Papers

In this paper we introduce a generalization of Helly's theorem closely connected to Bárány-Gromov overlap theorems (also called selection lemmas). Our main result implies both the topological colorful Helly of Kalai and Meschulam and Karasev's topological centerpoint theorem. We further investigate the topological fractional Helly theorem from this overlap perspective, and show an overlap theorem for dense complexes (a continuous second selection lemma for tame maps).

For any two links in the $3$-sphere, we provide a visual construction of a stable map $f$ from the $3$-sphere to the Euclidean plane such that $f$ has no cusp points, the set of definite (resp. indefinite) fold points of $f$ is isotopic to the first (resp. second) given link, and $f$ does not have a certain type of singular fibers. Moreover, we obtain a similar result for any $3$-manifold by setting the target space to the $2$-sphere.

We determine $2,540$ unknotting numbers of prime knots with at most $13$ crossings, that are unknown in the KnotInfo snapshot of 9 September 2026. The lower-bound calculations use Heegaard Floer correction-term obstructions and Greene's spanning-tree model. We also give explicit crossing-change constructions for upper bounds. For $959$ alternating knots with range $[2,3]$, we verify that no crossing change in a minimal diagram gives a knot of unknotting number one. We determine the unknotting numbers of all four knots in Brittenham and Hermiller's construction and thereby identify 13n3370 as an explicit counterexample to the original Bernhard-Jablan conjecture.

Under explicit hypotheses on the spectral Looijenga construction, we compute the hyperbolic-plane value of the zero-section invariant of Gukov, Krushkal, Meier, and Pei in periodic topological modular forms. The value is the Hopf element eta, and adjoining a hyperbolic plane acts by multiplication by this element. Earlier work obtains the hyperbolic value under an additional cobordism-duality assumption; here we derive it directly from the Looijenga restriction maps without that assumption. Together with annihilation and vanishing results for definite lattices, this gives a complete value formula on smooth closed simply connected spin four-manifolds. In particular, nonzero signature forces the invariant to vanish, so both orientations of a K3 surface have value zero.

Starting from the known differential expansions, we express the HOMFLY-PT polynomials of twist and antiparallel double-braid knots in rectangular representations \([r^s]\) as determinants of \(r\times r\) and \(s\times s\) matrices. We first give an elementary derivation for the figure-eight knot and then re-sum the KNTZ formulas for twist and double-braid evolution. The resulting matrices are constructed from single-hook evolution coefficients and explicit representation-dependent weights. These formulas provide a starting point for investigating possible extensions of knot polynomials to KP/Toda \(\tau\)-functions. Extensions to non-rectangular representations and to more general knot families remain open problems.

Jones described how the Ising and Potts models from statistical mechanics give rise, with appropriate choices of Boltzmann weights, to invariants of an oriented link. The Boltzmann weights that Jones proposed, however, work only with a correction factor that he does not mention. We fill in the missing details in two different ways. We also show that all of these invariants, which we call the Ising and Potts invariants, are given by evaluating the Jones polynomial at $t$ equal to one of these Boltzmann weights.

In this paper, we give the dual characterization of conformal tetrahedra in constant curvature spaces, which develops a geometric technique to obtain the global rigidity of (ideal) hyperbolic, Euclidean, spherical sphere packings on a tetrahedron. Moreover, we introduce ideal hyperbolic sphere packings and extend the variational principle to obtain the global rigidity of (ideal) hyperbolic sphere packings on 3-manifolds. We also provide a characterization of conformal $n(\ge3)$-simplices in constant curvature spaces, which is useful for studying higher-dimensional sphere packings.

We give conditions ensuring that an asymptotically rigid mapping class group, specifically a surface Houghton group $\mathcal{H}(S)$, has finite index in the asymptotically conformal modular group $\text{Mod}_0(X)$, where $X$ is a hyperbolic structure on $S$. These include geometric conditions on the pieces of the underlying rigid structure, as well as the existence in $\mathcal{H}(S)$ of an end-periodic homeomorphism which is asymptotically conformal. As a consequence, if $S$ has $n\geq 3$ ends, then $\text{Mod}_0(X)$ has type $F_{n-1}$ but not $FP_n$. We also establish analogous results for $L^p$ modular groups.

