Papers

Let $M_f$ be the closed hyperbolic $3$–manifold which is the mapping torus of a pseudo-Anosov homeomorphism $f$ of a closed surface $S$ of genus at least two. The inclusion of the fiber subgroup \(\pi_1(S)\hookrightarrow \pi_1(M_f)\) has a Cannon–Thurston boundary map \(\partial\pi_1(S)\cong S^1\to S^2\cong\partial\pi_1(M_f)\), which is a Peano curve. We relate this map to two variational representatives of the fiber homotopy class: the harmonic map of Eells–Sampson and Hartman, and a least-area embedded minimal surface. Equivariant lifts of homotopic maps are a bounded distance apart; hence, for every marked conformal structure on $S$, the lift of the harmonic representative extends continuously to the closed disk with boundary value the Cannon–Thurston map, and a least-area fiber lifts to a properly embedded minimal plane in $\Hh^3$ with the same boundary map. We give area and curvature bounds for a least-area fiber, show that its largest principal curvature is at least one, and study the energy of the harmonic representatives as a function on Teichmüller space, its critical points, which are the zeros of the Hopf differential, and its invariance under the monodromy.

Let $n \geq 4$ and $\rho: \pi_1S \rightarrow PSL(n,\mathbb{R})$ be a Hitchin representation. We study the topology of a cocompact domain of discontinuity $\Omega_{\rho}$ in the flag manifold $\mathcal{F}_{1,n-1}$ of line-hyperplane pairs in $\mathbb{R}^n$ defined by Guichard-Wienhard. In particular, we lift $\Omega$ to its $(\mathbb{Z}_2\times \mathbb{Z}_2)$-cover $\hat{\Omega}$ in the Stiefel manifold $V_2(\mathbb{R}^n)$. The domain $\hat{\Omega}$ is highly connected and smoothly fibers over hyperbolic space $\mathbb{H}^2$ with unknown fiber $\hat{\mathfrak{F}}$, a closed $(2n-5)$-manifold. We determine the homeomorphism type of $\hat{\mathfrak{F}}$ as well as its diffeomorphism type up to connected sum with a homotopy sphere. The classification of $\hat{\mathfrak{F}}$ involves computing both its homology and a certain differential topology invariant needed to invoke Wall's classification of highly connected odd-dimensional manifolds.

We study the profinite non-rigidity of arithmetic lattices and its implications for the Kähler property. In the first part, we characterize the absolute Dynkin types that admit non-isomorphic real forms of higher-rank Lie groups containing torsion-free arithmetic lattices with isomorphic profinite completions. In the second part, as a geometric application, we answer a question asked independently by Arapura and Libgober, by showing that the Kählerness of finitely presented, residually finite groups is not determined by its profinite completion.

The goal of this paper is to clarify the relation between the Kerler-Lyubashenko TQFT $J_\mathcal{E}$ associated with the adjoint end $\mathcal{E}$ of a (not necessarily semisimple) modular category $\mathcal{C}$ and its renormalized version $V_\mathcal{E}$ based on the modified trace supported by the ideal $\mathrm{Proj}(\mathcal{C})$ of projective objects of $\mathcal{C}$. More precisely, we construct a $3$-dimensional ETQFT $\boldsymbol{\hat{A}}_\mathcal{E}$ that contains both $J_\mathcal{E}$ and $V_\mathcal{E}$. The ETQFT $\boldsymbol{\hat{A}}_\mathcal{E}$ is given by a $2$-functor whose source is the $2$-category of admissible $3$-dimensional cobordisms, and whose target is the $2$-category of finitely complete linear categories. We improve on earlier versions of the construction by dropping the admissibility condition for surfaces. By doing so, we obtain an ETQFT whose circle category $\boldsymbol{\hat{A}}_\mathcal{E}(\boldsymbol{S}^1)$ is equivalent to the category $\mathcal{C}$, as opposed to the ideal $\mathrm{Proj}(\mathcal{C})$.

