Latest in math.gt

total 17670took 0.10s
The Geometric Index and Attractors of Homeomorphisms of $\mathbb{R}^3$Sep 18 2019In this paper we focus on compacta $K \subseteq \mathbb{R}^3$ which possess a neighbourhood basis that consists of nested solid tori $T_i$. We call these sets toroidal. In \cite{hecyo1} we defined the genus of a toroidal set as a generalization of the ... More
Doubly slice knots and metabelian obstructionsSep 17 2019For $\ell >1$, we develop $L^{(2)}$-signature obstructions for $(4\ell-3)$-dimensional knots with metabelian knot groups to be doubly slice. For each $\ell>1$, we construct an infinite family of knots on which our obstructions are non-zero, but for which ... More
Representation stability for pure braid group Milnor fibersSep 17 2019We prove a representation stability result for the Milnor fiber associated to the pure braid group. Our result connects previous work of Simona Settepenella to representation stability in the sense of Church--Ellenberg--Farb, answering a question of Graham ... More
Natural maps for measurable cocycles of compact hyperbolic manifoldsSep 17 2019Let $\text{G}(n)$ be equal either to $\text{PO}(n,1),\text{PU}(n,1)$ or $\text{PSp}(n,1)$ and let $\Gamma \leq \text{G}(n)$ be a uniform lattice. Denote by $\mathbb{H}^n_K$ the hyperbolic space associated to $\text{G}(n)$, where $K$ is a division algebra ... More
An Asymptotic Analysis on Generalized Secretary ProblemSep 16 2019As a famous result, the ``37\% Law'' for Secretary Problem has widely influenced peoples' perception on online decision strategies about choice. However, using this strategy, too many attractive candidates may be rejected in the first 37\%, and in practice ... More
Subgroups of the mapping class group of the torus generated by powers of Dehn twistsSep 16 2019We study subgroups of the mapping class group of the torus generated by powers generated by powers of Dehn twists. We give a criterion to show when a collection of powers Dehn twists generates a free group using the ping pong lemma. We show that the subgroup ... More
Arbitrarily large torsion in Khovanov cohomologySep 16 2019For any positive integer $k$ and $p\in \{3,5,7\}$ we construct a link which has a direct summand $\mathbb{Z}/p^k\mathbb{Z}$ in its Khovanov cohomology.
Quadratic differentials and circle patterns on complex projective toriSep 16 2019Given a triangulation of a closed surface, we consider a cross ratio system that assigns a complex number to every edge satisfying certain polynomial equations per vertex. Every cross ratio system induces a complex projective structure together with a ... More
The octonionic projective planeSep 16 2019This small note, without claim of originality, constructs the projective plane over the octonionic numbers and recalls how this can be used to rule out the existence of higher-dimensional real division algebras, using Adams' solution of the Hopf invariant ... More
2-knot homology and Roseman moveSep 16 2019Ng constructed an invariant of knots in ${\mathbb{R}}^3$, a combinatorial knot contact homology. Extending his study, we construct an invariant of surface-knots in ${\mathbb{R}}^4$ using diagrams in ${\mathbb{R}}^3$.
The dimension of the image of the Abel map associated with normal surface singularitiesSep 16 2019Let $(X,o)$ be a complex normal surface singularity with rational homology sphere link and let $\widetilde{X}$ be one of its good resolutions. Fix an effective cycle $Z$ supported on the exceptional curve and also a possible Chern class $l'\in H^2(\widetilde{X},\mathbb{Z})$. ... More
2-knot homology and Yoshikawa moveSep 16 2019Ng constructed an invariant of knots in ${\mathbb{R}}^3$, a combinatorial knot contact homology. Extending his study, we construct an invariant of surface-knots in ${\mathbb{R}}^4$ using marked graph diagrams.
Link homology theories and ribbon concordancesSep 16 2019It was recently proved by several authors that ribbon concordances induce injective maps in knot Floer homology, Khovanov homology, and the Heegaard Floer homology of the branched double cover. We give a simple proof of a similar statement in a more general ... More
Knots with Exactly 10 SticksSep 16 2019We prove that the knots $13n_{592}$ and $15n_{41,127}$ both have stick number 10. These are the first non-torus prime knots with more than 9 crossings for which the exact stick number is known.