We study which compact flat manifolds of dimensions $3 \leq n \leq 6$ occur as cusp cross-sections in commensurability classes of non-compact arithmetic hyperbolic $(n+1)$-manifolds. Using the classification of low-dimensional flat manifolds together with the relationship between their holonomy representations and rational quadratic forms, we determine the possible arithmetic commensurability classes for every flat manifold in these dimensions. We show that every flat $n$-manifold for $n=3,4,5$ occurs as a cusp cross-section in infinitely many distinct arithmetic commensurability classes. In dimension six, the same holds with exactly eight exceptions: eight non-orientable flat $6$-manifolds occur in a unique arithmetic commensurability class. We also show that, for every $3 \leq n \leq 6$, there is a single commensurability class of arithmetic hyperbolic $(n+1)$-manifolds containing every flat $n$-manifold as a cusp cross-section.

We define a construction of Legendrians inside contact manifolds that arise by doubling an exact Lagrangian filling in the page of an open book decomposition. This can be seen as a generalization of a previous construction by Courte and Ekholm to arbitrary open books. These Legendrians, called Legendrian doubles, are shown to admit regular flexible exact Lagrangian fillings, and they are thus classified up to Legendrian isotopy by classical data. Finally, we show that the Legendrian suspension construction, as defined by Arikan and the author in previous work,-this is a Legendrian contained inside a page of an open book that is obtained by using Seidel's suspension of Lefschetz fibrations- is a Legendrian double.

A knot invariant is a fingerprint shared by equivalent knots: unequal values certify inequivalence, while equal values may conceal different knots. Under two rooted random-diagram models, we prove that the incomplete normalized Alexander polynomial separates random pairs but collides in a growing database. If $D_n$ and $D_n'$ are independent $n$-crossing diagrams, then $\Pr\{\Delta_{K(D_n)}=\Delta_{K(D_n')}\}=O(n^{-1/2})$, and every fixed normalized Alexander polynomial is exponentially rare. With exponentially high probability, the diagram shadow contains linearly many disjoint opened-trefoil slots. Conditional on the shadow and exterior crossing signs, their indicators are independent Bernoulli$(1/4)$ variables whose sum is a binomial coordinate in the $3$-adic determinant valuation. Consequently, $\sup_{a\geq 1}\Pr\{\det K(D_n)=a\}=O(n^{-1/2})$. For a discrete invariant $I$, let $\alpha_n(I)=\Pr\{I(D_n)=I(D_n')\}$. Its collision entropy is $H_2(I(D_n))=-\log\alpha_n(I)$. An independent sample of size $M$ has $\binom{M}{2}\alpha_n(I)$ colliding pairs on average and birthday scale $\alpha_n(I)^{-1/2}$. If $M_n^2\alpha_n(I)\to\infty$, a repeated value occurs with probability tending to one even when $\alpha_n(I)\to0$. For the determinant, $M=o(n^{1/4})$ suffices for collision freedom with high probability; a matching lower bound is open. We also give exact finite-population formulas for fixed censuses, calculate the expected cost of invariant cascades, and relate collision probability to the frequency of calls to a complete equivalence procedure. On a balanced pair-classification benchmark, the normalized-Alexander rule has balanced accuracy $1-O(n^{-1/2})$ but cannot distinguish inequivalent pairs in one fiber.

Let $K\subset S^3$ be a non-trivial knot and let $\mathrm{IS}(K)$ be the simplicial complex whose vertices are the ambient isotopy classes of incompressible Seifert surfaces in the exterior $E(K)$, a finite set of distinct vertices spanning a simplex exactly when its classes admit simultaneously pairwise disjoint representatives. Kakimizu proved that $\mathrm{IS}(K)$ is connected; whether it is contractible was asked by Przytycki and Schultens, who also identified the obstruction, namely that the projection used in the connectedness proof is not known to be well defined on isotopy classes. We prove that $\mathrm{IS}(K)$ is contractible. We also prove contractibility of every genus truncation $\mathrm{IS}_\ell(K)$ with $\ell\ge g(K)$ and every non-empty lexicographic complexity sublevel. The argument factors through an unconditional combinatorial theorem: every non-empty connected flag exchange complex is contractible, where an exchange complex carries a complexity function subject to two axioms which require only that an exchanging vertex exist, never that a coherent selection rule be supplied. The combinatorial proof uses hereditary descending links and transfinite induction. We compare its attachment step with existing Morse criteria, including a formulation applicable to every countable exchange complex, and distinguish the given exchange order from a dismantling order. The geometric argument also applies to links satisfying a linking condition that forces every spanning surface to be connected. For the geometric input we prove fixed-boundary area attainment among smooth neat embeddings, using smooth convex replacement domains, area-controlled disc cleaning and boundary-preserving smoothing. All area complexities use this smooth competing class. No interpretation of an undefined piecewise smooth existence class is needed.