This paper studies Veech groups of dilation surfaces whose fundamental group is not finitely generated. We prove that if $ S$ is a surface with self-similar end space and every end is accumulated by genus, then every countable subgroup of $SL(2,\mathbb{R}) $ can be realized as the Veech group of a dilation surface homeomorphic to $ S$. Additionally, we can construct this dilation surface such that it realizes horizon saddle connections in a prescribed set of directions. This contrasts with the case of closed dilation surfaces, where horizon saddle connections impose restrictions on the algebraic structure of the Veech group.

We give an explicit quasi-isometry from the flip graph of triangulations $\mathscr{F}_{g,n}$ to the $\epsilon$-thick part of Teichmüller space equipped with the Teichmüller metric, by mapping each ideal triangulation to the hyperbolic surface whose shearing coordinates along that triangulation are all equal to zero. Extending this construction, we obtain a quasi-isometry $Q$ from the hexagon graph $\mathscr{H}_{g,n}$ to the augmented Teichmüller space $\overline{\operatorname{Teich}}_{g,n}$ equipped with the Weil-Petersson metric, and we estimate its width, that is, the Hausdorff distance between $Q(\mathscr{H}_{g,n})$ and $\overline{\operatorname{Teich}}_{g,n}$. By bounding the Weil-Petersson distance along grafting rays and the Teichmüller distance along shearing deformations, we show that the width is at most $\sqrt{g+n}\log(g+n)$. We also provide an explicit projection from any point in $\overline{\operatorname{Teich}}_{g,n}$ to the thick part of boundary strata, maintaining simultaneous control over the Weil-Petersson distance and the shearing coordinates.

Scientific data span heterogeneous structures, including coordinates, networks, surfaces, volumes and fields, yet their topology can be quantified within a common framework through graph connectivity, cycle structure, genus and spatial embedding. Graph- and homology-based summaries do not determine spatial embedding, while standard knot and link polynomials require extensions to accommodate branching graphs. Here, we introduce KnottedGraph, a computational framework that converts such scientific representations to knotted graphs that retain graph connectivity and spatial embedding together. It constructs projected diagrams and PD codes, enabling various topological analyses, including Yamada-polynomial evaluation for topological classification. For scalable exact evaluation, it combines partial resolutions that leave the same unresolved connections and optimizes their processing order; the resulting algorithm is verified against published topological invariants of knotted graphs with up to 500 crossings. This scalability enables us to introduce an LLM-assisted mathematical-discovery methodology, in which computational topological data generated across knotted-graph families are used to identify candidate closed-form formulas. With this approach, we identify analytical Yamada-polynomials for generic graph motif families exhibiting Abelian and non-Abelian word sequences. Together, these scalable capabilities make knotted-graph topology computationally accessible across scientific domains, enabling large-scale classification and introducing a route from topological data to LLM-assisted AI4Math discovery.

We study representations of compact oriented surface groups into $\mathrm U(p)$ with elliptic-unipotent boundary holonomies, meaning that no boundary holonomy has eigenvalue $1$. We prove that the signature of the associated flat Hermitian bundle completely determines the connected component, and that every component is path-connected. For genus $g\geq1$ and $n\geq1$ boundary components, there are $np-1$ components; for $g=0$ and $n\geq2$, there are $(n-2)p+1$. The disk case is empty, while the closed-surface representation spaces are connected. The same component classification holds after taking the quotient by conjugation. Our proofs use the boundary rho invariant and explicit matrix deformations. For a three-holed sphere, the representation components have the homotopy types of complex Grassmannians, and their conjugation quotients are contractible.

We introduce a notion of topological complexity of a group with respect to a family of subgroups, with the case of the trivial and diagonal families recovering the classical category and topological complexity, respectively. These invariants come equipped with a battery of upper and lower bounds derived from functoriality and Bredon cohomology. We introduce a new family of subgroups, the permutational family, and show that the corresponding invariant is a lower bound for the distributional topological complexity of Dranishnikov-Jauhari and Knudsen-Weinberger. As a first application, we show that the permutational and diagonal families coincide, and thus that classical and distributional topological complexity are equal, for a large class of torsion-free groups, extending work of Dranishnikov. Second, we show that the lower bounds of Grant-Lupton-Oprea are in fact lower bounds on distributional topological complexity; in particular, it follows that Farber's conjecture holds distributionally.