Fundamental domains in ${\rm PSL}(2,{\mathbb R})$ for Fuchsian groupsSep 15 2019In this paper, we provide a necessary and sufficient condition for a set in ${\rm PSL}(2,{\mathbb R})$ or in $T^1{\mathbb H}^2$ to be a fundamental domain of a given Fuchsian group via its respective fundamental domain in the hyperbolic plane ${\mathbb ... More
WWPD elements of big mapping class groupsSep 14 2019We study mapping class groups of infinite type surfaces with isolated punctures and their actions on the loop graphs introduced by Bavard-Walker. We classify all of the mapping classes in these actions which are loxodromic with a WWPD action on the corresponding ... More
Degeneration of 3-dimensional hyperbolic cone structures with decreasing cone anglesSep 14 2019For deformation of 3-dimensional hyperbolic cone structures about cone angles $\theta$, the local rigidity is known for $0 \leq \theta < 2\pi$, but the global rigidity is known only for $0 \leq \theta \leq \pi$. The proof of the global rigidity by Kojima ... More
Null, recursively starlike-equivalent decompositions shrinkSep 13 2019A subset $E$ of a metric space $X$ is said to be starlike-equivalent if it has a neighbourhood which is mapped homeomorphically into $\mathbb{R}^n$ for some $n$, sending $E$ to a starlike set. A subset $E\subset X$ is said to be recursively starlike-equivalent ... More
Volume entropy and lengths of homotopically independent loopsSep 13 2019This paper presents a new inequality for closed Riemannian manifolds involving the volume entropy and the set of lengths of any family of homotopically independent loops based at the same point. This inequality implies a curvature free collar theorem, ... More
Strategic Inference with a Single Private SampleSep 13 2019Motivated by applications in cyber security, we develop a simple game model for describing how a learning agent's private information influences an observing agent's inference process. The model describes a situation in which one of the agents (attacker) ... More
Twist left-veering open books and overtwisted contact structuresSep 13 2019We introduce a notion of $N$-twist left-veering mapping classes and prove that an open book with $N$-twist left-veering monodromy contains an overtwisted disk that intersects the binding at $N$ points. As an application we show that an open book is $2$-twist ... More
On the entropy norm on $Ham(S^2)$Sep 12 2019In this note we prove that for each positive integer $m$ there exists a bi-Lipschitz embedding $Z^m\to Ham(S^2)$, where $Ham(S^2)$ is equipped with the entropy metric. In particular, the same result holds when the entropy metric is substituted with the ... More
Harmonic Forms, Price Inequalities, and Benjamini-Schramm ConvergenceSep 12 2019We study Betti numbers of sequences of Riemannian manifolds which Benjamini-Schramm converge to their universal covers. Using the Price inequalities we developed elsewhere, we derive two distinct convergence results. First, under a negative Ricci curvature ... More
Quasi-morphisms on surface diffeomorphism groupsSep 12 2019We show that the identity component of the group of diffeomorphisms of a closed oriented surface of positive genus admits many unbounded quasi-morphisms. As a corollary, we also deduce that this group is not uniformly perfect and its fragmentation norm ... More
Choreography of divisors on algebraic curvesSep 12 2019For a non-singular real algebraic projective curve, topological restrictions on a closed motion of a simple real divisor in its linear equivalence class are found.
Topologically flat embedded 2-spheres in specific simply connected 4-manifoldsSep 12 2019In this note we study whether specific elements in the second homology of specific simply connected closed $4$-manifolds can be represented by smooth or topologically flat embedded spheres.
Connected sums of knots do not admit purely cosmetic surgeriesSep 11 2019Two Dehn surgeries on a knot are called purely cosmetic if their surgered manifolds are homeomorphic as oriented manifolds. Gordon conjectured that non-trivial knots in $S^3$ do not admit purely cosmetic surgeries. In this article, we confirm this conjecture ... More
Symplectic 4-Manifolds on the Noether Line and between the Noether and Half Noether LinesSep 10 2019We will construct minimal, simply connected and symplectic 4-manifolds on the Noether line and between the Noether and half Noether lines by star surgeries introduced by Karakurt and Starkston, and by using complex singularities. We will show that our ... More
Vanishing cycles of matrix singularitiesSep 10 2019The paper is on the vanishing topology of singular Milnor fibres of holomorphic families of arbitrary square, symmetric and skew-symmetric matrices with sufficiently many parameters. We define vanishing cycles on such fibres, prove an extended form of ... More
A Stochastic Knapsack Game: Revenue Management in CompetitionsSep 10 2019We study a mathematical model for revenue management under competition with multiple sellers. The model combines the stochastic knapsack problem, a classic revenue management model, with a non coorperative game model that characterizes the sellers' rational ... More
Equivariant Morse theory on Vietoris-Rips complexes & universal spaces for proper actionsSep 10 2019We formalize an equivariant version of Bestvina-Brady discrete Morse theory, and apply it to Vietoris-Rips complexes in order to exhibit finite universal spaces for proper actions for all asymptotically CAT(0) groups.
