms are strictly identified and whose first Chern classes agree, but whose exact symplectic forms are not symplectomorphic. Moreover, no diffeomorphism makes the associated Weinstein structures homotopic. The examples are obtained by removing a tubular neighborhood of a smooth bicanonical curve from a fake projective plane and comparing the induced Stein structure with its conjugate. Their canonical Spinc structures differ by a nonzero class of order two. After the boundary forms are normalized using a common Boothby-Wang connection form, a hypothetical symplectomorphism preserves the oriented circle fiber and extends over the divisor cap, contradicting canonical-class rigidity. For the Weinstein statement, the required fiber-preservation is obtained from a monopole Floer grading asymmetry, using the Nelson-Weiler computation and Taubes's ECH-Seiberg-Witten correspondence. This gives an affirmative solution to a normalized version of the relative filling problem following Problem 4.96 in the K3 problem list.

Toric real loci are distinguished lagrangian submanifolds of toric symplectic manifolds. We develop a polyhedral model for toric real loci, called a kaleidoscope, based on the restriction of the moment map. This construction provides a direct geometric description of the topology of toric real loci and leads to simple criteria for orientability, together with transparent formulas for the Euler characteristic. We illustrate the theory with numerous examples and count, in every complex dimension up to 9, the smooth toric Fano varieties whose real loci are orientable. The kaleidoscope construction offers a simple, visual, and effective framework for understanding the topology of toric real loci through the combinatorics of their moment polytopes.

We show that in any Haken 3-manifold $M$, the dimensions of $\mathcal{ML}(M)$ and $\mathcal{ML}_0(M)$ can be calculated using normal surfaces. As a corollary, we prove the number of closed orientable essential surfaces of Euler characteristic at least $-2n$ in a closed orientable hyperbolic 3-manifold with $H_2(M,\mathbb{Z}/2\mathbb{Z}) = 0$ is asymptotic to $n^{\dim \mathcal{ML}_0(M)}$.

We assemble numerical algorithms to search for $J$-holomorphic curves in symplectic manifolds. Each algorithm employs several different numerical techniques, each technique addressing a different aspect of the geometric problem. We separately consider both classical Fourier expansion and deep neural networks in our algorithms and compare their performance. Our algorithms take as input a smooth curve in a given homology class and search for a $J$-holomorphic curve in the same homology class. We first verify we can produce explicitly known holomorphic curves in complex manifolds, for example the Weierstrass $\wp$ function on the torus and curves in $S^2\times S^2$ with the standard complex structure. Then we search for $J$-holomorphic curves in $S^2\times S^2$ with non-integrable almost complex structures: essentially we start with a known holomorphic curve in an integrable almost complex structure $J_0$, deform $J_0$ to a nearby nonintegrable almost complex structure $J_\epsilon$, and use our methods to find the nearby $J_\epsilon$-holomorphic curve.

Let $\Sigma$ be a compact connected oriented surface of genus at least $3$ with at most one boundary component. Santharoubane associated to certain presentations of the mapping class group modulo its center a finitely presented algebra equipped with a canonical surjection onto the generic Kauffman bracket skein algebra of $\Sigma$, and conjectured that a suitable choice yields an algebra isomorphic to the skein algebra. We show that every algebra arising from this construction admits an augmentation character, whereas the generic skein algebra of $\Sigma$ admits no unital character over $\mathbb Q(A)$. The latter obstruction follows from the intersection-one Dehn-twist identity together with a $4$-holed-sphere skein relation. Consequently, the conjectured isomorphism does not hold as stated.

We study the monodromy action of the mixed braid group $B_{n,\mathcal{P}}$ on the first cohomology of cyclic branched covers of $\mathbb{P}^1$, which are mutually determined by a partition of branch points by equal ramification. The monodromy representation splits into irreducible representations on the $t$-eigenspaces of the deck transformation. For each, we construct an explicit spanning set using lifts of Pochhammer contours and figure-eight curves, and compute the Hermitian intersection form. The representation factors through a reduced mixed braid group by dropping $t$-invisible parts of the partition (those with trivial local monodromy). In this reduced representation, each generator acts by a complex reflection when the corresponding spanning class is non-isotropic, and by a unitary transvection when it is isotropic. Provided $\infty$ has non-trivial local monodromy, the factored representation is isomorphic to the reduced multivariate Burau representation evaluated at $t$.

