Papers

We study the roots and the Mahler measure of the $Q$-polynomial of links. We first consider links obtained by adding twists to a pair of parallel strands. We prove that the Mahler measure of the transformed $Q$-polynomial converges as the number of twists increases. We also show that all but a uniformly bounded number of distinct roots of the $Q$-polynomial approach the real interval $[-2,2]$. This behavior is different from that of the roots of the Jones polynomial under twisting. Numerical experiments on prime knots lead us to a conjecture about the roots of the transformed $Q$-polynomial of alternating knots. Finally, we compare real and unit-circle roots of the Alexander, Jones, and $Q$-polynomials, and give an infinite family of $2$-bridge links whose $Q$-polynomials have only real nonzero roots.

Quandles can be regarded as generalizations of symmetric spaces. In the theory of symmetric spaces developed by Chen and Nagano, there is an interesting relationship between the two-number and the Euler characteristic. The two-number is a Riemannian geometric invariant that can also be characterized in terms of point symmetries, whereas the Euler characteristic is a topological invariant. The aim of this paper is to initiate the study of quandle analogues of Chen–Nagano theory. In particular, we investigate relationships between the two-numbers and the Euler characteristics of quandles, and provide examples of finite quandles that either satisfy or fail to satisfy properties analogous to those of symmetric spaces. These examples are constructed from directed simple graphs labeled by abelian groups.

We give a presentation for the Kauffman bracket skein algebra of a closed or one-holed oriented surface of genus $2$. This is the first such result for genus greater than $1$.

We introduce a local move defining a welded extension of 4-equivalence. It induces the same quandle relations as 4-equivalence, giving a common algebraic framework for both the classical and welded settings. We also develop a version of arrow calculus adapted to this welded extension and apply it to the study of virtual tangles. This leads to a finite classification of algebraic virtual tangles without closed components under the resulting equivalence relation. As an application, we prove that the closure of any such tangle can be reduced to the unknot, giving a partial positive answer to a welded analogue of Nakanishi's conjecture.

We investigate a topological version of the Cartan-Hadamard theorem that allows one to study asphericity of spaces via distinguished families of paths. A topological space has the distinguished path property (dpp) if there exists a continuous map from the space $\Pi(X)$ of homotopy classes of paths (relative endpoints) to the path space $X^I$ that is a right inverse to the canonical quotient map. For complete metric spaces endowed with a locally convex metric, the distinguished paths are precisely the local geodesics. We show that if $X$ has the dpp, then its universal cover is contractible and, in particular, $X$ is aspherical. The space $\Pi(X)$ is a fiber bundle over $X$ whose fiber is the universal cover of $X$, and in the case of Riemannian manifolds of non-positive curvature it is naturally isomorphic to the tangent bundle. We prove that every aspherical CW-complex that is locally finite or countable has the dpp, and this allows us to reinterpret asphericity of CW-complexes in terms of the existence of continuous sections. We also define and study the notion of asphericity complexity of spaces by means of local sections from $\Pi(X)$ to $X^I$ and relate it to the concept of equivariant topological complexity introduced by Colman and Grant.

For any link, we define a monodromy map from the motion group of the link to the group of automorphisms of the link's Khovanov homology. The image of this map is the unoriented monodromy group of the link. This map allows us to convert results concerning motion groups of links into ones about their Khovanov monodromy. In particular, we use a characterisation of the motion groups of split links to write their unoriented monodromy groups as an explicit semidirect product in terms of their unsplit pieces. We give some demonstrative examples, computing the unoriented monodromy groups of unlinks, Hopf links, and split links composed of pieces thereof.

The topological mating of two postcritically finite polynomials with dendritic Julia sets is encoded by a pair of circle laminations $\Lambda^{\pm}$, whose collapse produces a sphere-filling curve. When a leaf of $\Lambda^{+}$ and a leaf of $\Lambda^{-}$ share an endpoint they form a perfect fit. An unpublished proposition of Epstein, recorded by Petersen and Meyer, shows that for matings of honest degree-$d$ polynomials an infinite-diameter ray equivalence class, i.e. an infinite chain of perfect fits is impossible. We show that this finiteness is a genuinely holomorphic phenomenon. Allowing the dynamics to carry a periodic critical orbit, we construct combinatorially expanding Thurston maps-realized by no expanding rational map-that admit invariant sphere-filling curves yet whose laminations $\Lambda^{\pm}$ contain infinite chains of perfect fits, in fact infinitely many of them.