On symmetries of peculiar modules; or, $δ$-graded link Floer homology is mutation invariantSep 10 2019We investigate symmetry properties of peculiar modules, a Heegaard Floer invariant of 4-ended tangles which the author introduced in [arXiv:1712.05050]. In particular, we give an almost complete answer to the geography problem for components of peculiar ... More
Golden gamesSep 10 2019We consider extensive form win-lose games over a complete binary-tree of depth $n$ where players act in an alternating manner. We study arguably the simplest random structure of payoffs over such games where 0/1 payoffs in the leafs are drawn according ... More
The Chiral Anomaly of the Free Fermion in Functorial Field TheorySep 10 2019When trying to cast the free fermion in the framework of functorial field theory, its chiral anomaly manifests in the fact that it assigns the determinant of the Dirac operator to a top-dimensional closed spin manifold, which is not a number as expected, ... More
Two-Dimensional Extended Homotopy Field TheoriesSep 09 2019We define $2$-dimensional extended homotopy field theories (E-HFTs) with aspherical targets and classify them. When target is a $K(G,1)$-space, oriented E-HFTs taking values in the symmetric monoidal bicategory of algebras, bimodules, and bimodule maps ... More
Sensitivity Analysis for Markov Decision Process Congestion GamesSep 09 2019We consider a non-atomic congestion game where each decision maker performs selfish optimization over states of a common MDP. The decision makers optimize for their own expected costs, and influence each other through congestion effects on the state-action ... More
Sensitivity Analysis for Markov Decision Process Congestion GamesSep 09 2019Sep 12 2019We consider a non-atomic congestion game where each decision maker performs selfish optimization over states of a common MDP. The decision makers optimize for their own expected costs, and influence each other through congestion effects on the state-action ... More
The Multi-variable Affine Index PolynomialSep 09 2019We define a multi-variable version of the Affine Index Polynomial for virtual links. This invariant reduces to the original Affine Index Polynomial in the case of virtual knots, and also generalizes the version for compatible virtual links recently developed ... More
Boundary N=2 Theory, Floer Homologies, Affine Algebras, and the Verlinde FormulaSep 09 2019Generalizing our ideas in [arXiv:1006.3313], we explain how topologically-twisted N=2 gauge theory on a four-manifold with boundary, will allow us to furnish purely physical proofs of (i) the Atiyah-Floer conjecture, (ii) Munoz's theorem relating quantum ... More
Not even Khovanov homologySep 09 2019We construct a supercategory that can be seen as a skew version of (thickened) KLR algebras for the type $A$ quiver. We use our supercategory to construct homological invariants of tangles and show that for every link our invariant gives a link homology ... More
Crossing-changeable braids from chromatic configuration spacesSep 09 2019Motivated by the work in [15], this paper deals with the theory of the braids from chromatic configuration spaces. This kind of braids possess the property that some strings of each braid may intersect together and can also be untangled, so they are quite ... More
Relating tangle invariants for Khovanov homology and knot Floer homologySep 09 2019Ozsvath and Szabo recently constructed an algebraically defined invariant of tangles which takes the form of a DA bimodule. This invariant is expected to compute knot Floer homology. The authors have a similar construction for open braids and their plat ... More
Path-connectivity of the set of uniquely ergodic and cobounded foliationsSep 09 2019We show that for a closed surface of genus at least 5, or a surface of genus at least 2 with at least 3 marked points, the set of uniquely ergodic foliations and the set of cobounded foliations is path-connected and locally path-connected.
