Papers

For an ordered ideal triangulation of a knot exterior in $S^3$, we give a cochain construction relating its face matrix $C$ to its preferred-longitude gluing matrices $A,B$. The obstruction to uniqueness is the first cohomology of the ideal compactification. Its vanishing yields nullity doubling and the FAMED matrix identity, including a constrained formulation for singular face matrices. A separate rank argument under a strict angle structure supplies a proposed proof of Wong's generalized FAMED conjecture. Over the integers, the construction gives a direct-sum decomposition of $\ker_{\mathbb{Z}}B$ and an exact sequence from the two face cokernels to the gluing cokernel, with quotient $\mathbb{Z}/2\mathbb{Z}$. In the unhalved longitude convention this implies $|\det B|=2|\det C|^2$. We also obtain cyclic face torsion, order-reversal invariance of Smith normal form, and affine face coordinates for fixed lifted gluing systems. The determinant-one assertion and the full Andersen–Kashaev volume conjecture remain open here.

In this paper we consider groups of complex hyperbolic isometries generated by three complex involutions. Our first main result states that if the group is discrete and at least two of the mirrors of complex reflections intersect, then the group may be put into a normal form involving the ordering of traces of short, even length words. Our second main result refines a conjecture of Schwartz (proved by Grossi) that determines the type of the group in terms of the angles between the mirrors. That is, in the parameter space of complex triangles with given angles, it determines which of two given words becomes elliptic first.

For a closed orientable surface of genus at least 2, we study the large-scale geometry of the (symmetrized) Lipschitz metric on its space of projective filling geodesic currents. We show that the rank and, hence, asymptotic dimension of the space of filling geodesic currents is infinite. This gives the first example of a natural proper space associated to a closed surface, with a metrically proper isometric action of the mapping class group, that has infinite asymptotic dimension. In contrast, we show that any finite simplex of geodesic currents has rank uniformly bounded above by $8g-9$. We then show a sufficient criterion for its rank and asymptotic dimension to be equal to $8g-9$ and construct examples satisfying it, thus showing the bound is sharp. We further prove that Teichmüller space equipped with the Lipschitz metric has rank equal to $3g-3$. In particular, the space of projective filling geodesic currents and the Teichmüller space are not quasi-isometric.

Inspired by work of Gaster, Greene, and Vlamis on the chromatic number of curve graphs, we prove that every endomorphism of the total Kneser graph is an automorphism. We also show that the automorphism group of the total Kneser graph is the product of a pair of permutation groups.

We develop two methods in quantitative homological algebra: A declarative approach to integral cohomological filling problems and a quantitatively controlled replacement of normed chain complexes by their homology complex. Both methods are connected through so-called chain extensions. As applications, we obtain transparent proofs for uniform integral simplicial cohomological fillings for subdivisions of simplicial complexes, for uniform integral cohomological fillings of certain finite coverings, and for lower bounds of the $\ell^1$-semi-norm on product spaces.

A flag complex $L$ determines a locally CAT(0) cube complex $P_L$ with the links of all vertices isomorphic to $L$. The fundamental group of $P_L$ is the commutator subgroup of the right-angled Coxeter group $W_L$. We observe that an edge contraction $L \rightarrow L/e$ determines a $2$-fold branched cover $P_L \rightarrow P_{L/e}$ (up to a homotopy equivalence). We use this branched cover to give an inequality between the $\mathbb{F}_2$-$L^2$-Betti numbers of $W_L$ and $W_{L/e}$. This reduces an $\mathbb{F}_2$-version of the Singer Conjecture for right-angled Coxeter groups to minimal flag triangulations of even dimensional spheres. We also use this inequality to compute the $L^2$-Betti numbers of $W_L$ when $L$ is a flag triangulation of the projective plane or the torus.

