Papers

We study foliations which force every transverse flow to have a closed orbit. We fully characterize this property among $C^2$ finite depth foliations on atoroidal 3-manifolds. We also show that for many foliations, this property can be broken by a $C^0$-small modification of the foliation. We produce closed orbits using the theory of endperiodic maps, and we construct $C^1$ flows without closed orbits by inserting dynamical plugs transverse to foliations along curves of attracting holonomy.

Surface braids are a generalization of braids to surfaces in the 4-disk. They are described by planar graphs called charts, and sequences of braid words called braid systems, that represent monodromies of branch points in projections. Yang-Baxter operators (YBOs) have been used extensively in knot theory for constructions of Jones and other polynomials, defined by braid group representations. We propose to use YBOs for constructing surface braid invariants through braid systems. The intersections of eigenspaces are used for this construction. We further use 3-cocycles of the Yang-Baxter cohomology theories via charts, in an analogue of quandle cocycle invariants but more generally utilizing tensor products of modules. An approach towards defining knotted surface invariants through closures of surface braids is discussed.

Hambleton and Kreck described the group of based homotopy self-equivalences of a closed oriented $4$-manifold by a commutative braid of exact sequences built from bordism groups and the Postnikov $2$-type. We construct the analogous braid for closed oriented $BO\langle n+1\rangle$-manifolds $M$ of dimension $2n$ or $2n-1$ whose fundamental group is finite or infinite cyclic and which satisfy one condition on the action of $\pi_1$ on the homotopy of the fibre of $M\to P_n(M)$. We compute $\operatorname{Aut}_\bullet(S^1\times S^{m-1})\cong(\mathbb{Z}/2)^2$ for $m\ge4$, and $\operatorname{Aut}_\bullet(S^3\times S^3)\cong(\mathbb{Z}/12)^2\rtimes SL_2(\mathbb{Z})$ with the standard action modulo $12$. Further, $\operatorname{Aut}_\bullet(S^3\times S^4)\cong(\mathbb{Z}/2\oplus\mathbb{Z}/24)\rtimes(\mathbb{Z}/2)^2$ has order $192$, that $\operatorname{Aut}_\bullet(S^4\times S^4)\cong D_4\times\mathbb{Z}/2$ has order $16$, and that for connected sums of $r\ge2$ copies of $S^3\times S^3$ the kernel contains a subgroup $(\mathbb{Z}/12)^{2r}$ on which the permutations of the summands act non-trivially. For $S^p\times S^p$ the kernel is a sum of two copies of $\pi_{2p}(S^p)$ modulo Whitehead products with $\iota_p$, and the image is determined by an obstruction on the Whitehead square. In particular $\operatorname{Aut}_\bullet(S^5\times S^5)$ is isomorphic to the theta subgroup of $SL_2(\mathbb{Z})$ and $\operatorname{Aut}_\bullet(S^7\times S^7)$ is an extension of $SL_2(\mathbb{Z})$ by $(\mathbb{Z}/120)^2$.

We consider the multi-spinor Seiberg-Witten equations on a 4-manifold in a general setting, where we allow the auxiliary bundle to be non-trivial. We show that when the metric is Kähler and the auxiliary connection is unobstructed anti-self-dual Yang-Mills, the moduli space of solutions is compact for all parameters in some open neighborhood of the given metric and auxiliary connection. We use this fact to compute a count of solutions on some examples of Kähler surfaces. These computations have a surprising purely algebro-geometric corollary which may be of independent interest.

We determine the intrinsic mirror of an orbifold cubic surface with three nodes and interpret the resulting geometry through mirror symmetry between the $PGL(2,\mathbb{C})$ and $SL(2,\mathbb{C})$ character varieties of the once-punctured torus. We further construct a quantization of the mirror using higher genus Gromov-Witten theory and show that it recovers the $A_1$ spherical double affine Hecke algebra.

A recent manuscript produced by Claude under the direction of Levent Alpöge constructs a 1-parameter family of compact complex threefolds diffeomorphic to $S^6$. Our goal is to describe the geometric ideas behind the construction, give a self-contained proof, and remark on possible future research directions.

The Thurston spine of the genus 2 Teichmüller space of closed compact surfaces is obtained. In particular, we identify all mapping class group-orbits of filling sets of systoles in genus 2. Thereby, surprisingly rich cases already appear in this simplest case which is genus 2. This is contrasted with a cell decomposition of a bordification of Teichmüller space obtained computationally using Rivin's angle coordinates, on which the cells are labelled by Delaunay graphs closely related to our systolic graphs.