Optimal Lipschitz Maps on One-holed Tori and the Thurston Metric Theory of Teichmueller SpaceSep 09 2019We study Thurston's Lipschitzand curve metrics, as well as the arc metric on the Teichmueller space of the torus equipped with hyperbolic metrics eith one boundary component of fixed length. We construct natural Lipschitz maps between two such hyperbolic ... More
Decomposability of the Higson coronae of finitely generated groups with one endSep 08 2019We characterize a space which is coarsely equivalent to the space of natural numbers using the indecomposability of its Higson corona. This leads to the characterization that a finitely generated group has exactly one end if and only if its Higson corona ... More
Geometric intersections of loops on surfacesSep 08 2019Based on Nielsen fixed point theory and \gr basis, we give a simple method to compute geometric intersection number and self-intersection of loops on surfaces.
Annular link invariants from the Sarkar-Seed-Szabó spectral sequenceSep 08 2019For a link in a thickened annulus $A \times I$, we define a $\mathbb{Z} \oplus \mathbb{Z} \oplus \mathbb{Z}$ filtration on Sarkar-Seed-Szab\'o's perturbation of the geometric spectral sequence. The filtered chain homotopy type is an invariant of the isotopy ... More
First Passage Percolation on Hyperbolic groupsSep 08 2019We study first passage percolation (FPP) on a Gromov-hyperbolic group $G$ with boundary $\partial G$ equipped with the Patterson-Sullivan measure $\nu$. We associate an i.i.d.\ collection of random passage times to each edge of a Cayley graph of $G$, ... More
Circle patterns on surfaces of finite topological typeSep 08 2019This paper investigates circle patterns with obtuse exterior intersection angles on surfaces of finite topological type. We characterise the images of the curvature maps and establish several equivalent conditions regarding long time behaviors of Chow-Luo's ... More
Characterizing the interplay between information and strength in Blotto gamesSep 08 2019In this paper, we investigate informational asymmetries in the Colonel Blotto game, a game-theoretic model of competitive resource allocation between two players over a set of battlefields. The battlefield valuations are subject to randomness. One of ... More
Characterizing the interplay between information and strength in Blotto gamesSep 08 2019Sep 13 2019In this paper, we investigate informational asymmetries in the Colonel Blotto game, a game-theoretic model of competitive resource allocation between two players over a set of battlefields. The battlefield valuations are subject to randomness. One of ... More
The Roger-Yang skein algebra and the decorated Teichmuller spaceSep 06 2019Based on hyperbolic geometric considerations, Roger and Yang introduced an extension of the Kauffman bracket skein algebra that includes arcs. In particular, their skein algebra is a deformation quantization of a certain commutative curve algebra, and ... More
Modelling Cooperation in a Dynamic Healthcare SystemSep 06 2019Our research is concerned with studying behavioural changes within a dynamic system, i.e. health care, and their effects on the decision-making process. Evolutionary Game theory is applied to investigate the most probable strategy(ies) adopted by individuals ... More
A Reinforcement Learning Based Approach for Joint Multi-Agent Decision MakingSep 06 2019Reinforcement Learning (RL) is being increasingly applied to optimize complex functions that may have a stochastic component. RL is extended to multi-agent systems to find policies to optimize systems that require agents to coordinate or to compete under ... More
Dynamics for discrete subgroups of $\operatorname{SL}_2(\mathbb C)$Sep 06 2019Margulis wrote in the preface of his book Discrete subgroups of semisimple Lie groups that "A number of important topics have been omitted. The most significant of these is the theory of Kleinian groups and Thurston's theory of 3-dimensional manifolds: ... More
Dynamics for discrete subgroups of $\operatorname{SL}_2(\mathbb C)$Sep 06 2019Sep 09 2019Margulis wrote in the preface of his book Discrete subgroups of semisimple Lie groups that "A number of important topics have been omitted. The most significant of these is the theory of Kleinian groups and Thurston's theory of 3-dimensional manifolds: ... More
Identification of 2-bridge linksSep 06 2019We find all 2-Bridge links up to 11 crossings and locate them in Thistlethwaite's link table. The splitting numbers of some links are calculated by the help of this identification.