Let $S$ be a closed translation surface of genus $g\ge 2$ with $\mathop{area}(S)= 4\pi g$. We show that for any $\lambda \in (0, 1)$ there exist $\lfloor \lambda \cdot g\rfloor$ homologically independent simple closed curves of length at most $ C(\lambda) \cdot \log(g)$, where $C(\lambda)$ is a constant that depends only on $\lambda$. This result is obtained from a mixed graph construction based on a Voronoi graph of the surface with the cone points as seeds and its dual graph. We also give a complementary result using only the Voronoi graph that produces $2g$ short loops that form a homology basis. This construction, however, depends on the length of a shortest saddle connection of the surface $S$.

We construct a multi-filtered smooth knot concordance invariant, $\beta(t^1, a, t^2)$, derived from the knot Floer complex $CFK^\infty$. $\beta$ is a three-variable piecewise linear function defined on $(0,2) \times \mathbb{R} \times (0,2)$. We demonstrate that $\beta$ satisfies a subadditivity relation. We apply $\beta$ to prove concordance results for a certain family of L-space knots. We show that the positive cones spanned by Teragaito's knots $K_{1,n}$ and $K_{2,n}$ are disjoint in the smooth concordance group $\mathcal{C}$. For positive L-space knots, we show $\beta$ completely determines the Alexander polynomial and distinguishes knots with identical $\Upsilon$ invariants.

We study planar openbooks obtained by lifting braids through branched covers of D^2, together with the quantum operations in the Ising representation. We give a criterion for the Stein fillings to be Calabi-Yau (CY). Among positive factorizations of a fixed monodromy, a CY one has minimal length and its filling minimizes \chi and b_2. Applied to Baykur's recent examples, this gives a planar contact 3-manifold with infinitely many non-homeomorphic CY fillings, the cover there having degree \ge 6. At degree 4 a single CY filling forces every filling to be CY, and in one case the filling is unique; at degree \le 3 it is unique, and CY under a mild condition. Admissible cuts of D^2 decompose the openbook into subopenbooks. Every positive factorization then localizes, and the CY condition holds exactly when it holds locally. The state space is then a direct sum of tensor products indexed by the compatible parity choices, with at most two qubits per factor when the pieces have degree $\le 4$, the same range in which the CY condition depends only on the monodromy. For one degree 4 cover, the points, lines and flags of the two-qubit doily are realized by its subopenbooks. Fifteen liftable braids there share the same quantum operation and Stein filling, and are separated only by the subopenbook systems they admit. A choice of tensor-product structure is thus carried by the lift, not by the braid group representation, suggesting a link between contact topology and quantum entanglement.

We introduce a new structural framework for constructing multivariate Alexander quandles based on a groupoid ${\cal G}$, composed of a disjoint union of delooping groupoids of deck groups. Compared to standard Alexander modules over group rings ${\mathbb{Z}}[{\mathbb{Z}}]$, the multivariate version involves modules over groupoid rings ${\mathbb{Z}}[{\cal G}]$. A manifestly categorical description is given in terms of ${\mathbb{Z}}$-Algebroids, whose functorial module theory yields an equivalent definition of multivariate Alexander modules. Using the category of elements construction, we define multivariate Alexander quandles from this groupoid framework. Three classes of multivariate quandle operations follow. In contrast to earlier approaches, we abstract the algebraic structure from explicit link-dependence. As a consequence, we have a structural framework for: (i) enumerating new classes of Alexander quandles, and (ii) finding new coloring invariants of links. We show that maps between ${\mathbb{Z}}$-modules, facilitated via multivariate quandle operations, realize a quiver presentation of quandles, which makes manifest the oidification of Alexander modules. Oidified Alexander modules, based on groupoids, provide a constructive framework for composing univariate quandles to obtain new multivariate ones. Finally, we comment on the possibility of new link invariants coming from groupoid-based formulations, including the fundamental groupoid.

We import the theory of $R$-links and derivative links into contact topology in both the Legendrian and transverse setting. This framework is used to characterize various forms of Lagrangian and symplectic sliceness, and more generally establishes one approach to what we call the Lagrangian slice-ribbon conjecture. Our main theorem asserts that a Legendrian knot with Thurston-Bennequin invariant $-1$ is Lagrangrian slice (resp. regularly Lagrangian slice) if and only if it supports a tight transverse (resp. Legendrian) $R$-link derivative.