We prove that the exotic smooth $8$-sphere and both oriented homotopy $10$-spheres representing elements of order three admit Riemannian metrics with strictly positive sectional curvature. The construction combines Sperança's special $S^3$-$S^3$ bundle models with the compatible-disk construction of He, Liu and Yau. We show that the relevant bundles admit equivariant polar normal forms, with transition functions that are constant along meridians and conjugation-equivariant. We also show that the representation-dependent part of the He–Liu–Yau construction requires only a uniform bound on the infinitesimal action fields. Sperança's $8$- and $10$-dimensional examples satisfy this bound. The resulting northern and southern metrics have matching boundary metrics and compatible second fundamental forms, so the Reiser–Wraith gluing theorem gives the required positively curved metrics.

In [OS05] the authors proved an exact triangle for the Heegaard Floer homology of the double branched covers associated to a skein sequence of links, and constructed a spectral sequence from the reduced Khovanov homology to the Heegaard Floer homology of the double branched cover of a link. For $L \subset S^3$ a link and $\Sigma(L)$ its double branched cover, the invariant $HF(\Sigma(L))$ doesn't capture the information of the natural involution on $\Sigma(L)$. In this paper we study the equivariant version of $HF(\Sigma(L))$ and prove that similar exact triangle and spectral sequence hold for this equivariant version.

We show that a closed spin four-manifold with nonzero signature, even after being punctured, does not admit a complete metric with uniformly positive scalar curvature. In particular, the punctured $K3$ surface does not admit such a metric. This is known to be false in higher dimensions. A key new observation, which could be of independent interest, is that Gromov's $\mu$-bubbles are a convenient class of boundaries for the APS boundary value problem for the Dirac operator.

We discuss some contexts in which the topological dynamics of certain Anosov subgroups of higher-rank Lie groups on "bad" subsets of flag manifolds can be analyzed using results from homogeneous dynamics in the infinite-covolume rank-one setting. For example, we show that a group of projective transformations dividing a strictly convex domain in projective space and intersecting Zariski-densely the stabilizer of an ellipsoid has a dense orbit in the complement of the domain, and acts minimally on the space of full projective flags tangent to the domain. This accounts for all known examples of divisible strictly convex domains in sufficiently high dimensions. We also provide examples of Zariski-dense groups that are Anosov in a partial flag manifold but such that the equivariant projection from the Benoist-Guivarc'h limit set in the Furstenberg boundary to the Anosov limit set is not a fibration.

Dunfield, Friedl, and Jackson showed that if the $SL(2, \mathbb{C})$-character variety of a knot has an irreducible curve component that contains the character of an irreducible representation and a character with nonmonic twisted Alexander polynomial, then this component has only finitely many characters with monic twisted Alexander polynomials. In this paper, we give explicit upper bounds on the number of such characters in terms of a presentation of the knot group. In particular, we give upper bounds in terms of the crossing number of the knot.

We develop Picard Farey symbols for finite-index subgroups of \(G=\PSL_2(\ZZ[i])\) from the Gaussian Farey tessellation of \(\HH^3\) by ideal octahedra. The ambient geometry naturally involves two different index-two extensions of \(G\): the orientation-preserving group \(\PGL_2(\ZZ[i])\) and a reflection extension. This distinction explains the two \(G\)-orbits of Farey faces, which merge into one \(\PGL_2\)-orbit, and the classical extended modular tessellation appears as a totally geodesic slice. A Picard Farey symbol records finitely many Gaussian-rational octahedral occurrences, their local stabilizers, and cooriented face-pairing transformations. We prove that such a symbol reconstructs the finite Picard-cell action and hence the subgroup; moreover, when subgroup membership is decidable, the symbol can be found by a terminating finite double-coset procedure. For torsion-free subgroups we obtain whole-octahedron fundamental domains. The marked symbol also supplies finite orbit relations for the established Gaussian Steinberg edge–face presentation, with its corresponding Bianchi modular-symbol relations. When the natural modular-to-Picard coset map is bijective, a side-connected Kulkarni special polygon lifts, using the same modular representatives, to a Picard fundamental polyhedron whose intersection with the invariant real plane is exactly that polygon; the level-$(2+i)$/level-$5$ family is worked out explicitly. Comparison examples show that octahedral boundary minimization, absolute geometric face minimization, and preservation of such a modular section are distinct problems. Canonical reduction and effective Hecke reduction remain open.

