Mathematics - Geometric Topology Publications (50)

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Mathematics - Geometric Topology Publications

We refine some classical estimates in Seiberg-Witten theory, and discuss an application to the spectral geometry of three-manifolds. In particular, we show that on a rational homology three-sphere $Y$, for any Riemannian metric the first eigenvalue of the laplacian on coexact one-forms is bounded above explicitly in terms of the Ricci curvature, provided that $Y$ is not an $L$-space (in the sense of Floer homology). The latter is a purely topological condition, and holds in a variety of examples. Read More


We extend techniques due to Pardon to show that there is a lower bound on the distortion of a knot in $\mathbb{R}^3$ proportional to the minimum of the bridge distance and the bridge number of the knot. We also exhibit an infinite family of knots for which the minimum of the bridge distance and the bridge number is unbounded and Pardon's lower bound is constant. Read More


In 1992, Reid asked whether hyperbolic 3-manifolds with the same geodesic length spectra are necessarily commensurable. While this is known to be true for arithmetic hyperbolic 3-manifolds, the non-arithmetic case is still open. Building towards a negative answer to this question, Futer and Millichap recently constructed infinitely many pairs of non-commensurable, non-arithmetic hyperbolic 3-manifolds which have the same volume and whose length spectra begin with the same first $m$ geodesic lengths. Read More


A natural class of coloring complexes $X$ on closed manifold $M^n$ is investigated that gives a holonomy map $\mbox{Hol}_X: \pi_1(M) \to S_{n+1}$. By a $k$-multilayer complex construction the holonomy map may be defined to any finite permutation group $\mbox{Hol}_X: \pi_1(M) \to S_{n+k}$, $k>0$. Under isotopy of $X$ and surgery on $B^n \subset M^n$ a holonomy class of complexes $[X]$ is defined with $[X]=[Y] \iff \mbox{Hol}_X = \mbox{Hol}_Y$. Read More


The mapping class group $\Gamma^k(N_g)$ of a non-orientable surface with punctures is studied via classical homotopy theory of configuration spaces. In particular, we obtain a non-orientable version of the Birman exact sequence. In the case of $\mathbb R {\rm P}^2$, we analize the Serre spectral sequence of a fiber bundle $F_k(\mathbb R {\rm P}^2)/\Sigma_k \to X_k \to BSO(3)$ where $X_k$ is a $K(\Gamma^k(\mathbb R {\rm P}^2),1)$ and $F_k(\mathbb R {\rm P}^2)/\Sigma_k$ denotes the configuration space of unordered $k$-tuples of distinct points in $\mathbb R {\rm P}^2$. Read More


We study here global and local entanglements of open protein chains by implementing the concept of knotoids. Knotoids have been introduced in 2012 by Vladimir Turaev as a generalization of knots in 3-dimensional space. More precisely, knotoids are diagrams representing projections of open curves in 3D space, in contrast to knot diagrams which represent projections of closed curves in 3D space. Read More


We consider the link obtained by replacing each component of the given link with several parallel strands, which we call a parallel of a link. We show that an even parallel of a link is $\mathbb{Z}$-colorable except for the case of 2 parallels with non-zero linking number. We then determine the minimal coloring number for such $\mathbb{Z}$-colorable even parallels of links. Read More


We study the homotopy type of the harmonic compactification of the moduli space of a 2-cobordism S with one outgoing boundary component, or equivalently of the space of Sullivan diagrams of type S on one circle. Our results are of two types: vanishing and non-vanishing. In our vanishing results we are able to show that the connectivity of the harmonic compactification increases with the number of incoming boundary components. Read More


We consider partial matchings, which are finite graphs consisting of edges with degree zero or one. We consider transformations between two states of partial matchings. We introduce a method of presenting a transformation between partial matchings. Read More


We introduce a method of computing biquandle brackets of oriented knots and links using a type of decorated trivalent spatial graphs we call trace diagrams. We identify algebraic conditions on the biquandle bracket coefficients for moving strands over and under traces and identify a new stop condition for the recursive expansion. In the case of monochromatic crossings we show that biquandle brackets satisfy a Homflypt-style skein relation and we identify algebraic conditions on the biquandle bracket coefficients to allow pass-through trace moves. Read More


We prove that $Out(F_N)$ is boundary amenable. This also holds more generally for $Out(G)$, where $G$ is either a toral relatively hyperbolic group or a right-angled Artin group. As a consequence, all these groups satisfy the Novikov conjecture on higher signatures. Read More