We develop a framework for studying discrete subgroups of $\mathsf{PGL}(d+1,\mathbb{R})$ via the paracomplex hyperbolic space $\mathbb{H}_\tau^d$, a rank-$1$ pseudo-Riemannian symmetric space. We characterize projective transverse, relatively Anosov, and Anosov subgroups in terms of properly discontinuous, geometrically finite, and convex-cocompact actions respectively on their weak hulls, which are canonical flow spaces in the spacelike unit tangent bundle of $\mathbb{H}_\tau^d$. A key ingredient is the construction of a Busemann-type horofunction on the spacelike unit tangent bundle with the properties needed to describe cuspidal geometry. We further prove for relatively Anosov subgroups that the geodesic flows on their weak hulls are uniformly hyperbolic, giving a relative analogue of the Axiom A property.

For a crossed module $\chi: G \to H$, we introduce the notion of $\chi$-crossed braided (resp. ribbon) categories, where the categories are graded by group $G$ and carry an $H$-action. Our definition unifies and generalises several familiar notions: taking $\chi = id: G \to G$ with the conjugation action recovers $G$-crossed braided categories; taking $\chi: G \to \{*\}$ for abelian $G$ yields $G$-graded braided categories; taking $\chi: \{*\} \to G$ leads to braided categories equipped with a $G$-action. The equivalence relation between $\chi$-crossed braided categories is typically finer than that between $G$-crossed braided ones. We classify $\chi$-crossed braided structures on the category of $G$-graded vector spaces in terms of cohomological data, and give explicit examples for cyclic groups. Given a doubly central algebra with $G$- and $H$-actions in a braided monoidal category, we define a notion of twisted-local modules and show how they give rise to $\chi$-crossed braided categories. We furthermore give sufficient conditions so that these categories are additionally $\chi$-crossed ribbon or admit an orthogonal $G$-decomposition.

As is already known, there is a correspondence between circle patterns and the dimer model, which is invariant under Euclidean transformations. We establish a new correspondence that is invariant under the group of Möbius transformations PO(3,1), which is the natural symmetry group of circle patterns. We show that Miquel dynamics preserve the partition function, and that convexity of the t-embedding guarantees positivity of the face weights. We also show that the correspondence has a somewhat hidden symmetry group PO(3,3), which relates to the Lorentz lift of the t-embedding. Finally, we consider isoradial graphs, Doyle spirals, the once-punctured torus and isothermic circle patterns.

We prove a Legendrian Reidemeister theorem for Legendrian knots in thickened convex surfaces.

Let $\Sigma$ be a closed oriented surface of genus $>1$ and $M$ a complete hyperbolic 3-manifold with a marking $i:\Sigma\longrightarrow M$. We consider the case that $M$ has no parabolic cusps and at least one of the two ends is simply degenerate. For $\varGamma=\pi_1(\Sigma)$, let $\rho_M:\varGamma\longrightarrow \mathrm{PSL}_2(\mathbb{C})$ be the holonomy of $M$ and $\rho:\varGamma\longrightarrow \mathrm{PSL}_2(\mathbb{C})$ any representation in $\mathrm{PSL}_2(\mathbb{C})$. We will show that, if $\rho$ is discrete and non-faithful, then \[ \|[\mathrm{Vol}(\rho)]-[\mathrm{Vol}(\rho_M)]\|_\infty\geq \boldsymbol{v}_3 \] holds, where $[\mathrm{Vol}(\rho)]$ denotes the bounded fundamental class of $\rho$ in the bounded cohomology $H_b^3(\varGamma,\mathbb{R})$ of $\varGamma$ and $\boldsymbol{v}_3$ is the volume of a regular ideal 3-simplex in $\mathbb{H}^3$. As an application, we present a rigidity theorem for $\rho_M$ in the set of representations $\rho$ of $\varGamma$ in $\mathrm{PSL}_2(\mathbb{C})$ in terms of $[\mathrm{Vol}(\rho)]$. The rigidity theorem implies that $\boldsymbol{v}_3$ is the optimal separating constant.

