Isotopy Classes Research Articles

Rationality of Real Conic Bundles With Quartic Discriminant Curve

Abstract We study real double covers of $\mathbb P^{1}\times \mathbb P^{2}$ branched over a $(2,2)$-divisor, which are conic bundles with smooth quartic discriminant curve by the second projection. In each isotopy class of smooth plane quartics, we construct examples where the total space is $\mathbb R$-rational. For five of the six isotopy classes, we construct $\mathbb C$-rational examples with obstructions to rationality over $\mathbb R$, and for the sixth class, we show that the models we consider are all rational. Moreover, for three of the five classes with irrational members, we characterize rationality using the real locus and the intermediate Jacobian torsor obstruction of Hassett–Tschinkel and Benoist–Wittenberg. These double cover models were introduced by Frei, Sankar, Viray, Vogt, and the first author, who determined explicit descriptions for their intermediate Jacobian torsors.

Relative Khovanov–Jacobsson classes

To a smooth, compact, oriented, properly-embedded surface in the $4$-ball, we define an invariant of its boundary-preserving isotopy class from the Khovanov homology of its boundary link. Previous work showed that when the boundary link is empty, this invariant is determined by the genus of the surface. We show that this relative invariant: can obstruct sliceness of knots; detects a pair of slices for $9_{46}$; is not hindered by detecting connected sums with knotted $2$-spheres.

The classification of linked 3–manifolds in 6–space

Let $M_1$ and $M_2$ be closed connected orientable $3$-manifolds. We classify the sets of smooth and piecewise linear isotopy classes of embeddings $M_1\sqcup M_2\rightarrow S^6$.

Genus two Goeritz equivalence in S3

The Goeritz group of a genus g Heegaard splitting of a 3-manifold is the group of isotopy classes of orientation-preserving automorphisms of the manifold that preserve the Heegaard splitting. In the context of the standard genus 2 Heegaard splitting of S3, we introduce the concept of Goeritz equivalence of curves, present two algebraic obstructions to Goeritz equivalence of simple closed curves that are straightforward to compute, and provide families of examples demonstrating how these obstructions may be used.

Annular Link Invariants from the Sarkar–Seed–Szabó Spectral Sequence

For a link in a thickened annulus A×I, we define a Z⊕Z⊕Z filtration on Sarkar–Seed–Szabó’s perturbation of the geometric spectral sequence. The filtered chain homotopy type is an invariant of the isotopy class of the annular link. From this, we define a two-dimensional family of annular link invariants and study their behavior under cobordisms. In the case of annular links obtained from braid closures, we obtain a necessary condition for braid quasi-positivity and a sufficient condition for right-veeringness, as well as Bennequin-type inequalities.

Point-pushing actions for manifolds with boundary

Given a manifold M and a point in its interior, the point-pushing map describes a diffeomorphism that pushes the point along a closed path. This defines a homomorphism from the fundamental group of M to the group of isotopy classes of diffeomorphisms of M that fix the basepoint. This map is well-studied in dimension d=2 and is part of the Birman exact sequence. Here we study, for any d\geq 3 and k\geq 1 , the map from the k -th braid group of M to the group of homotopy classes of homotopy equivalences of the k -punctured manifold M \smallsetminus z , and analyse its injectivity. Equivalently, we describe the monodromy of the universal bundle that associates to a configuration z of size k in M its complement, the space M\smallsetminus z . Furthermore, motivated by our earlier work (2021), we describe the action of the braid group of M on the fibres of configuration-mapping spaces.

Braid loops with infinite monodromy on the Legendrian contact DGA

We present the first examples of elements in the fundamental group of the space of Legendrian links in ( S 3 , ξ st ) $(\mathbb {S}^3,\xi _{\text{st}})$ whose action on the Legendrian contact DGA is of infinite order. This allows us to construct the first families of Legendrian links that can be shown to admit infinitely many Lagrangian fillings by Floer-theoretic techniques. These new families include the first-known Legendrian links with infinitely many fillings that are not rainbow closures of positive braids, and the smallest Legendrian link with infinitely many fillings known to date. We discuss how to use our examples to construct other links with infinitely many fillings, and in particular give the first Floer-theoretic proof that Legendrian ( n , m ) $(n,m)$ torus links have infinitely many Lagrangian fillings if n ⩾ 3 , m ⩾ 6 $n\geqslant 3,m\geqslant 6$ or ( n , m ) = ( 4 , 4 ) , ( 4 , 5 ) $(n,m)=(4,4),(4,5)$ . In addition, for any given higher genus, we construct a Weinstein 4-manifold homotopic to the 2-sphere whose wrapped Fukaya category can distinguish infinitely many exact closed Lagrangian surfaces of that genus in the same smooth isotopy class, but distinct Hamiltonian isotopy classes. A key technical ingredient behind our results is a new combinatorial formula for decomposable cobordism maps between Legendrian contact DGAs with integer (group ring) coefficients.