We prove the rectangular cyclotomic expansion conjecture of Kameyama, Nawata, Tao, and Zhang for reduced colored HOMFLY-PT polynomials of arbitrary zero-framed knots. We also prove their explicit double-twist formula for every rectangular color and all pairs of nonzero integer twist parameters. For each knot, we construct a double Newton expansion with coefficients in $\mathbb Z[A^{\pm1},q^{\pm1}]$. These coefficients are uniquely determined and independent of the width and height of the rectangular color. The construction uses finite-rank representation symmetries and integral interpolation. The double-twist formula follows from a simultaneous exterior-power comparison of the two four-point Casimir operators and the known one-row formula. Finite transition formulas express the double Newton coefficients of double-twist knots in terms of the prescribed twist coefficients.

We present formulas for the higher Alexander polynomials of satellite links.

In this paper, building on the work of Dai–Sasahira–Stoffregen, we derive constraints on the intersection forms of smooth, compact, indefinite, spin $4$-manifolds bounded by AR homology spheres using two methods. First, we use the constraints on the existence of $\operatorname{Pin}(2)$-equivariant maps between representation spheres, established by Hopkins–Lin–Shi–Xu in their proof of the $10/8 + 4$-inequality. Second, we compute Lin's $\kappa o$-invariant for AR homology spheres. Combined with Lin's version of the relative $10/8$-inequality, this computation yields constraints different from those based on Hopkins–Lin–Shi–Xu. As applications, we obtain new lower bounds on the minimal genus of surfaces bounded by knots whose double branched covers are AR homology spheres in various $4$-manifolds, and provide an obstruction to smooth $4$-manifold decompositions related to the $11/8$-conjecture.

In this paper we investigate the relation between the bonded braid monoid BBn and the group of pure braids $P_n$, realised by an algebraic analogue of the topological tangle insertion, namely by replacing a bond by (a power of) a pure braid generator. To do so, we provide appropriate presentations for $P_n$ using cyclic relations and clasp generators, and extend them to presentations of $B_n$. We construct the monoid epimorphism $\Phi: BB_n \to Bn$, mapping bonds to pure braid generators, which is the main result of the paper. Using the isomorphism from the singular braid monoid $SB_n$ to the tight bonded braid monoid $BB_n$, we extend $\Phi$ to a new type of epimorphism $s \Phi: SB_n \to B_n$. The epimorphisms $\Phi$ and $s \Phi$ are extended to families of epimorphisms $\Phi_p$, $s \Phi_p$, $p \in \mathbb{Z}$.

We build a Floer theory on the space of braids of periodic orbits of a Hamiltonian flow on a closed symplectic surface. The differential and continuation maps are defined by counting Floer isotopies of braids: tuples of Floer cylinders with pairwise disjoint graphs. With this new perspective, we prove a quantitative braid stability result that shows the persistence of braids under possibly large Hamiltonian perturbations. As an application, we show that for every $\alpha\geq 0$ there is a sequence of Hamiltonian diffeomorphisms $\phi_k$ on the two-torus $T^2$ such that $$d_H(\operatorname{Ent}_{\leq \alpha}(T^2,\omega),\phi_k)\to \infty \quad (k \to \infty),$$ where $\text{Ent}_{\leq \alpha}(T^2,\omega)\subset \text{Ham}(T^2,\omega)$ denotes the set of Hamiltonian diffeomorphisms with topological entropy at most $\alpha$ and $d_H$ the Hofer metric. We prove this result by studying only contractible periodic orbits, whereas analogous higher-genus statements were previously obtained using non-contractible orbits.

We consider the three-strand pretzel knot $P_a=P(a,-a-2,-(a+1)^2/2)$ and prove that, when $a\equiv1\pmod4$, its Alexander polynomial fails the Fox–Milnor condition. The same remains true after adding any number of opposite odd parameter pairs $q,-q$ and reordering the parameters. Combined with work of Kim–Lee–Song, this gives the same conclusion for every odd $a\ge3$. In particular, no knot in Lecuona's exceptional family is algebraically or topologically slice. Together with earlier work of Lecuona, Miller, and Kim–Lee–Song, this completes the slice–ribbon and topological-sliceness classifications for three-strand pretzel knots and removes the nonexceptional hypothesis from the Lecuona–Wand classification of prime fibered ribbon pretzel knots.

We give local combinatorial criteria for realizing prescribed ideal triangulations by nondegenerate hyperbolic truncated tetrahedra. A local bichromatic criterion allows valence-7 edges when tetrahedra meeting low-valence edges use at most two quotient-edge classes. Using the extended combinatorial Ricci flow, we obtain parameter-dependent criteria for triangulations of minimum valence 6. Symmetric and pair-min angle estimates yield explicit valence conditions, including mixed conditions on entire edge stars. The proofs use analytic angle inequalities and rigorous interval evaluations. Explicit face pairings realize the criteria and distinguish their scope, while cyclic constructions and finite covers give infinite families. Each criterion yields a unique zero-curvature hyper-ideal metric and exponential convergence from any positive initial length vector.