We improve the Micallef–Moore index estimate for harmonic two-spheres in $n$-manifolds with positive isotropic curvature to the sharp bound $n-3$. Combining with recent work, this completes the diffeomorphism classification of closed manifolds with positive isotropic curvature in the remaining dimensions five and six.

Let $V$ be a compact connected manifold and $G$ a simply connected solvable Lie group. We study $G$-Lie foliations on $V$ from the point of view of $\mathcal{D}$-module theory, following the approach initiated by Dathe. To each singular foliation $\mathcal{I}$, we associate the $\mathcal{D}_X$-module $\mathcal{M}_{\mathcal{I}} = \mathcal{D}_X / \mathcal{D}_X \cdot \mathcal{I}$ and the derived ring $\mathcal{D}_{\mathcal{I}} := R\mathcal{H}om_{\mathcal{D}_X}(\mathcal{M}_{\mathcal{I}}, \mathcal{M}_{\mathcal{I}})$. We compute the D-irregularity $\mathrm{D\text{-}irr}(\mathcal{I})$ for three classes of solvable Lie foliations: regular homogeneous foliations, the Meigniez foliation with non-polycyclic holonomy group, and an explicit foliation on a compact 5-dimensional manifold. We show that the non-polycyclicity of the holonomy group $\Gamma$ is reflected in the non-vanishing of higher cohomology groups of $\mathcal{D}_{\mathcal{I}}$, establishing a new connection between the geometry of the holonomy group and $\mathcal{D}$-module invariants.

We study bridge indices of graphs, virtual graphs, and their handlebody classes. We construct virtual ravels for which the difference between bridge index and weighted overpass bridge index is unbounded. The lower bound uses exponentially many colorings for one fixed flow in a $\Z_2$-family of biquandles. For each negative Euler characteristic we construct classical almost unknotted graphs of one fixed trivalent graph with arbitrarily large bridge index. We also obtain unbounded even bridge indices for ravels of each fixed bouquet rank. A separate almost unknotted family has an unbounded difference between its spatial-graph bridge index and the bridge index of its regular neighborhood. Finally, edge connected sums of Suzuki curves give exact constrained and unconstrained bridge indices whose difference is unbounded.

The Lusternik–Schnirelmann category of a closed symplectic manifold admits various estimates. Locally conformally symplectic (LCS) geometry is a generalization of symplectic geometry involving a closed one-form, called the Lee form. In this paper, we introduce Farber's Lusternik–Schnirelmann category for closed one-forms into LCS geometry by applying it to the Lee class. In particular, this provides a new topological approach to LCS structures of the second kind. We then obtain lower bounds for this invariant under LCS blow-ups.

For the unordered discrete configuration space $\mathrm{UD}_n(\mathsf{\Gamma})$ of $n$ particles on a connected finite graph $\mathsf{\Gamma}$, we construct an explicit factor system on its universal cover. Its factors are encoded by legal pairs, namely subgraphs equipped with particle distributions. The nesting, orthogonality, and product regions in the resulting hierarchically hyperbolic group (HHG) structure admit explicit descriptions in terms of configuration-space geometry, and we show that this structure satisfies the additional properties needed for constructing and obstructing subgroups isomorphic to right-angled Artin groups (RAAGs). Using sufficiently subdivided models, we apply this hierarchy to graph braid groups. We give a finite combinatorial formula for the maximal rank of a free abelian subgroup and show that every RAAG occurs as an undistorted subgroup of some graph braid group. For graph $2$-braid groups, we obtain stronger restrictions: every RAAG subgroup has bipartite defining graph, and the embedding problem is characterized by an induced-subgraph condition in the expanded core graph of the hierarchy. For the RAAG defined by the four-vertex path, this condition is equivalent to a finite graphical criterion on the underlying graph.

It is shown that every locally flatly embedded genus $g \in \{1,2\}$ surface in the $4$-sphere with knot group $\mathbb{Z}$ is unknotted. The same proof establishes that any two genus $g \in \{1,2\}$ surfaces in $D^4$ with knot group $\mathbb{Z}$ and common boundary an Alexander polynomial one knot are isotopic rel. boundary. In previous work, the author and Powell reduced such unknotting problems to a question concerning the cancellation of $(-t)$-quadratic forms over $\mathbb{Z}[t^{\pm 1}]$, which was solved in genus $g \geq 3$ using work of Bass. In genus $g=2$, we observe that the same proof goes through using further work of Bass. In genus $g=1,$ the result instead follows from a statement in commutative algebra which was proved with the assistance of AI. Combined with earlier work of Freedman on locally flat spheres with knot group $\mathbb{Z}$, this shows that a locally flatly embedded orientable surface in $S^4$ is unknotted if and only if its knot group is $\mathbb{Z}$.