A regular complex bundle on an orbispace has fibres that are positive multiples of the stabilizers' regular representations. For stabilizers of order at most two, the sharp uniform rank bound in dimension five is $16$ for paracompact orbispaces and $4$ for closed oriented smooth orbifolds. Rank $16$ occurs with dense free locus and minimum regular rank $4$ on the nonfree locus. We also prove the sharp bound $16$ for closed oriented smooth six-orbifolds. We compute the stable regular index of cyclic gerbes over finite connected CW complexes as the least common multiple of the indices of their character twists. For paracompact orbispaces, optimal integral Adams multipliers give explicit rank bounds in terms of dimension and stabilizer orders. The least positive rank of an integral exterior-power expression with regular fibre and zero reduced class for every regular rank-four bundle on $S^4\times B\mu_2$ is $32$.

We extend the classification of manifolds with positive isotropic curvature to all dimensions $n \geq 5$. This result also completes the classification of uniformly positive isotropic curvature on complete non-compact $\kappa$-noncollapsed ancient solutions and gradient shrinking Ricci solitons. The proof involves a construction of pinching cones in all dimensions, refining and extending to all dimensions the results of the first author. We have used MATHEMATICA to find viable parameters for the cone families. The final presentation is self-contained and can be verified manually. No stage of this work involved LLMs.

We give explicit positive factorizations for a genus-19 Lefschetz fibration over the torus and a family of genus-4 Lefschetz fibrations over a genus-2 surface. All these examples have only three nodes, and their total spaces are symplectic 4-manifolds of general type on the Bogomolov-Miyaoka-Yau line. In the course of deriving the monodromy relation in genus 19, we obtain several interesting identities, including a factorization of a Dehn twist into a product of three order-3 mapping classes. We show that the total space of this fibration is topologically s-cobordant to the Cartwright-Steger surface, and we determine its real quotient up to homeomorphism. The genus-4 family consists of variations of the Bauer-Catanese fibrations and contains 65 isomorphism classes of Lefschetz fibrations on 45 diffeomorphism types of compact complex ball quotients. We describe their topology and that of the real quotients of the three original surfaces. The handle monodromies in our main factorizations are pseudo-Anosov, and we show more generally that for holomorphic genus-4 Lefschetz fibrations on the BMY line, this holds for every choice of handle generators.

The Ahlfors–Weill construction explicitly extends a univalent function $f$ with Schwarzian norm less than $1/2$ to a quasiconformal homeomorphism of the Riemann sphere, an extension that can be understood geometrically through Epstein surfaces. For asymptotically conformal curves, the uniformization map can be extended to a larger disk, and Loewner theory can be used to verify that this extension is quasiconformal. In this paper, we construct a local quasireflection with such an extension. Using this local quasireflection, we modify classical estimates of Ahlfors and show that domains bounded by asymptotically conformal curves have asymptotically similar hyperbolic and quasihyperbolic metrics. Furthermore, by combining these estimates with a geometric reinterpretation of the extension through Epstein surfaces, we show that the Epstein-Poincaré surfaces of the complementary domains are arbitrarily close to one another near the curve.

We undertake a systematic comparison of cluster algebras associated to Legendrian knots of the same smooth knot type. We describe a cluster structure on the m-graded augmentation variety of Legendrian representatives of a family of double twist knots that includes all orientably Lagrangian fillable twist knots. For different Legendrian representatives of the same smooth knot type, we show that these cluster structures are equivalent. Along the way, we characterize the representatives that admit Maslov-m exact Lagrangian fillings, construct a Catalan number of such fillings for each representative, and show that any Maslov-0 fillable representative is Legendrian isotopic to the (-1)-framed closure of a positive braid. Finally, we give two Legendrian representatives of a cabled twist knot that yield equivalent cluster structures of infinite type.

We study webs on surfaces for the $\rm{PGL}_2$ skein theory specialized at roots of unity. We construct the $\rm{PGL}_2$ analogue of Chebyshev polynomials and use them to construct central elements in the skein algebra. We then focus on the cases of the 3-punctured sphere and the closed torus. We compute the Frobenius homomorphism for these two surfaces and show that the $\rm{PGL}_2$ Frobenius homomorphism for the torus is not injective when the quantum parameter $q^2$ is a root of unity of even order $n.$

We show that all closed geodesics in any arithmetic complex hyperbolic surface of second type are simple. These include the fake projective planes studied by Mumford, Klingler, Prasad-Yeung, and Cartwright-Steger.