We prove that every acylindrical action on uniformly locally finite product of trees induces the hyperfinite orbit equivalence relation on the Roller boundary. As a byproduct, we construct an example of a standard Borel space and two commuting bounded-to-one surjective partial Borel maps that generate a universal countable Borel equivalence relation. This contrasts to Shinko-Weilacher-Yu's theorem on hyperfiniteness of bounded-to-one actions of commutative monoids.

We determine the vanishing cycles of Xiao's genus-two fibrations $X_d\to P^1$, $d=3,4,5$, whose fibers admit degree-$d$ maps to a fixed elliptic curve $E$. We tile the base $P^1$ by triangles and realize the surface $X_d$ as a degree-$d$ branched cover of $P^1\times E$. The covering description gives an explicit algorithm for the vanishing cycles. A vanishing path is recorded by the sides it crosses in the tiling; each crossing changes a labelled picture of the fiber by a local hexagon move, and the vanishing cycle is read from the terminal picture. We obtain the $7$, $13$, and $31$ vanishing cycles as explicit curves in a marked reference fiber. For $d=4,5$, these give new positive factorizations of the identity in the genus-two mapping class group, of types $(6,7)$ and $(12,19)$.

We study codimension-one holomorphic foliations admitting invariant singular real-analytic Levi-flat varieties. Our main problem is to determine when the local meromorphic first integrals furnished by the Levi-flat geometry can be continued coherently and globalized. We introduce a pencil-singularity condition and a projective compatibility condition for local meromorphic parameters. In the projective plane these yield a characterization of foliations admitting a primitive rational first integral, while in higher-dimensional projective space they give a rational integrability criterion under a generic local-product hypothesis along the dicritical locus. We also obtain semi-global and Stein versions of these results. In a different direction, we study foliations admitting infinitely many invariant Levi-flat varieties and prove a Darboux-type alternative: either the foliation admits a rational first integral, or infinitely many Levi-flats contain a common invariant algebraic curve whose virtual holonomy is abelian. Under additional assumptions on the singularities, this leads to the existence of a closed rational, and in the absence of saddle-nodes a closed logarithmic, defining \(1\)-form.

We study the asymptotic and TQFT-type properties of the Kashaev–Luo–Vartanov (KLV) partition function. We show that the partition function depends on the prescribed angle structure only through its peripheral angular holonomies. For any geometric triangulation of a cusped 3-manifold realizing a possibly incomplete hyperbolic structure, we express the exponential decay rate and the 1-loop term of the partition function in terms of, respectively, the volume and the adjoint twisted Reidemeister torsion of the corresponding incomplete hyperbolic structure. Consequently, these asymptotic formulas hold for any angle structure having the same peripheral angular holonomies as the geometric one. We further introduce a Reshetikhin–Turaev-type function associated with certain admissible Neumann–Zagier data, extending the Jones function in Teichmüller TQFT previously studied by Ben Aribi and the author. We show that, after a combinatorial normalization, for any triangulation, the KLV partition function can be expressed as a weighted integral of the squared norm of the Reshetikhin–Turaev-type function, where the weight is determined by the angular holonomies of peripheral curves. In particular, for FAMED triangulations, the Reshetikhin–Turaev-type function agrees with the Jones function, so that the normalized KLV partition function is obtained by integrating the squared norm of the integrand defining the Teichmüller TQFT partition function. This provides a noncompact analogue of the relationship between Turaev–Viro and Reshetikhin–Turaev invariants.

We prove log-concavity for the determinant-weighted external semi-activity polynomials of all real flat arrangements, strengthening their known trapezoidality. In fact, we establish a quadratic coefficient inequality that, in rank at least two, implies power concavity with an explicit rank-dependent exponent. The proof uses a new mixed-volume representation of the coefficients and the Alexandrov–Fenchel inequality. A more general formula gives a factorization and log-concavity for related mixed-volume sequences. As applications, we establish the conjectured log-concavity for spanning-tree polynomials of Eulerian digraphs, extend it to positive circulation weights, and strengthen the coefficient inequalities for Alexander polynomials of special alternating links.