A meander is a configuration of two simple plane curves intersecting transversely. The orders of their intersection points define a permutation that determines the configuration. When tangencies are allowed, however, different configurations can share the same permutation. We study the combinatorial and algebraic structures arising from this non-uniqueness. We give a realization criterion and show that the realizations of each realizable permutation form an affine space over the two-element field. We describe this space using an associated graph, called the component spine. We prove that the component spine of every permutation is a cactus. We also introduce the tangency polynomial, which counts realizations by their number of tangencies, investigate its properties, and prove that it factors over the cycles and bridges of the component spine. We derive a central limit theorem for tangency counts and obtain asymptotic formulas for the number of distinct tangency polynomials.

We show that the evaluations of combinatorial 1-cocycles defined by Gauss diagrams with a triangle on the canonical loops in the space of long knots are finite type invariants. Using Gauss diagrams, we construct two $\mathbb{Z}$-valued combinatorial 1-cocycles $\beta_1, \beta_2$ and a $\mathbb{Z}/2\mathbb{Z}$-valued 1-cocycle $\beta_3$ on the space of long knots and prove their cocyclicity by verifying their invariance under higher Reidemeister moves coming from the codimension-two singularities of plane curves. We show that they represent genuinely new 1-cohomology classes and compute their pairings with the rotation, rolling, half rolling, bracket and half bracket loops. A key new feature is that $\beta_1$ and $\beta_2$ can pair nontrivially with bracket and half-bracket loops. We conjecture that $(\alpha_3^1,\beta_1,\beta_2)$ over $\mathbb{Q}$, and their mod 2 reductions together with $\beta_3$ over $\mathbb{Z}/2\mathbb{Z}$, form bases of degree-one cohomology up to order 4 in the sense of Vassiliev. In addition, we show that the reparametrization loop is homotopic to the rolling loop concatenated with the rotation loop in $\operatorname{Emb}(S^1, S^3)$. Finally, we give the criteria for a 1-cohomology class in the long knot space to descend to 1-cohomology classes in $\operatorname{Emb}(S^1, S^3)$ and $\operatorname{Emb}(S^1, S^3)/\operatorname{Diff}^{+}(S^1)$. Using these criteria, we show that $\beta_1$ descends to a nontrivial 1-cohomology class in $\operatorname{Emb}(S^1,S^3)$, while $\beta_1 \bmod 2$ and $\beta_3$ descend to linearly independent nontrivial 1-cohomology classes in $\operatorname{Emb}(S^1, S^3)/\operatorname{Diff}^{+}(S^1)$.

This paper shows that one internal stabilization is enough to resolve all the self-referential tubes on an embedding. As an application, this paper gives a complete isotopy classification for embedded surfaces with a common geometric dual in orientable $S^2$-bundle over a surface, provided that the embedded surfaces have greater genus than the base surface of the bundle.

We study smooth fibred two-knots whose fundamental quandle is the prism quandle of order fifty-six. The candidates arise from framed circle surgery along a section of a mapping torus, using the two possible normal-framing classes. We prove that both ambient homotopy four-spheres are diffeomorphic to the standard four-sphere. The candidates have the same knot group and diffeomorphic exteriors, but they are inequivalent even as topological pairs. Every smooth fibred realization of this quandle in a homotopy four-sphere is equivalent, up to knot orientation, to exactly one of these two knots. Standardness follows from a Whitehead-link model for the orbit orbifold together with circle-action recognition. Inequivalence is detected by the third cyclic branched covers, on which a spin-bordism secondary invariant takes different values. Thus the parameter-seven prism quandle admits smooth fibred realizations in the standard four-sphere.

We show that non-Fuchsian geometrically finite hyperbolic structures on a given hyperbolizable 3-manifold M are uniquely determined, up to isotopy, by their bending laminations. Consequently, the bending map from the space of non-Fuchsian geometrically finite structures on M, endowed with the strong topology, to the space of bending laminations, endowed with Lecuire's tubular topology, is a homeomorphism. This extends the convex co-compact case established with Schlenker. The proof combines hyperbolic Dehn filling, continuity and properness of the bending map (Lecuire) and real-analyticity of its fibres (Bonahon). We also establish a criterion for contractibility of the boundary fibres of a continuous extension of a homeomorphism, extending Finney's theorem to a boundary setting. This topological result may be of independent interest.