We show that not every symplectic Lefschetz fibration over the $2$–sphere admits a smooth section, settling a long-standing open problem. To prove this, we develop a method for studying sections via lifts of monodromy factorizations with point-pushing maps and the induced handle decomposition of the total space, and apply it to an infinite family of Lefschetz fibrations. In contrast, we show that every Lefschetz fibration over $S^2$ admits positive multisections. Some of these examples admit no multisections with spherical components. We also construct signature-zero Lefschetz fibrations of every odd genus $g\geq 7$, which in turn give symplectic Lefschetz fibrations of any prescribed signature in each of these genera.

Let $S_g$ be a closed oriented surface of genus $g$. Ivanov conjectured that if $G$ a finitely generated subgroup of $\mathrm{Mod}(S_g)$ its finite-index Frattini subgroup $\Phi_f(G)$ is nilpotent. We prove this conjecture, using recent partial results of the first author about the Andersen-Masbaum-Ueno conjecture.

A petal diagram of a knot is a projection with a single multi-crossing and no nested loops; it is encoded by a permutation of the heights of the strands through the multi-crossing. Colton, Glover, Hughes and Sandberg proved a Reidemeister-type theorem for petal diagrams: two petal permutations represent the same knot if and only if they are related by trivial petal additions and deletions and by crossing exchanges. We ask whether every petal diagram of the unknot can be reduced to the one-petal diagram without ever increasing the number of petals, in analogy with Dynnikov's monotonic simplification theorem for rectangular diagrams. By an exhaustive, certified computer search we show that this is true for diagrams with at most 7 petals and false for 9 petals. Of the 40320 petal diagrams with 9 petals, 24992 represent the unknot, and exactly 108 of them are hard: none of them admits a crossing exchange or a trivial petal deletion, even if two natural petal-number-preserving symmetries are allowed. Up to these symmetries and mirror image there are three hard diagrams. Two of them can be untangled by passing through 11 petals; the third cannot be untangled through diagrams with at most 11 petals, but can through 13. We explain why the phenomenon differs from the rectangular case: a petal diagram is an arc presentation whose cyclic order of pages is determined by the order of its vertices on the binding, and no elementary move of Cromwell and Dynnikov preserves this rigid structure.

We study the genus-$19$ Albanese fibration of the Cartwright–Steger surface and its order-three symmetry. Passing to the orbifold quotient of the elliptic base gives a relation among the two handle monodromies and the three Dehn twists about the vanishing cycles. The two real vanishing paths give cycles whose real fixed points lie on different ovals of the invariant fiber. For a cyclic triple cover, we express the intersection of a curve with its image under the deck transformation as a signed crossing count on the quotient. We determine the two local branch values of a degree-$72$ bicanonical pencil near a node and study a symmetric bicanonical pencil. A general pencil also gives a finite degree-$72$ map from a blowup of the surface to the product of the elliptic curve and a projective line. The monodromy translates of the nodal vanishing classes span the first homology of a smooth fiber.

We equip Khovanov's colored $\mathfrak{gl}(2)$-homology groups with a degree-2 derivation. Over a field of characteristic $p>2$, the derivation specializes to a $p$-nilpotent differential, which leads to a categorification of colored $\mathfrak{gl}(2)$-homology at a $2p$th root of unity.

We characterize admissible BLD remetrizations of branched self-covers of spheres by linear local contractibility of their canonical pullback length metrics. A relative homology obstruction involving the total degree of an inverse image component yields uniform porosity of branch values. Explicit degree-two cusp covers of $\mathbb S^n$, $n\ge3$, have uniformly porous branch values of Hausdorff dimension $n-2$ but admit no compatible Ahlfors $n$-regular, linearly locally contractible BLD source metric. Short homologically essential loops provide the obstruction, showing that porosity is necessary but insufficient for admissibility. We also prove quasisymmetric invariance and sharp degree dependence of porosity for sphere-to-sphere covers in dimension two.