The Impact of Complex and Informed Adversarial Behavior in Graphical Coordination GamesSep 05 2019Sep 13 2019How does system-level information impact the ability of an adversary to degrade performance in a networked control system? How does the complexity of an adversary's strategy affect its ability to degrade performance? This paper focuses on these questions ... More
The Impact of Complex and Informed Adversarial Behavior in Graphical Coordination GamesSep 05 2019How does system-level information impact the ability of an adversary to degrade performance in a networked control system? How does the complexity of an adversary's strategy affect its ability to degrade performance? This paper focuses on these questions ... More
On the top dimensional cohomology groups of congruence subgroups of $\text{SL}_n(\mathbb{Z})$Sep 05 2019Let $\Gamma_n(p)$ be the level-$p$ principal congruence subgroup of $\text{SL}_n(\mathbb{Z})$. Borel-Serre proved that the cohomology of $\Gamma_n(p)$ vanishes above degree $\binom{n}{2}$. We study the cohomology in this top degree $\binom{n}{2}$. Let ... More
Connected sum decompositions of high-dimensional manifoldsSep 05 2019The classical Kneser-Milnor theorem says that every closed oriented connected 3-dimensional manifold admits a unique connected sum decomposition into manifolds that cannot be decomposed any further. We discuss to what degree such decompositions exist ... More
Two-bridge knots admit no purely cosmetic surgeriesSep 05 2019We show that two-bridge knots and alternating fibered knots admit no purely cosmetic surgeries, i.e., no pair of distinct Dehn surgeries on such a knot produce 3-manifolds that are homeomorphic as oriented manifolds. Our argument, based on a recent result ... More
$C^0$ stability of boundary actions and inequivalent Anosov flowsSep 05 2019We give a $C^0$--structural stability result for the action of the fundamental group of a compact manifold of negative curvature on its boundary at infinity: any nearby action of this group by homeomorphisms of the sphere is semi-conjugate to the standard ... More
State sums for some super quantum link invariantsSep 05 2019We present state sums for quantum link invariants arising from the representation theory of $U_q(\mathfrak{gl}_{N|M})$. We investigate the case of the $N$-th exterior power of the standard representation of $U_q(\mathfrak{gl}_{N|1})$ and explicit the ... More
Knots with Prism Manifold SurgeriesSep 05 2019Ballinger et al. have determined the list of all prism manifolds that are possibly realizable by Dehn surgeries on knots in $S^3$. In this paper, we explicitly find braid words of primitive/Seifert-fibered knots on which surface slope surgeries yield ... More
Sublinearly Morse Boundary I: CAT(0) SpacesSep 04 2019To every Gromov hyperbolic space X one can associate a space at infinity called the Gromov boundary of X. Gromov showed that quasi-isometries of hyperbolic metric spaces induce homeomorphisms on their boundaries, thus giving rise to a well-defined notion ... More
Trimmed sums of twists and the area Siegel-Veech constantSep 04 2019We relate trimmed sums of twists in cylinders along a typical Teichmuller geodesic to the area Siegel-Veech constant.
Existence of minimal hypersurfaces with non-empty free boundary for generic metricsSep 04 2019For almost all Riemannian metrics (in the $C^\infty$ Baire sense) on a compact manifold with boundary $(M^{n+1},\partial M)$, $3\leq (n + 1)\leq 7$, we prove that, for any open subset $V$ of $\partial M$, there exists a compact, properly embedded free ... More
Stability phenomena for Martin boundaries of relatively hyperbolic groupsSep 04 2019Let $\Gamma$ be a relatively hyperbolic group and let $\mu$ be an admissible symmetric finitely supported probability measure on $\Gamma$. We extend Floyd-Ancona type inequalities up to the spectral radius of $\mu$. We then show that when the parabolic ... More
A Surgery Formula for the Casson-Seiberg-Witten Invariant of Integral Homology $S^1 \times S^3$Sep 04 2019We prove a surgery formula of the Casson-Seiberg-Witten invariant of integral homology $S^1 \times S^3$ along an embedded torus, which could either be regarded as an extension of the product formula for Seiberg-Witten invariants or a manifestation of ... More
Local topology of a deformation of a function-germ with a one-dimensional critical setSep 04 2019The Brasselet number of a function $f$ with nonisolated singularities describes numerically the topological information of its generalized Milnor fibre. In this work, we consider two function-germs $f,g:(X,0)\rightarrow(\mathbb{C},0)$ such that $f$ has ... More
Separating subgroups of mapping class groups in homological representationsSep 03 2019Let $\Gamma$ be either the mapping class group of a closed surface of genus $\geq 2$, or the automorphism group of a free group of rank $\geq 3$. Given any homological representation $\rho$ of $\Gamma$ corresponding to a finite cover, and any term $\mathcal{I}_k$ ... More