In 2023, the first author, Nunez, Singh, and Swain [10] formulated an analogue of Kaplansky's idempotent conjecture for integral quandle rings. In this paper, we develop a new Fourier-analytic approach to quandle rings and use it to establish the conjecture for two major classes of quandles: Takasaki quandles, including dihedral quandles, and medial commutative quandles. A central contribution of the paper is the introduction of Fourier analysis on Alexander quandles, which provides a new framework for studying quandle rings and, in particular, their idempotents. Using this Fourier-analytic framework, we also prove that a previously known sufficient condition for the existence of counterexamples to the conjecture is in fact necessary, thereby resolving a problem of Jablonowski [12].

In dimension 3, following Eliashberg-Thurston and Mitsumatsu, one can associate a transverse pair of oppositely oriented contact structures to any Anosov flow. We show that when the ambient manifold is hyperbolic, the isotopy classes of these contact structures completely determine the flow up to isotopy orbit equivalence. This then implies that on any closed hyperbolic 3-manifold, the number of Anosov flows modulo isotopy orbit equivalence is finite. This approach also yields an explicit bound on the number of orbit equivalence classes of Anosov flows in terms of the number of universally tight contact structures. In addition, we show finiteness of pseudo-Anosov flows for which the complement of the singular orbits is atoroidal and, in the non- hyperbolic case, we obtain a control on the dynamics of Anosov flows on the hyperbolic pieces of the JSJ-decomposition.

We give a presentation of a groupoid of transformations of quilt decompositions of a surface, which are pants decompositions of the surface with the addition of "seams". We consider colored quilt decompositions of a surface, where one side of a quilt decomposition is colored. We also give a presentation of a groupoid of transformations of colored quilt decompositions of a surface. These presentations of such groupoids are realized by constructing connected and simply-connected two-dimensional cell complexes whose 0-cell are these up-to-isotopy structures on the surface. In the latter part of this paper, we show that Torelli group (or more precisely, a torsion-free subgroup of a mapping class group of a surface) acts on these complexes freely.

Using the newly developed $3$d quantum trace map, we compile more evidence for the length conjecture, a refinement of the all-order volume conjecture that incorporates insertions of additional links in the skein module of the knot complement. In particular, we prove that the length conjecture holds up to first order for an infinite family of twist knots. Along the way, we form a conjecture that the $3$d quantum trace map behaves naturally with respect to Dehn filling.

An Euler circuit of a graph is a closed path that visits every edge of the graph exactly once. Euler circuit and circuit decomposition problems can also be formulated for higher dimensional simplicial complexes. An Euler k-circuit in K is a cyclic sequence of vertices v_1...v_n such that every k+1 adjacent terms { v_i,v_{i+1},...,v_{i+k} } (indexed modulo n) form a k-simplex, and every k-simplex of K appears exactly once in the sequence v_1v_2...v_n(v_1v_2...v_k). We investigate the 2-circuit decomposition problem for triangulated closed compact surfaces. For an orientable triangulated surface we use interior angles of paths to define an obstruction that lives in the first cohomology of the surface. It vanishes if and only if the surface has a 2-circuit decomposition. We also show that a non-orientable triangulated surface has a 2-circuit decomposition if and only if its orientable 2-fold cover does.

In this paper, we derive some new relative volume comparison theorems and Bishop-Gromov volume comparisons on Finsler metric measure manifolds, all of which are controlled by the integral weighted Ricci curvature. In particular, we establish a Bishop-Gromov volume comparison theorem for nonconcentric balls. Based on these, we prove a theorem of Bonnet-Myers type on Finsler metric measure manifolds with integral weighted Ricci curvature bounds.

We introduce the notion of Frobenius Alexander quandles associated with abelian varieties over finite fields, and study them from both knot-theoretic and arithmetic viewpoints. We determine exactly for which rational primes a knot admits a nonconstant coloring by these quandles, in terms of the resultant of its Alexander polynomial and the Frobenius characteristic polynomial of the abelian variety. This leads to the notion of persistent colorability and its characterization, as well as applications to fibered knots and knots of small genus. We also prove a rigidity result showing that the isomorphism class of a Frobenius Alexander quandle determines the similarity class of the Frobenius endomorphism on the corresponding torsion subgroup. As a consequence, one sufficiently large Frobenius Alexander quandle determines the characteristic polynomial of the Frobenius endomorphism, and hence, determines the isogeny class of the abelian variety over the given finite base field.