We study the $\rm{PGL}_2$ skein algebras of small surfaces, focusing on the closed torus and the once-punctured torus. We provide explicit finite presentations by generators and relations for the $\rm{PGL}_2$ skein algebra of the torus and once-punctured torus, analogous to those of Bullock and Przytycki for the Kauffman bracket skein algebras. We give explicit bases of the $\rm{PGL}_2$ skein algebras of the torus and once-punctured torus and describe the multiplicative structure of the basis for the torus, analogous to the product-to-sum formula of Frohman and Gelca. We twist the basis for the torus to obtain one with structure constants in $\mathbb{N}[q,q^{-1}].$ We compute the kernels of natural maps between the $\rm{PGL}_2$ and Kauffman bracket skein algebras associated to the once-punctured torus and closed torus. We compute the center of the $\rm{PGL}_2$ skein algebra of the torus, which depends on the specialization of the quantum parameter $q.$

In this paper we define the ascending number and bridge number for virtual and welded spatial graphs. We prove that the former admits a lower bound in terms of the latter. This relationship allows us to compare different notions of unknotting virtual spatial graphs by crossing changes.

The $\hat{Z}$ invariant of a three-manifold $M_{3}$, a half-index of the $3d$ $\mathcal{N}=2$ theory $T[M_{3}]$, is well understood for negative-definite plumbed three-manifolds. For orientation-reversed manifolds, however, the known formulas fail to produce $q$-series, and making sense of the operation $q \rightarrow q^{-1}$ remains a central open problem. Seifert manifolds, whose $\hat{Z}$ invariants are built from false theta functions, offer a natural testing ground. We study “dual false theta functions”, the objects that should replace false theta functions under orientation reversal. Two characterizations of these duals exist in the literature: a modular one, based on mock modularity with a prescribed shadow, and a resurgent one, based on Borel–Mordell integrals and transseries. We prove that the two characterizations are equivalent. We construct a new family of dual false theta functions for values of positive integer $p$ satisfying a Pell-equation condition. Its members are built from Zwegers' indefinite theta functions, have integer coefficients, and have effective central charge $c_{\text{eff}} \leq 1$. We also show that the mock theta functions of Li and Schwagenscheidt satisfy these characterizations and arise from a natural regularization of the divergent product $\eta^{3}(\tau)\psi_{p, a}(- \tau)$.

Let $K$ be the trefoil or figure-eight knot and let $M_K$ denote zero surgery on $K$. We prove that the fiberwise elliptic involution gives a two-fold branched cover \[ M_K\times S^1\longrightarrow T^2\times S^2 \] whose branch locus is a section together with a connected braided three-section. The corresponding braid words are computed explicitly for both knots and make visible the relation with Luttinger surgery along product Lagrangian tori. More generally, for a genus-$g$ fibered knot with hyperelliptic monodromy, the same construction gives a braided $(2g+2)$-section; this includes every genus-two fibered knot. We also state the corresponding surface-bundle version over $T^2$. We then track the induced peripheral data and gluing diffeomorphisms in the constructions of $Y_K$ and its twisted double $X_K$, as studied by the author in \cite[Section~4]{Akh07}, and through the constructions of the small exotic manifolds homeomorphic to $\CP^2\#k\CPb^2$, $k=3,5$ \cite{AkhmedovSmall,ABP,AP2008}. In the symmetric construction of a $k=5$ example \cite[Section~3.3]{ABP}, each summand admits a double cover branched along a surface, although the marked complements used in the twisted double need not be invariant. Finally, we prove boundary-action, relative-homology, and local-smoothing obstructions to extending these local involutions across the gluing regions to a branched cover of the closed manifold.

Let $\Gamma<\mathrm{Isom}(\mathbb H^n)$, $n\geq2$, be a cocompact lattice containing a convex cocompact codimension-one subgroup in the sense of Sageev. We prove that every finitely supported admissible random walk on $\Gamma$ has hitting measure singular with respect to Lebesgue measure on $\partial\mathbb H^n$. The proof extends the method of Kosenko–Tiozzo, replacing Fourier analysis for a cyclic subgroup by a von Neumann dimension argument that also applies to nonabelian subgroups. As corollaries, we prove the singularity conjecture of Kaimanovich and Le Prince for cocompact hyperbolic lattices admitting a proper cocompact cubulation, cocompact Kleinian groups, and cocompact hyperbolic lattices with totally geodesic sublattices of codimension one. In particular, our result covers all cocompact hyperbolic reflection groups, cocompact arithmetic lattices of simplest type and cocompact nonarithmetic lattices arising from the hybrid and inbreeding constructions.