We provide the first information on diffeotopy groups of exotic smoothings of R^4: For each of uncountably many smoothings, there are uncountably many isotopy classes of self-diffeomorphisms. We realize these by various explicit group actions. There are also actions at infinity by nonfinitely generated groups, for which no nontrivial element extends over the whole manifold. Read More


T. Kobayashi [36th Geometry Symposium (1989)] conjectured that a homogeneous space G/H of reductive type does not admit a compact Clifford-Klein form if rank G - rank K < rank H - rank K_H. We solve this conjecture affirmatively. Read More


We construct an infinite collection of knots with the property that any knot in this family has $n$-string essential tangle decompositions for arbitrarily high $n$. Read More


We consider complex projective structures on Riemann surfaces and their groups of projective automorphisms. We show that the structures achieving the maximal possible number of projective automorphisms allowed by their genus are precisely the Fuchsian uniformizations of Hurwitz surfaces by hyperbolic metrics. More generally we show that Galois Bely\u{\i} curves are precisely those Riemann surfaces for which the Fuchsian uniformization is the unique complex projective structure invariant under the full group of biholomorphisms. Read More


We show that the monodromy for a genus one, fibered knot can have at most two monodromy equivalence classes of once-unclean arcs. We use this to classify all monodromies of genus one, fibered knots that possess once-unclean arcs, all manifolds containing genus one fibered knots with generalized crossing changes resulting in another genus one fibered knot, and all generalized crossing changes between two genus one, fibered knots. Read More


We introduce a new quasi-isometry invariant of 2-dimensional right-angled Coxeter groups, the hypergraph index, that partitions these groups into infinitely many quasi-isometry classes, each containing infinitely many groups. Furthermore, the hypergraph index of any right-angled Coxeter group can be directly computed from the group's defining graph. The hypergraph index yields an upper bound for a right-angled Coxeter group's order of thickness, order of algebraic thickness and divergence function. Read More


Tautological classes, or generalised Miller-Morita-Mumford classes, are basic characteristic classes of smooth fibre bundles, and have recently been used to describe the rational cohomology of classifying spaces of diffeomorphism groups for several types of manifolds. We show that rationally tautological classes depend only on the underlying topological block bundle, and use this to prove the vanishing of tautological classes for many bundles with fibre an aspherical manifold. Read More


We consider two manifestations of non-positive curvature: acylindrical actions (on hyperbolic spaces) and quasigeodesic stability. We study these properties for the class of hierarchically hyperbolic groups, which is a general framework for simultaneously studying many important families of groups, including mapping class groups, right-angled Coxeter groups, most 3-manifold groups, right-angled Artin groups, and many others. A group that admits an acylindrical action on a hyperbolic space may admit many such actions on different hyperbolic spaces. Read More


We study Riemannian metrics on compact, torsionless, non-geometric $3$-manifolds, i.e. whose interior does not support any of the eight model geometries. Read More


We survey the problem of separation under conjugacy and malnormality of the abelian peripheral subgroups of an orientable, irreducible $3$-manifold $X$. We shall focus on the relation between this problem and the existence of acylindrical splittings of $\pi_1(X)$ as an amalgamated product or HNN-extension along the abelian subgroups corresponding to the JSJ-tori. Read More


We show that there is a knot whose unknotting number cannot be determined using data only obtained from minimal crossing projections, thus giving a counterexample to the Bernhard-Jablan Conjecture. Read More


We relate some terms on the boundary of the Newton polygon of the Alexander polynomial $\Delta(x,y)$ of a rational link to the number and length of monochromatic twist sites in a particular diagram that we call the standard form. Normalize $\Delta(x,y)$ so that no $x^{-1}$ or $y^{-1}$ terms appear, but $x^{-1}\Delta(x,y)$ and $y^{-1}\Delta(x,y)$ have negative exponents, and so that terms of even total degree are positive and terms with odd total degree are negative. If the rational link has a reduced alternating diagram with no self crossings, then $\Delta(-1, 0) = 1$. Read More


Extended welded links are a generalization of Fenn, Rim\'{a}nyi, and Rourke's welded links. Their braided counterpart are extended welded braids, which are closely related to ribbon braids and loop braids. In this paper we prove versions of Alexander and Markov's theorems for extended welded braids and links, following Kamada's approach to the case of welded objects. Read More