Let $X$ be a closed hyperbolic surface. We prove that the number of topological types of primitive closed geodesics of length at most $L$ is asymptotic to \[ \frac{1}{|\Isom(X)|}\frac{e^L}{2L}. \] as $L$ grows. Thus Huber's asymptotic remains unchanged after quotienting by topological type, up to the finite symmetry factor coming from the isometry group of $X$.

For punctured surfaces $\Sigma_{g,p}$ of genus $g\geqslant 2$, we explore the dynamics of the mapping class group action on the relative $\mathrm{PSL}(2,\mathbb{R})$-character varieties of type-preserving representations. For such relative character varieties, Goldman's conjecture predicts that the mapping class group acts ergodically on their non-Teichmüller components. A related question of Bowditch asks whether every non-elementary type-preserving representation that is non-Fuchsian sends some non-peripheral simple closed curve to a non-hyperbolic element. We show that, on the components of the relative character variety indexed by fixed signs of images of peripheral elements and relative Euler classes non-extremal in the generalized Milnor-Wood inequality, an affirmative answer to Bowditch's question implies Goldman's conjecture.

We study finite-volume convex hyperbolic $4$-polytopes whose dihedral angles are all equal to a fixed strictly acute angle $\alpha$. Put $\alpha_0=\arccos(1/3)$ and $\alpha_1=\arccos(1/4)$. We first show that the class is empty for $\alpha<\alpha_0$. For $\alpha_0\leq\alpha<\alpha_1$, the unique polytope of minimum volume is the regular hyperbolic $4$-simplex with dihedral angle $\alpha$. As $\alpha$ increases to $\alpha_1$, this simplex degenerates to a Euclidean one and no hyperbolic simplex exists at $\alpha_1$. For $\alpha_1\leq\alpha<\pi/2$, the unique minimum is the regular equiangular hyperbolic $4$-cube. The proof combines the hyperbolic Gram–Euler relation with the Davis–Okun theorem on the Charney–Davis inequality for flag triangulations of the $3$-sphere. The geometric step is a missing-face argument showing that the boundary of the dual of every nonsimplex in the class is flag.

Let $S$ be an oriented surface, possibly of infinite type, endowed with a complete Riemannian metric with pinched negative curvature. We prove that the area form defines a non-trivial class in the second bounded cohomology group of $S$, unless $S$ is diffeomorphic to the disc or the cylinder. This is in sharp contrast with the $n$-dimensional case, $n>2$, where the non-triviality of the volume form in bounded cohomology is related to the Cheeger constant of the manifold. We also discuss how the bounded area class depends on the metric: we prove that, for compact surfaces, it recognizes constant curvature metrics among pinched negatively curved ones, while, even in the case of surfaces of infinite type, it does not distinguish non-isometric hyperbolic structures. More precisely, when $S$ is compact we show that, among the negatively curved structures of fixed area, the ones with constant curvature provide the unique minimizers for the norm of the bounded area class.

We classify lens spaces for which the canonical negative-definite plumbing has minimal $b_2$ among all symplectic fillings of the standard contact structure. The classification is given by 4 infinite families of "forbidden subgraphs" in a manner similar to Aceto-McCoy-Park's work on smooth negative-definite fillings of lens spaces. In particular, no classification of this form can be given using a finite list of forbidden configurations, highlighting a difference between symplectic fillings and smooth negative-definite fillings of lens spaces.

This paper establishes necessary and sufficient conditions for locally flat knotted surfaces in simply-connected $4$-manifolds to be equivalent. For surfaces with knot group $\mathbb{Z}_d$, we extend results of Lee-Wilczynski from spheres to surfaces of arbitrary genus; the surfaces are permitted to be nonorientable and have boundary. We prove that most projective planes with knot group $\mathbb{Z}_2$ and the same Euler number are determined by the equivariant intersection form of their exterior. We also prove that knots with prime power determinants bound at most one Moebius band in $D^4$ with knot group $\mathbb{Z}_2$ and a given Euler number. Cancellation results lead to new criteria for homologous discs to be equivalent rel. boundary. Finally, we determine the topological extendable mapping class group of knotted surfaces with abelian knot group.

We present the knot polynomial associated to the 12-dimensional sporadic Nichols algebra $\mathrm{FK}_3$ with automorphism, compute it for all knots with $\leq$ 16 crossings and discuss the found patterns. This knot polynomial is not a Vassiliev power series. We ponder how it compares to the knot polynomials that come from Lie algebras and their representations and what the other knot polynomials associated with the higher dimensional sporadic Nichols algebras are.