Building on recent work of Hom and Wan, we construct additional examples of exotic knot traces, providing partial progress toward their conjecture. Our approach uses the immersed-curve techniques in Heegaard Floer homology developed by Chen and Hanselman.

Motivated by the study of asymptotics of quantum invariants, Dimofte and Garoufalidis introduced a power series associated to a suitable ideal triangulation of a cusped hyperbolic $3$-manifold. They proved that its constant term can be written in terms of Neumann-Zagier data and the complex shape parameters of ideal tetrahedra, and conjectured that it equals the adjoint twisted Reidemeister torsion. On the other hand, in the study of asymptotics of Turaev-Viro type invariants of cusped $ 3$-manifolds, the authors, together with Liu, Sun and Yang, found that the one-loop terms of their asymptotic expansions could be written in terms of Gram matrices and decorated edge lengths of ideal tetrahedra. In this paper, we prove the conjecture of Dimofte and Garoufalidis and relate the one-loop term appearing in the Turaev-Viro type invariant to the adjoint twisted Reidemeister torsion.

In this paper, we conduct a systematic study of quandle colorings and quandle coloring quivers for pretzel links using the dihedral quandle $\mathbb{Z}_{n}$. First, we systematically investigate all possible colorings of 3-pretzel links, determining the number of distinct colorings in each case as well as the structure of their quandle coloring quivers. In order to obtain more general conclusions, we impose restrictions on $n$ based on the properties of the coefficient matrix of the system of congruence equations. So we examine the number of quandle colorings and the quandle coloring quivers for 4-pretzel links in the case where $n$ is prime. Finally, Combining the results of 4-pretzel links we rigorously derive both the coloring numbers and quandle coloring quivers for general $m$-pretzel links in the case where $n$ is prime, with full proofs provided.

Let $K$ be a hyperbolic knot and let $\rho_n$ be the $n$-dimensional irreducible representation induced from a lift of its holonomy representation. Motivated by Goda's asymptotic volume formula and the complexified Volume Conjecture, we study whether the higher-dimensional twisted Alexander invariants associated with $\rho_n$ detect the complex volume of the knot complement. We compute $$ \frac{\pi}{2} \log \left( \frac{A_{K,n-2}(1)A_{K,n+2}(1)} {A_{K,n}(1)^2} \right) $$ for all hyperbolic knots with at most six crossings. Our numerical experiments indicate that these values approach $$ \operatorname{Vol}(S^3\setminus K) +i\,2\pi^2\operatorname{CS}(S^3\setminus K) $$ modulo $i\pi^2\mathbb{Z}$. Based on these computations, we propose a complexified analogue of Goda's asymptotic volume formula.

Links in projective space $\mathbb{R}P^3$ can be represented by diagrams in $\mathbb{R}P^2$ where the projective plane $\mathbb{R}P^2$ is represented by a disk with antipodal identifications on the boundary. In this context if we place an $n-$strand braid $\beta$ on this disk keeping its end points on the boundary then due to the identification of antipodal points it naturally represents a link diagram in $\mathbb{R}P^3$. We call this the projective closure of the braid $\beta$. In this paper we show that not all links in $\mathbb{R}P^3$ possess a diagram represented by projective closure of some braid.

We prove that a lattice $\Gamma$ in $\mathrm{PSL}_2(\mathbb{R})$ or $\mathrm{PSL}_2(\mathbb C)$ has positive trace gap, meaning that its traces are uniformly separated, if and only if it is derived from an admissible quaternion algebra. For cocompact Fuchsian groups, this proves the positive trace gap conjecture attributed to Sarnak by Geninska and Leuzinger in 2008. The same characterization by quaternion algebras holds if the difference set of traces is not dense. Our method also gives a similar result for lattices in $\mathrm{SL}_d(\mathbb R)$, for every $d\ge3$: a lattice $\Gamma<\mathrm{SL}_d(\mathbb R)$ has positive trace gap if and only if all its traces are integers. Equivalently, after conjugation, it has finite index in the norm-one group of an order in a central simple algebra of degree $d$ over $\mathbb Q$ that splits over $\mathbb R$. We also discuss spectral consequences and other applications of the results and techniques.