Topological horseshoes for surface homeomorphisms

In this work, we develop a new criterion for the existence of topological horseshoes for surface homeomorphisms in the isotopy class of the identity. Based on our previous work on forcing theory, this new criterion is purely topological and can be expressed in terms of equivariant Brouwer foliations and transverse trajectories. We then apply this new tool in the study of the dynamics of homeomorphisms of surfaces with zero genus and null topological entropy, and we obtain several applications. For homeomorphisms of the open annulus A with zero topological entropy, we show that rotation numbers exist for all points with nonempty ω-limit sets, and that if A is a generalized region of instability, then it admits a single rotation number. We also offer a new proof of a recent result of Passeggi, Potrie, and Sambarino, showing that zero entropy dissipative homeomorphisms of the annulus having a circloid as an attractor have a single rotation number. Our work also studies homeomorphisms of the sphere without horseshoes. For these maps, we present a structure theorem in terms of fixed point free invariant subannuli, as well as a very restricted description of all possible dynamical behavior in the transitive subsets. This description ensures, for instance, that transitive sets can contain at most two distinct periodic orbits and that, in many cases, the restriction of the homeomorphism to the transitive set must be an extension of an odometer. In particular, we show that any nontrivial and stable transitive subset of a dissipative diffeomorphism of the plane is always infinitely renormalizable in the sense of Bonatti et al.

Planar Legendrian Θ ${\rm{\Theta }}$‐graphs

AbstractWe classify topologically trivial Legendrian ‐graphs in and identify the complete family of nondestabilizable Legendrian realizations in this topological class. In contrast to all known results for Legendrian knots, this is an infinite family of graphs. We also show that any planar graph that contains a subdivision of a ‐graph or as a subgraph will have an infinite number of distinct, topologically trivial nondestabilizable Legendrian embeddings. Additionally, we introduce two moves, vertex stabilization and vertex twist, that change the Legendrian type within a smooth isotopy class.

The symplectic isotopy problem for rational cuspidal curves

We define a suitably tame class of singular symplectic curves in 4-manifolds, namely those whose singularities are modeled on complex curve singularities. We study the corresponding symplectic isotopy problem, with a focus on rational curves with irreducible singularities (rational cuspidal curves) in the complex projective plane. We prove that every such curve is isotopic to a complex curve in degrees up to five, and for curves with one singularity whose link is a torus knot. Classification results of symplectic isotopy classes rely on pseudo-holomorphic curves together with a symplectic version of birational geometry of log pairs and techniques from four-dimensional topology.

Framed Knotoids and Their Quantum Invariants

We modify the definition of spherical knotoids to include a framing, in analogy to framed knots, and define a further modification that includes a secondary ‘coframing’ to obtain ‘biframed’ knotoids. We exhibit topological spaces whose ambient isotopy classes are in one-to-one correspondence with framed and biframed knotoids respectively. We then show how framed and biframed knotoids allow us to generalize quantum knot invariants to a knotoid setting, leading to the construction of general Reshetikhin-Turaev type biframed knotoid invariants.

A Lagrangian Klein bottle you can’t squeeze

Suppose you have a nonorientable Lagrangian surface L in a symplectic 4-manifold. How far can you deform the symplectic form before the smooth isotopy class of L contains no Lagrangians? I solve this question for a particular Lagrangian Klein bottle. I also discuss some related conjectures.

Attractors of dissipative homeomorphisms of the infinite surface homeomorphic to a punctured sphere

A class of dissipative orientation preserving homeomorphisms of the infinite annulus, pairs of pants, or generally any infinite surface homeomorphic to a punctured sphere is considered. We prove that in some isotopy classes the local behavior of such homeomorphisms at a fixed point, namely the existence of so-called inverse saddle, impacts the topology of the attractor — it cannot be arcwise connected.

Ergodicity of partially hyperbolic diffeomorphisms in hyperbolic 3-manifolds

We study conservative partially hyperbolic diffeomorphisms in hyperbolic 3-manifolds. We show that they are always accessible and deduce as a result that every conservative C1+ partially hyperbolic in a hyperbolic 3-manifold must be ergodic, giving an affirmative answer to a conjecture of Hertz-Hertz-Ures in the context of hyperbolic 3-manifolds. We also get some results for general partially hyperbolic diffeomorphisms homotopic to the identity and in some isotopy classes on Seifert manifolds.

Lorentzian distance functions in contact geometry

An important tool to analyse the causal structure of a Lorentzian manifold is given by the Lorentzian distance function. We define a class of Lorentzian distance functions on the group of contactomorphisms of a closed contact manifold depending on the choice of a contact form. These distance functions are continuous with respect to the Hofer norm for contactomorphisms defined by Shelukhin [The Hofer norm of a contactomorphism, J. Symplectic Geom. 15 (2017) 1173–1208] and finite if and only if the group of contactomorphisms is orderable. To prove this, we show that intervals defined by the positivity relation are open with respect to the topology induced by the Hofer norm. For orderable Legendrian isotopy classes we show that the Chekanov-type metric defined in [D. Rosen and J. Zhang, Chekanov’s dichotomy in contact topology, Math. Res. Lett. 27 (2020) 1165–1194] is nondegenerate. In this case, similar results hold for a Lorentzian distance functions on Legendrian isotopy classes. This leads to a natural class of metrics associated to a globally hyperbolic Lorentzian manifold such that its Cauchy hypersurface has a unit co-tangent bundle with orderable isotopy class of the fibres.