We study the classifying map $c\colon BX\to K(\As(X),1)$ of a rack $X$ of finite type. Let $t = \Type(X)$. We prove that $t c_{n*}=0$ for every $n\ge2$ when $X$ is connected, and that $t^{n-1}c_{n*}=0$ on the torsion subgroup $\Tor H_n^\mathbb{R}(X)$ without any connectedness assumption. For a finite rack, under our sign conventions, the rationalized classifying map in degree $n$ is given by $(-1)^n$ times the canonical projection from the $n$-fold tensor power of the orbit module to its $n$-th exterior power. For an arbitrary rack of finite type, we determine $H_2^{\mathrm{gr}}(\As(X);\mathbb{Z}[1/t])$. We also derive low-dimensional applications to symplectic and Alexander structures.

Given a spherical finite tensor category C, not necessarily semisimple, and a two-sided modified trace on its projective ideal, we define an open-closed three-dimensional topological field theory with values in vector spaces. The bordism category has as morphisms three-dimensional bordisms with corners, whose boundary is partitioned into the gluing boundary, parametrised by source and target surface, and the unparametrised free boundary. The free boundary is equipped with an embedded C-coloured graph satisfying an admissibility condition. Our construction starts from a new three-manifold invariant based on the multi-handlebody invariant of [arXiv:1809.07991] and the chromatic maps of [arXiv:2302.04509]. The open-closed topological field theory is then obtained via the universal construction.

In this work, we construct a generalization of the $G_n^k$-theory to the case of an arbitrary hypergraph. The case of spherical braids is considered separately, using the stratification of the moduli space $\mathcal{M}_n(S^2)$ and the hypergraph $\Gamma_n^{\mathrm{sph}}$ encoding projective constraints. In contrast to the original $G_{n}^{k}$ theory, where codimension-one properties are determined by exactly $k$ particles, the present work considers various cases corresponding to strata of codimension~$1$. These groups admit nice maps to free products of cyclic groups. Among unsolved problems, we emphasize the question how the above construction works for abelian varieties and, in particular, for elliptic curves.

We establish a topological analogue of Ismagilov's theorem concerning the first continuous cohomology of volume-preserving diffeomorphisms. For a closed oriented manifold \(M\) with volume form \(\Omega\), we consider the group \(\mathbb{G}^{\Omega}(M)\) of homeomorphisms obtained as uniform limits of smooth volume-preserving isotopies. We show that its first continuous cohomology with values in the Banach space of zero-mean continuous functions is isomorphic to the first de Rham cohomology of \(M\). The proof develops a theory of transport for volume-preserving isotopies, producing a topological volume flux homomorphism and its associated transport cocycle. Under the additional hypothesis that \((M,\Omega)\) satisfies the local \(C^0\)-generation property, and assuming the Müller-Sikorav approximation theorem (known for \(n\neq 4\)), the isomorphism extends to the full identity component \(\operatorname{Homeo}_0^\Omega(M)\) of the group of volume-preserving homeomorphisms, and the topological flux conjecture follows in that setting.

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.

A finitely generated group is profinitely rigid among a class of finitely generated groups if it can be distinguished among this class by its set of finite quotient groups. This paper proves that all lattices in $\mathrm{PSL}(2,\mathbb{C})$ are profinitely rigid among themselves. In addition, for any lattice $\Gamma\le \mathrm{PSL}(2,\mathbb{C})$, it is proven that $\mathrm{Out}(\widehat{\Gamma})\cong \mathrm{Out}(\Gamma)$, where $\widehat{\Gamma}$ denotes the profinite completion of $\Gamma$.

Cericola proved that every knot diagram can be transformed into an ascending unknot diagram by arc crossing changes. We refine this result for all diagrams on a fixed R1-reduced classical knot shadow with more than three crossings. Apart from at most two source diagrams, any two diagrams of the same state parity are mutually reachable. Each source diagram reaches every non-source diagram of its parity but cannot be reached from another diagram. A local five-occurrence condition on the marked Gauss word characterizes the sources and thereby determines the full directed reachability relation.