Profinite rigidity for twisted Alexander polynomialsSep 03 2019We formulate and prove a profinite rigidity theorem for the twisted Alexander polynomials up to several types of finite ambiguity. We also establish torsion growth formulas of the twisted homology groups in a $\mathbb{Z}$-cover of a 3-manifold with use ... More
The full spectrum of scl on recursively presented groupsSep 03 2019We show that the set $SCL^{rp}$ of stable commutator lengths on recursively presented groups equals the set of non-negative right-computable numbers. Hence all non-negative algebraic or computable numbers are in $SCL^{rp}$ and $SCL^{rp}$ is not closed ... More
On symplectic fillings of virtually overtwisted torus bundlesSep 03 2019We use Menke's JSJ-type decomposition theorem for symplectic fillings to reduce the classification of strong and exact symplectic fillings of virtually overtwisted torus bundles to the same problem for tight lens spaces. For virtually overtwisted structures ... More
Stick number of non-paneled knotless spatial graphsSep 03 2019We show that the minimum number of sticks required to construct a non-paneled knotless embedding of $K_4$ is 8 and of $K_5$ is 12 or 13. We use our results about $K_4$ to show that the probability that a random linear embedding of $K_{3,3}$ in a cube ... More
The dualizing module and top-dimensional cohomology group of $\text{GL}_n(\mathcal{O})$Sep 03 2019For a number ring $\mathcal{O}$, Borel and Serre proved that $\text{SL}_n(\mathcal{O})$ is a virtual duality group whose dualizing module is the Steinberg module. They also proved that $\text{GL}_n(\mathcal{O})$ is a virtual duality group. In contrast ... More
Untwisting 3-strand torus knotsSep 03 2019We prove that the signature bound for the topological 4-genus of 3-strand torus knots is sharp, using McCoy's twisting method. We also show that the bound is off by at most 1 for 4-strand and 6-strand torus knots, and improve the upper bound on the asymptotic ... More
Spaces of knotted circles and exotic smooth structuresSep 03 2019Suppose that $N_1$ and $N_2$ are closed smooth manifolds of dimension $n$ that are homeomorphic. We prove that the spaces of smooth knots $Emb(S^1, N_1)$ and $Emb(S^1, N_2)$ have the same homotopy $(2n-7)$-type. In the 4-dimensional case this means that ... More
Asymptotic translation lengths and normal generations of pseudo-Anosov monodromies for fibered 3-manifoldsSep 03 2019Let $M$ be a hyperbolic fibered 3-manifold. We study properties of sequences $(S_{\alpha_n}, \psi_{\alpha_n})$ of fibers and monodromies for primitive integral classes in the fibered cone of $M$. The main tool is the asymptotic translation length $\ell_{\mathcal{C}} ... More
Corrigendum to "Graphs of hyperbolic groups and a limit set intersection theorem"Sep 03 2019The purpose of this article is to point out a mistake in the published paper "Graphs of hyperbolic groups and limit set intersection theorem- Proc AMS, vol 146, no 5, pp 1859--1871, which subsequently weakens the main theorem of that paper. We state and ... More
Lightlike and ideal tetrahedraSep 03 2019We give a unified description of tetrahedra with lightlike faces in 3d anti-de Sitter, de Sitter and Minkowski spaces and of their duals in 3d anti-de Sitter, hyperbolic and half-pipe spaces. We investigate the geometry of these tetrahedra and show that ... More
New Stick Number Bounds from Random Sampling of Confined PolygonsSep 03 2019The stick number of a knot is the minimum number of segments needed to build a polygonal version of the knot. Despite its elementary definition and relevance to physical knots, the stick number is poorly understood: for most knots we only know bounds ... More
A Matsumoto-Mostow result for Zimmer's cocycles of hyperbolic latticesSep 02 2019As for the theory of maximal representations, we introduce the volume of a Zimmer's cocycle $\Gamma \times X \rightarrow \mbox{PO}^\circ(n, 1)$, where $\Gamma$ is a torsion-free (non-)uniform lattice in $\mbox{PO}^\circ(n, 1)$, with $n \geq 3$, and $X$ ... More
Brasselet number and function-germs with a one-dimensional critical setSep 02 2019The Brasselet number of a function $f$ with nonisolated singularities describes numerically the topological information of its generalized Milnor fibre. In this work, using the Brasselet number, we present several formulas for germs $f:(X, 0) \rightarrow ... More
Asymmetric L-space knots by experimentSep 02 2019Dunfield found 9 manifolds in the SnapPy census that are both asymmetric and complements of L-space knots in $S^3$. Using SnapPy and KLO we find presentations of these knots as closures of positive braids. The smallest of these has genus 12 and braid ... More
Hierarchically hyperbolic groups and uniform exponential growthSep 01 2019We give several sufficient conditions for uniform exponential growth in the setting of virtually torsion-free hierarchically hyperbolic groups. For example, any hierarchically hyperbolic group that is also acylindrically hyperbolic has uniform exponential ... More
Actions of $2$-groups of bounded exponent on manifoldsAug 31 2019In this paper, we show that an infinite 2-group of bounded exponent cannot act faithfully and smoothly on compact manifolds.