The topology of $SU(3)$-representation varieties of the fundamental groups of planar webs so that the meridians are sent to matrices with trace equal to $-1$ are explored, and compared to data coming from spider evaluation of the webs. Corresponding to an evaluation of a web as a spider is a rooted tree. We associate to each geodesic $\gamma$ from the root of the tree to the tip of a leaf an irreducible component $C_{\gamma}$ of the representation variety of the web, and a graded subalgebra $A_{\gamma}$ of $H^*(C_{\gamma};\mathbb{Q})$. Read More


A meander is a topological configuration of a line and a simple closed curve in the plane (or a pair of simple closed curves on the 2-sphere) intersecting transversally. In physics, meanders provide a model of polymer folding, and their enumeration is directly related to the entropy of the associated dynamical systems. We combine recent results on Masur-Veech volumes of the moduli spaces of meromorphic quadratic differentials in genus zero and our previous result that horizontal and vertical separatrix diagrams of integer quadratic differentials are asymptotically uncorrelated to derive two applications to asymptotic enumeration of meanders. Read More


The interior polynomial is an invariant of bipartite graphs, and a part of the HOMFLY polynomial of a special alternating link coincides with the interior polynomial of the Seifert graph of the link. We extend the interior polynomial to signed bipartite graphs, and we show that, in the planar case, it is equal to a part of the HOMFLY polynomial of a naturally associated link. We also establish some other, more basic properties of this new notion. Read More


It is a well known result of Gromov that all manifolds of a given dimension with positive sectional curvature are subject to a universal bound on the sum of their Betti numbers. On the other hand, there is no such bound for manifolds with positive Ricci curvature: indeed, Perelman constructed positive Ricci metrics on $\#_k\mathbf{C}P^2$. In this paper, we revisit and extend Perelman's construction to show that $\#_k\mathbf{C}P^n$, $\#_k\mathbf{H}P^n$, and $\#_k\mathbf{O}P^2$ all admit metrics of positive Ricci curvature. Read More


We give an explicit formula for a 2-fold branched covering from $\mathbb{CP}^2$ to $S^4$, and relate it to other maps between quotients of $S^2\times{S^2}$. Read More


Periodic geodesics on the modular surface correspond to periodic orbits of the geodesic flow in its unit tangent bundle $\mathrm{PSL}_2(\mathbb{Z})\backslash\mathrm{PSL}_2(\mathbb{R})$. The complement of any finite number of orbits is a hyperbolic $3$-manifold, which thus has a well-defined volume. We present strong numerical evidence that, in the case of the set of geodesics corresponding to the ideal class group of a real quadratic field, the volume has linear asymptotics in terms of the total length of the geodesics. Read More


If the face-cycles at all the vertices in a map $X$ on a surface are of same type then the map $X$ is said to be a semi-equivelar map. There are eleven types of Archimedean tilings on the plane. All the Archimedean tilings are semi-equivelar maps. Read More


Let K be a nontrivial knot in the 3-sphere with the exterior E(K). A slope element u in the knot group G(K) is a nontrivial element represented by a simple closed curve on the boundary of E(K). Each slope element u defines a normal subgroup, the normal closure of u. Read More


We show that if a cusped hyperbolic manifold is Platonic, i.e., can be decomposed into isometric Platonic solids, it can also be decomposed into geodesic ideal tetrahedra. Read More


Let $S_{0,n}$ be an $n$-punctured sphere. For $n\geq 4$, we construct a sequence $(\mathcal{X}_i)_{i\in\mathbb{N}}$ of finite rigid sets in the pants graph $\mathcal{P}(S_{0,n})$ such that $\mathcal{X}_1 \subset \mathcal{X}_2 \subset .. Read More


We construct a family of right-angled Coxeter groups which provide counter-examples to questions about the stable boundary of a group, one-endedness of quasi-geodesically stable subgroups, and the commensurability types of right-angled Coxeter groups. Read More


The category of finite dimensional module over the quantum superalgebra U_q(sl(2|1)) is not semi-simple and the quantum dimension of a generic U_q(sl(2|1))-module vanishes. This vanishing happens for any value of q (even when q is not a root of unity). These properties make it difficult to create a fusion or modular category. Read More


In an $n$-manifold $X$ each element of $H_{n-1}(X; \mathbb{Z}_2)$ can be represented by an embedded codimension-1 submanifold. Hence for any two such submanifolds there is a third one that represents the sum of their homology classes. We construct such a representative explicitly. Read More