Enumerating Isotopy Classes of Tilings Guided by the Symmetry of Triply Periodic Minimal Surfaces

We present a technique for the enumeration of all isotopically distinct ways of tiling a hyperbolic surface of finite genus, possibly nonorientable and with punctures and boundary. This generalizes the enumeration using Delaney--Dress combinatorial tiling theory of combinatorial classes of tilings to isotopy classes of tilings. To accomplish this, we derive an action of the mapping class group of the orbifold associated to the symmetry group of a tiling on the set of tilings. We explicitly give descriptions and presentations of semipure mapping class groups and of tilings as decorations on orbifolds. We apply this enumerative result to generate an array of isotopically distinct tilings of the hyperbolic plane with symmetries generated by rotations that are commensurate with the three-dimensional symmetries of the primitive, diamond, and gyroid triply periodic minimal surfaces, which have relevance to a variety of physical systems.

Counting essential surfaces in 3-manifolds

We consider the natural problem of counting isotopy classes of essential surfaces in 3-manifolds, focusing on closed essential surfaces in a broad class of hyperbolic 3-manifolds. Our main result is that the count of (possibly disconnected) essential surfaces in terms of their Euler characteristic always has a short generating function and hence has quasi-polynomial behavior. This gives remarkably concise formulae for the number of such surfaces, as well as detailed asymptotics. We give algorithms that allow us to compute these generating functions and the underlying surfaces, and apply these to almost 60,000 manifolds, providing a wealth of data about them. We use this data to explore the delicate question of counting only connected essential surfaces and propose some conjectures. Our methods involve normal and almost normal surfaces, especially the work of Tollefson and Oertel, combined with techniques pioneered by Ehrhart for counting lattice points in polyhedra with rational vertices. We also introduce a new way of testing if a normal surface in an ideal triangulation is essential that avoids cutting the manifold open along the surface; rather, we use almost normal surfaces in the original triangulation.

The complex of essential surfaces is contractible

Let M be a 3-manifold and S be a component of ∂M. We introduce a simplicial complex E(M,S), whose vertices are isotopy classes of compact essential surfaces in M with at least one boundary component lying on S. E(M,S) can be thought as an analogy to the arc complex of a surface. We show that if M is irreducible and S is compressible in M, then E(M,S) is contractible.

Algorithmic aspects of branched coverings III/V. Erasing maps, orbispaces, and the Birman exact sequence

Let \tilde{f}\colon(S^2,\widetilde{A})\circlearrowleft be a Thurston map and let M(\tilde{f}) be its mapping class biset: isotopy classes rel \widetilde{A} of maps obtained by pre- and post-composing \tilde{f} by the mapping class group of (S^2,\widetilde{A}) . Let A\subseteq\widetilde{A} be an \tilde{f} -invariant subset, and let f\colon(S^2,A)\circlearrowleft be the induced map. We give an analogue of the Birman short exact sequence: just as the mapping class group \mathbf{Mod}(S^2,\widetilde{A}) is an iterated extension of \mathbf{Mod}(S^2,A) by fundamental groups of punctured spheres, M(\tilde{f}) is an iterated extension of M(f) by the dynamical biset of f . Thurston equivalence of Thurston maps classically reduces to a conjugacy problem in mapping class bisets. Our short exact sequence of mapping class bisets allows us to reduce in polynomial time the conjugacy problem in M(\tilde{f}) to that in M(f) . In case \tilde{f} is geometric (either expanding or doubly covered by a hyperbolic torus endomorphism) we show that the dynamical biset B(f) together with a “portrait of bisets” induced by \widetilde{A} is a complete conjugacy invariant of \tilde{f} . Along the way, we give a complete description of bisets of (2,2,2,2) -maps as a crossed product of bisets of torus endomorphisms by the cyclic group of order 2 , and we show that non-cyclic orbisphere bisets have no automorphism. We finally give explicit, efficient algorithms that solve the conjugacy and centralizer problems for bisets of expanding or torus maps.

On the Number of Conjugate Classes of Derangements

The number of conjugate classes of derangements of order n is the same as the number h n of the restricted partitions with every portion greater than 1. It is also equal to the number of isotopy classes of 2 × n Latin rectangles. Sometimes the exact value is necessary, while sometimes we need the approximation value. In this paper, a recursion formula of h n will be obtained and also will some elementary approximation formulae with high accuracy for h n be presented. Although we may obtain the value of h n in some computer algebra system, it is still meaningful to find an efficient way to calculate the approximate value, especially in engineering, since most people are familiar with neither programming nor CAS software. This paper is mainly for the readers who need a simple and practical formula to obtain the approximate value (without writing a program) with more accuracy, such as to compute the value in a pocket science calculator without programming function. Some methods used here can also be applied to find the fitting functions for some types of data obtained in experiments.