The first examples of asymmetric L-space knots were found by Baker and Luecke. Among Baker-Luecke knots, the simplest one has braid index 12. Later, it turned out that there are just 9 asymmetric hyperbolic L-space knots in the SnapPy census, and their braid indices take the values 4, 5, 6 and 7. It is known that asymmetric L-space knots have braid index at least 4. Recently, Baker and the second author give infinitely many asymmetric hyperbolic L-space knots with braid index 4. In this paper, we construct an asymmetric hyperbolic L-space knot with arbitrary braid index bigger than 4. In fact, there exist infinitely many such knots for each braid index.

For a surface with an odd number of punctures, the moduli space of flat $SU(2)$ connections with traceless holonomy around each puncture is a symplectic manifold. When the moduli space is nonempty, there is a natural homomorphism from the mapping class group of the punctured surface to the symplectic mapping class group of this moduli space. It is shown that this homomorphism is injective if and only if the dimension of the moduli space is greater than $2$. This generalizes work of Seidel and Wehrheim–Woodward. Also given is a complete classification of Lagrangian spheres in the projective plane blown up at $5$ points with its monotone symplectic structure, which is the moduli space for the 5-punctured sphere. Furthermore, it is determined when two such Lagrangian spheres can be displaced by a symplectic isotopy. Results are also obtained regarding Lagrangian spheres in the intersection of two quadrics in $\mathbb{C}\mathbb{P}^5$. The proofs involve instanton Floer theory and results on Heegaard splittings. A main technical result establishes the approximation of any Hamiltonian isotopy of the $SU(2)$ moduli space by holonomy perturbations which are used in instanton homology.

We prove that topological isotopy implies smooth isotopy for neat surfaces with common nonempty boundary in a smooth orientable 4-manifold, in the presence of a geometric dual in the boundary. Together with our earlier work on surface smoothing, this shows that the natural map from smooth isotopy classes to topological isotopy classes of such surfaces with prescribed boundary is bijective.

This is an English translation of a book originally published in Japanese by Saiensu-sha in 2015. The book is an introduction to the 3d-3d correspondence, the relation between 3d $\mathcal{N}=2$ supersymmetric gauge theories and the geometry of 3-manifolds, which arises from the compactification of the 6d $\mathcal{N}=(2,0)$ theory. The presentation is bottom-up rather than top-down: instead of starting from the 6d theory, we begin by asking what a quantum field theory is, and let the geometry emerge on its own. Along the way we discuss renormalization and low-energy effective theories, gauging as an operation which glues field theories together, the resulting $Sp(2n, \mathbb{Z})$-action on 3d theories, 3d $\mathcal{N}=2$ supersymmetric theories and their dualities, the squashed three-sphere partition function and supersymmetric localization, duality domain walls of 4d $\mathcal{N}=2$ theories, quantum Teichmuller theory, complex Chern-Simons theory and the quantization of the moduli space of flat $SL(2, \mathbb{C})$-connections on 3-manifolds, with quantum dilogarithm functions as a recurring thread. The emphasis throughout is on geometric and algebraic structures which are invisible in a single field theory, and appear only in the theory space of quantum field theories. Appendices summarize supersymmetry in various dimensions, classical and quantum dilogarithm functions, and cluster algebras. Exercises with difficulty ratings are included in each chapter.

For each $g\ge 2$, we show that there are finitely many cyclic commensurability classes of closed, fibered arithmetic hyperbolic $3$-manifolds with genus $g$ fiber. More generally, we show that there are only finitely many conjugacy classes of admissible surface subgroups of PSL$_2(\mathbb{C})$ whose image is contained in the fundamental group of some arithmetic hyperbolic $3$-manifold. We also give an effectively computable upper bound with effective asymptotic rate on both finiteness statements. This affirms a conjecture of Bowditch, Maclachlan, and Reid.