Let $M$ be a globally hyperbolic maximal compact $3$-dimensional spacetime locally modelled on Minkowski, anti-de Sitter or de Sitter space. It is well known that $M$ admits a unique foliation by constant mean curvature surfaces. In this paper we extend this result to singular spacetimes with particles (cone singularities of angles less than $\pi$ along time-like geodesics). Read More


We prove that, for any two finite volume hyperbolic $3$-manifolds, the amalgamation of their fundamental groups along any nontrivial geometrically finite subgroup is not LERF. This generalizes the author's previous work on nonLERFness of amalgamations of hyperbolic $3$-manifold groups along abelian subgroups. A consequence of this result is that closed arithmetic hyperbolic $4$-manifolds have nonLERF fundamental groups. Read More


In this article is we introduce a family of transverse invariants living in the deformations of Khovanov homology. This family include Plamenevskaya and Lipshitz-Ng-Sarkar invariants. In particular, we investigate two invariants coming from Bar-Natan deformation, which we call $\beta$-invariants. Read More


The computation of the fundamental group of the complement of an algebraic plane curve has been theoretically solved since Zariski-van Kampen, but actual computations are usually cumbersome. In this work, we describe the notion of Wirtinger presentation of such a group relying on the real picture of the curve and with the same combinatorial flavor as the classical Wirtinger presentation; we determine a significant family of curves for which Wirtinger presentation provides the required fundamental group. The above methods allow us to compute that fundamental group for an infinite subfamily of hypocycloids, relating them with Artin groups. Read More


We study the local invariants that a meromorphic $k$-differential on a Riemann surface of genus $g\geq0$ can have. These local invariants are the orders of zeros and poles, and the $k$-residues at the poles. We show that for a given pattern of orders of zeroes, there exists, up to a few exceptions, a primitive $k$-differential having these orders of zero. Read More


In this article, we use the recently developed mean curvature flow with surgery for 2 convex hypersurfaces to prove several isotopy existence and finally extrinsic finiteness results (in the spirit of Gromov's compactness theorem) for the space of 2 convex hypersurfaces in Rn1. Read More


We define and discuss fibered commensurability of outer automorphisms of the free groups, which lets us study symmetries of outer automorphisms. The notion of fibered commensurability is first defined by Calegari-Sun-Wang on mapping class groups. The Nielsen-Thurston type of mapping classes is a commensurability invariant. Read More


We consider commensurability of quadratic differentials on surfaces. Each commensurability class has a natural order by the covering relation. We show that each commensurability class contains a unique (orbifold) element. Read More


We consider C-homotopy classes of maps from manifolds to spaces and C-homotopy invariants of covers on spaces. We show that the C-homotopy invariants of covers on manifolds is equivalent to the C-homotopy classes of their associated maps. Moreover some C--homotopy groups of spheres and manifolds are determined with applications to homotopy theory of covers on spaces. Read More


We define the {\it Wirtinger number} of a link, an invariant closely related to the meridional rank. The Wirtinger number is the minimum number of generators of the fundamental group of the link complement over all meridional presentations in which every relation is an iterated Wirtinger relation arising in a diagram. We prove that the Wirtinger number of a link equals its bridge number. Read More


We construct a Hennings type logarithmic invariant for restricted quantum $\mathfrak{sl}(2)$ at a $2\mathsf{p}$-th root of unity. This quantum group $U$ is not braided, but factorizable. The invariant is defined for a pair: a 3-manifold $M$ and a colored link $L$ inside $M$. Read More


We show that for any C^0 Jordan curve C in the sphere at infinity of H^3, there exists an embedded $H$-plane P_H in H^3 with asymptotic boundary C for any H in (-1,1). As a corollary, we proved that any quasi-Fuchsian hyperbolic 3-manifold M=SxR contains an H-surface S_H in the homotopy class of the core surface S for any H in (-1,1). We also proved that for any C^1 Jordan curve J in the sphere at infinity, there exists a unique minimizing H-plane P_H with asymptotic boundary J for a generic H in (-1,1). Read More


This paper proves that every finite volume hyperbolic 3-manifold M contains a ubiquitous collection of closed, immersed, quasi-Fuchsian surfaces. These surfaces are ubiquitous in the sense that their preimages in the universal cover separate any pair of disjoint, non-asymptotic geodesic planes. The proof relies in a crucial way on the corresponding theorem of Kahn and Markovic for closed 3-manifolds. Read More