An invariant $v$ of $m$-component links is called "cableable" if there exists a $k$ such that whenever a link $L'$ is obtained from a link $L=(K_1,\dots,K_m)$ by replacing each knot $K_i$ with its $(p_i,q_i)$-cable for some $p_i$ and $q_i$, we have $v(L')=(p_1\cdots p_m)^kv(L)$. The following problem is implicit in a number of papers by P. M. Akhmetiev and originates from the Arnold-Moffatt program for finding topological lower bounds for the energy of a magnetic field: Does there exist a cableable finite type invariant of links in $S^3$ which is not a function of the pairwise linking numbers? A potential solution of this problem was proposed by Akhmetiev himself, with the desired invariant defined as an analytic expression involving a magnetic field modeled on the given link, but we note that basic properties that he claimed of his invariant cannot be all true. In any case, we offer a different solution, with the desired invariant being a function of the coefficients of the Conway polynomial of the link and its sublinks. Moreover, we show that the cables can be replaced by arbitrary satellites. Much of the proof is a study of low degree coefficients of the Conway potential function $\Omega_L(x_1,\dots,x_n)$ expanded as a formal power series in Conway's variables $z_i=x_i-x_i^{-1}$. We also discuss type $n$ invariants which are "cableable up to an invariant of type $n-1$", some cableable invariants which are not of finite type (particularly a certain modification of Milnor's $\bar\mu$-invariants), and applications to links of solenoids.

We study mixed symplectic sums along connected unbranched cyclic multisections of degrees $p,q\geq 2$ in ruled surfaces. With \[ p(g-1)=q(h-1)=n, \] the resulting four-manifold has signature zero and Euler characteristic \[ e=4n\left(1-\frac1p-\frac1q\right). \] We classify the types having the rational cohomology ring of $\#_k(S^2\times S^2)$. They occur only for odd $k$ and, with $t=(k+1)/2$, $a=g-1$, and $b=h-1$, are exactly the solutions of \[ n=a+b+t,\qquad a\mid b+t,\qquad b\mid a+t. \] For each odd $k$ there are finitely many types; we give an explicit parametrization and tables for $k\leq 15$. Every type admits an equatorially product-framed symplectic realization and, possibly with different framings, a spin realization whose free integral intersection form is $kH$. We prove minimality, boundary incompressibility, an integral Mayer–Vietoris presentation, and a regular cyclic cover of degree $\operatorname{lcm}(p,q)$. The boundary subgroup $\pi_1(\Sigma_G)\times\mathbb Z$ embeds in the fundamental group; this gives a nontrivial Bass–Serre splitting and rules out word-hyperbolicity, cocompact real- or complex-hyperbolic lattices, and torsion-free cocompact irreducible bidisk lattices. There are five types for $k=3$ and seven for $k=5$; for $k=1$ we recover the degree pairs $(2,3)$, $(2,4)$, and $(3,3)$.

In this paper, we show that a knot with a nontrivial Jones polynomial of span at most 11 does not admit purely cosmetic surgery.

Let $S$ be a closed, oriented surface of genus at least $2$, let $h$ be a smooth Riemannian metric on $S$ with curvature $K\in (-1,0]$, and let $c\in \cT_S$ be a conformal structure on $S$. There exists a unique equivariant isometric immersion of $(S,h)$ in $\HH^3$ such that the pull-back by the hyperbolic Gauss map of the conformal structure at infinity is $c$. Dually, if $h^*$ is a smooth metric on $S$ with curvature $K^*\in (-\infty, 0)$ and if $c\in \cT_S$, there exists a unique equivariant immersion of $S$ into $\HH^3$ with third fundamental form $h^*$ and such that the pull-back of the conformal class at infinity by the Gauss map is $c$. Equivalently, given $h$ and $c\in \cT_S$, there is a unique pair $(E,u)$ where $E$ is a hyperbolic end with conformal structure at infinity $c$ and $u$ is an isometric embedding of $(S,h)$ in $E$. Given $h^*$ and $c$, there exists a unique pair $(E,u^*)$, where $E$ is a hyperbolic end with conformal structure at infinity $c$, and $u^*:S\to E$ is an embedding inducing the third fundamental form $h^*$. Those statements can be considered as a smooth counterpart of known or conjectural statements on the grafting map and on circle patterns.

We prove the Bleiler-Litherland conjecture: every lens space obtained by a nontrivial Dehn surgery on a hyperbolic knot in $S^3$ has order at least 18. The key ingredient is a spectral obstruction: if a hyperbolic knot $K$ of genus $g$ admits a lens space surgery with slope $\pm(4g-2)$, then $\Delta_K(t)$ has a real root outside the unit circle. The proof combines Floer-theoretic restrictions on lens space surgeries, Gabai's degeneracy-slope bound and Gabai-Oertel's persistence theorem for essential laminations, Ni's fixed-point theorem for monodromy, and a mod-2 orientability criterion, together with the relation between homological monodromy and the Alexander polynomial. As a further application of this spectral obstruction, we obtain characterizing-slope results for torus knots.