Biquandle Brackets and KnotoidsAug 31 2019Biquandle brackets are a type of quantum enhancement of the biquandle counting invariant for oriented knots and links, defined by a set of skein relations with coefficients which are functions of biquandle colors at a crossing. In this paper we use biquandle ... More
Presentations for the Euclidean Picard modular groupsAug 31 2019Mark and Paupert devised a general method for obtaining presentations for arithmetic non-cocompact lattices, $\Gamma$, in isometry groups of negatively curved symmetric spaces. The method involves a classical theorem of Macbeath applied to a $\Gamma$-invariant ... More
Harmonic 2-forms and positively curved 4-manifoldsAug 31 2019We prove that if a compact Riemannian 4-manifold with positive sectional curvature satisfies a Kato type inequality, then it is definite. We also discuss some new insights for compact Riemannian 4-manifolds of positive sectional curvature.
Geometric non-commutative geometryAug 30 2019In a recent paper, the authors proved that no spin foliation on a compact enlargeable manifold with Hausdorff homotopy graph admits a metric of positive scalar curvature on its leaves. This result extends groundbreaking results of Lichnerowicz, Gromov ... More
On triple-crossing projections, moves on knots and links, and their minimal diagramsAug 30 2019In this paper we present a systematic method to generate prime knot and prime link minimal triple-point projections, and then classify all classical prime knots and prime links with triple-crossing number at most four. We also extend the table of known ... More
On the mapping class groups of strongly irreducible Heegaard splittingsAug 30 2019We show that for any $g \geq 3$ and $n \geq 2$, there exists a genus-$g$ Heegaard splitting of distance $n$ whose mapping class group is the trivial group or $\mathbb{Z} / 2 \mathbb{Z}$. We also show that there exist Heegaard splittings of distance $2$ ... More
A Generalization of the Tristram-Levine Knot Signatures as a Singular Furuta-Ohta Invariant for ToriAug 29 2019Given a knot $K$ inside an integer homology sphere $Y$, the Casson-Lin-Herald invariant can be interpreted as a signed count of conjugacy classes of irreducible representations of the knot complement into $SU(2)$ which map the meridian of the knot to ... More
The Lubin-Tate Theory of Configuration Spaces: IAug 29 2019We construct a spectral sequence converging to the Morava $E$-theory of unordered configuration spaces and identify its E$^2$-page as the homology of a Chevalley-Eilenberg-like complex for Hecke Lie algebras. Based on this, we compute the $E$-theory of ... More
The mapping class group is generated by two commutatorsAug 29 2019We show that the mapping class group of any closed connected orientable surface of genus at least five is generated by only two commutators, and if the genus is three or four, by three commutators.
The local-to-global property for Morse quasi-geodesicsAug 29 2019Sep 12 2019We show the mapping class group, CAT(0) groups, the fundamental groups of closed 3-manifolds, and certain relatively hyperbolic groups have a local-to-global property for Morse quasi-geodesics. This allows us to generalize combination theorems of Gitik ... More
The local-to-global property for Morse quasi-geodesicsAug 29 2019We show the mapping class group, CAT(0) groups, the fundamental groups of closed 3-manifolds, and certain relatively hyperbolic groups have a local-to-global property for Morse quasi-geodesics. This allows us to generalize combination theorems of Gitik ... More
Forbidden detour number on virtual knotAug 29 2019We show that the forbidden detour move, essentially introduced by Kanenobu and Nelson, is an unknotting operation for virtual knots. Then we define the forbidden detour number of a virtual knot to be the minimal number of forbidden detour moves necessary ... More