For classical knots, many characteristics are difficult to determine from arbitrary diagrams of the knot. Bridge number is of this type. It is possible to have a bridge presentation of a knot in which the number of bridges is only locally minimal, as shown by Ozawa and Takao. Here, we give a method for beginning with an arbitrary classical knot diagram having \(n\) crossings and producing a locally minimal bridge presentation of the knot with size \(O(n^2)\). This method requires only polynomial time and space. For some classes of knots, any local minimum is also the global minimum, which was shown for the trivial knot by Otal. Because the unknot is the only knot with bridge number \(1\), unknot recognition is in \(P\).

For every even integer $m\geq 2$, we construct a closed, orientable, smooth, aspherical $(2m+1)$-manifold whose fundamental group has nontrivial center and for which infinitely many tautological classes in the $\mathbb{F}_2$-cohomology of the classifying space of its group of homotopically trivial diffeomorphisms are nonzero and not nilpotent. Specializing the calculation to cohomological degree $0$ gives examples of such manifolds that do not bound compact smooth manifolds. These provide counterexamples to a conjecture of Hebestreit–Land–Lück–Randal-Williams.

Borisov and Yeung showed that the Cartwright-Steger surface $X$ is defined over ${\mathbb {Q}}$, hence complex conjugation gives an antiholomorphic involution on $X$. We show that the orbit space $Y$ is a closed oriented smooth four-manifold homeomorphic to ${\mathbb {CP}}^2\# (S^1 \times S^3)$, presenting $X$ as a double branched cover $X\to Y$, with branch locus diffeomorphic to the connected sum of three copies of the real projective plane.

We give a criterion for property $P_{\text{naive}}$ for subgroups of orientable big mapping class groups and apply it to mapping class groups of non-orientable infinite type surfaces through the orientation double cover.

Let $p,q$ be positive braids with Demazure products $u,v$. The endpoint stratum $\mathrm{Conf}(p,q)_{u,v}$ of the double Bott-Samelson cell splits as $\mathrm{Conf}(u,v) \times X(p) \times X(q^{\mathrm{op}})$, a double Bruhat cell times two braid varieties. We prove that the stratum's cluster structure is given by a cluster localization of the one on $\mathrm{Conf}(p,q)$, and that the splitting map is a quasi-cluster isomorphism. This comes from a mutation sequence along a double Demazure weave, one mutation per trivalent vertex, ending at an amalgamation of an extension of the double word quiver with the weave quiver. In the half-decorated case, this proves the conjecture of Gorsky-Kim-Scroggin-Simental for the splicing map $X(p) \times X(\Delta\Delta) \to X(p\Delta)$, where $\Delta$ is a reduced positive braid for $w_0$ and $\mathrm{dem}(p) = w_0$.

A tubular group has a graph of groups decomposition with $\mathbb{Z}^2$ vertex groups and $\mathbb{Z}$ edge groups. This paper gives a classification of one vertex one loop tubular groups $G_{(k,\ell),(m,n)} = \langle a,b,t: [a,b]=1, t a^m b^n t^{-1} = a^k b^\ell \rangle$ up to commensurability and quasi-isometry. We show that tubular groups where the images of the edge maps have nonzero “intersection number" are all commensurable and so quasi-isometric. When the intersection number is zero, there are two quasi-isometry classes and infinitely many commensurability classes. The nonzero and zero intersection number tubular groups are not quasi-isometric.

The group of polynomial automorphisms of the affine n-space is an interesting large group. A slightly simpler group is its subgroup of tame automorphisms. Natural problems about these groups include the existence of normal subgroups, the classification of finite subgroups, the Tits alternative, and the possible dynamical degrees of their elements. One method to investigate these questions is via some actions on some metric spaces, namely the coset complex and the valuation complex, that we introduce in detail. This paper is a survey focusing on the following three cases: dimension 2, dimension 3, and dimension 4 for tame automomorphism preserving a nondegenerate quadratic form.