We determine the torsion part of the higher smooth surgery structure sets of complex projective spaces (up to some extension problems) and complete the description of the forgetful map to their topological versions in low dimensions. In addition, an application to mapping class groups of complex projective spaces in low dimensions is presented.

On the once-punctured torus, we give an exact formula for the number of curves in any given mapping class group orbit of given word length. This settles a conjecture of Chas in [Cha16].

Lipshitz and Sarkar asked in an ICM 2018 question whether their homotopy-theoretic refinement of Khovanov homology can detect mirror reflection when ordinary integral Khovanov homology cannot. We give an affirmative answer by presenting a prime knot whose integral Khovanov homology agrees with that of its mirror in every bidegree, including torsion, and yet over $\mathbb{F}_2$, the second Steenrod square has rank one for the knot and rank zero for its mirror in a specified bidegree. We also give a composite example. The construction, search and computations are done with the help of chat-GTP 5.6 sol Pro combing over the Regina Census of knots.

Let $X$ be proper CAT$(-1)$ and let $\Gamma\le\Isom(X)$ be finitely generated and discrete. The sharp structural theorem states that, if $\Gamma$ is accessible over finite subgroups and $\delta_X(\Gamma)<1$, then $\Gamma$ is geometrically finite and virtually free, has a finite graph of groups with finite edge groups and virtually cyclic infinite vertex groups, and $\partial\Gamma\to\Lambda_\Gamma$ collapses exactly their conjugate two point boundaries. The result is hereditary, and below $1/2$ every finitely generated subgroup is convex-cobounded (\cref{thm:accessible-main}). Hence non-virtually-free accessible groups have $\delta_X(\Gamma)\ge1$ (\cref{cor:accessible-gap}). Consequences cover finitely presented groups, groups with uniformly bounded finite-subgroup orders, characteristic-zero linear groups, and Kleinian groups, also infinite parabolic-free Kleinian groups have finite-index classical Schottky subgroups (\cref{cor:accessibility-extension,cor:linear-groups,cor:kleinian-classical}). Hence we cover substantial larger class than \cite{LiuWang2023},\cite{Hou2001}, also see \cref{rem:strictness-sharpness}. Finally, we also state consequences for finite JSJ representatives and hierarchies (\cref{thm:JSJ,thm:hierarchy,cor:hierarchy-dimension}).

The multicrossing homology of a knot can be considered as a generalization of the homology of its fundamental quandle. We show that certain sums of knot secants define invariant classes in the multicrossing homology.

We survey computations and tools concerning the mapping class group of a compact, oriented, connected 3-manifold $M$. We provide a guide to the literature and sketch proofs for various families of irreducible and geometric 3-manifolds. We also consider JSJ and prime decompositions of 3-manifolds, and consequences for their mapping class groups.

We introduce combinatorial and diagrammatic methods for representing properly embedded symplectic surfaces in 4-dimensional Weinstein domains. We show that positive ascending surfaces, which include complex curves in Stein domains and multisections of Lefschetz fibrations, can be placed in bridge position with respect to Islambouli–Starkston's bisection-with-divides structure on the Weinstein domain. We algorithmically relate various decompositions of such surfaces, including transverse banded unlink diagrams, quasipositive factorizations, bridge bisections with divides, shadow diagrams (curves on surfaces), and pointed monodromy factorizations. We also develop a new way to present branched covers of Weinstein domains along positive ascending surfaces, which, combined with work of Loi–Piergallini, recovers Islambouli–Starkston's result that every compact Weinstein domain admits a bisection with divides.

We relate the dynamics on a closed 3-manifold to the topological invariant, homology guts, proposed by Agol and Zhang. Given a closed manifold $M$ that admits a pseudo-Anosov flow $\varphi$ without perfect fits, we construct a canonical semiflow on the homology guts associated to homology classes carried by $\varphi$. We show that, up to orbit equivalent outside half annuli, the semiflow on the guts is invariant on each Thurston open cone. As an application, the fundamental group and the sutured structure of homology guts encode closed orbits of $\varphi$ with particular type. Besides, for a positive class in $H^{1}(M)$, the canonical semiflow on the guts has a well-defined growth rate.

We extend the diagrammatic criterion for $(1,1)$ L-space knots by Greene, Lewallen, and Vafaee to $(1,1)$-knots in $S^1\times S^2$. We also discuss applications to L-space surgeries on two-bridge links.