We give many examples of surface bundles over surfaces whose fundamental groups are hierarchically hyperbolic by showing that extensions of surface groups constructed by the third author and Reid (LR surface groups) are hierarchically hyperbolic. This is facilitated by a new combination theorem for bundles of hyperbolic graphs over a hyperbolic base. This result differs from previous combination theorems for graph bundles by being applicable to bundles where the fibers are not properly embedded. We apply our combination theorem to establish the hyperbolicity of certain graph bundles that arise naturally from extensions of parabolically geometrically finite (PGF) subgroups of mapping class groups. These subgroups include LR surface groups, finitely generated Veech groups, free products of multi-twist groups, and many other examples created by a theorem of Udall. When the PGF groups have cyclic peripherals, hyperbolicity of our graph bundle is the key milestone towards showing that the extension groups are hierarchically hyperbolic.

We present a "myopic" Tutte polynomial for graphs on $\mathbb{R}P^2$ which takes only nullhomologous spanning subgraphs as input. It recovers the generalized Krushkal polynomial and Drobotukhina's analogue of the Jones polynomial for alternating, nullhomologous links in $\mathbb{R}P^3$. We use this myopic Tutte polynomial to prove an analogue of the Kauffman-Murasugi-Thistlethwaite Theorem, relating the Jones polynomial of an alternating link to certain refinements of the crossing number. Finally, we construct a spanning tree model for the Khovanov homology of nullhomologous links, mirroring work by Champanerkar-Kofman and Wehrli for links in $S^3$. For alternating links, we use our model to prove that the Khovanov homology in $\mathbb{Z}/2\mathbb{Z}$ coefficients is determined entirely by the Jones polynomial and signatures of the link.

In this paper, we determine the set of Euler classes of all taut foliations for a family of closed graph manifolds. As a consequence, for any closed aspherical graph manifold $Y$, every integral class in the closed dual Thurston unit ball can be virtually realized as the Euler class of some taut foliation. In particular, $Y$ virtually admits a taut foliation with vanishing Euler class. We present examples of graph manifolds with arbitrarily large first Betti number, which admit no taut foliations with vanishing real Euler classes. This shows that the virtual requirement is necessary. We also provide examples of graph manifolds with arbitrarily large first Betti number, for which every taut foliation has vanishing real Euler class, but none has vanishing integral Euler class. This illustrates the subtle difference between real Euler class zero and integral Euler class zero.

The $\Delta$-unknotting number for a knot is defined as the minimum number of $\Delta$-moves needed to deform the knot into the trivial knot. In this paper, we discuss Montesinos knots whose $\Delta$-unknotting number is equal to one. We propose a conjectural characterization of Montesinos knots with $\Delta$-unknotting number one and provide examples admitting distinct $\Delta$-moves, each of which deforms the knot into the trivial knot.

We prove that the $(-2,3,7)$ pretzel knot is the only hyperbolic knot of genus at most $5$ with a lens space surgery. Our key technical result asserts that if a genus-$g$ hyperbolic knot $K$ has an elliptic surgery of slope greater than $4g-3$, then the invariant foliations of its monodromy have an orientation that is reversed by the monodromy, and so the opposite of the dilatation must be a root of the Alexander polynomial of $K$. Combined with prior work of Moser, Wu, Bleiler–Litherland, Baker, and Greene, this verifies the Berge conjecture for knots of genus at most $5$.

We extend the definition of Khovanov-Lee homology to links in connected sums of interval bundles over surfaces and $S^1\times S^2$'s, and construct a Rasmussen-type invariant for links in these manifolds. As an application, we prove an inequality relating the Rasmussen-type invariant to the genus of surfaces with boundary in four-manifolds that are boundary connect sums of $D^2\times S^2$, $\mathbb{C} P^2\setminus B^4$, and $DTS^2$.