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Coarse free productsMay 16 2019We define a notion of free product for coarse spaces that generalizes the corresponding notion of a free product for groups. We show that free products preserve coarse properties such as coarse property C, finite coarse decomposition complexity, and coarse ... More

On planar Cayley graphs and Kleinian groupsMay 16 2019Let $G$ be a finitely generated group acting faithfully and properly discontinuously by homeomorphisms on a planar surface $X \subseteq \mathbb{S}^2$. We prove that $G$ admits such an action that is in addition co-compact, provided we can replace $X$ ... More

A note on chirally cosmetic surgery on cable knotsMay 16 2019We show that a $(p,q)$-cable of a non-trivial knot $K$ does not admit chirally cosmetic surgery for $q\neq 2$, or $q=2$ with additional assumptions. In particular, we show that $(p,q)$-cable of non-trivial knot $K$ does not admit chirally cosmetic surgery ... More

Legendrian Rack Invariants of Legendrian KnotsMay 15 2019We define a new algebraic structure called Legendrian racks or racks with Legendrian structure, motivated by the front-projection Reidemeister moves for Legendrian knots. We provide examples of Legendrian racks and use these algebraic structures to define ... More

Agol's theorem on hyperbolic cubulationsMay 15 2019The following theorem was proven by Agol: let $G$ be a hyperbolic group acting properly and cocompactly on a CAT(0) cube complex $X$, then $G$ has a finite index subgroup $G'$ that acts freely on $X$ such that the quotient $X/G'$ is special. The aim of ... More

Möbius invariant metrics on the space of knotsMay 15 2019We introduce a method to give M\"obius invariant weighted inner products on the tangent spaces of the space of non-circular knots by making use of M\"obius invariant energies of knots, by which we can obtain M\"obius invariant gradients of such energies. ... More

On O'hara knot energies I: Regularity for critical knotsMay 15 2019We develop a regularity theory for extremal knots of scale invariant knot energies defined by J. O'hara in 1991. This class contains as a special case the M\"obius energy. For the M\"obius energy, due to the celebrated work of Freedman, He, and Wang, ... More

Cusp cobordism group of Morse functionsMay 14 2019By a Morse function on a compact manifold with boundary we mean a real-valued function without critical points near the boundary such that its critical points as well as the critical points of its restriction to the boundary are all non-degenerate. For ... More

Capacity and Price Competition in Markets with Congestion EffectsMay 14 2019We study oligopolistic competition in service markets where firms offer a service to customers. The service quality of a firm (from a customer's perspective) depends on the level of congestion and the charged price. A firm can set a price for the service ... More

On a Poincaré polynomial from Khovanov homology and Vassiliev invariantsMay 14 2019We introduce a Poincar\'{e} polynomial with two-variable $t$ and $x$ for knots, derived from Khovanov homology, where the specialization $(t, x)$ $=$ $(1, -1)$ is a Vassiliev invariant of order $n$. Since for every $n$, there exist non-trivial knots with ... More

Double branched covers of tunnel number one knotsMay 14 2019We provide criteria ensuring that a tunnel number one knot $K$ is not determined by its double branched cover, in the sense that the double branched cover is also the double branched cover of a knot $K'$ not equivalent to $K$.

Desingularizing positive scalar curvature 4-manifoldsMay 13 2019We show that the bordism group of closed 3-manifolds with positive scalar curvature (psc) metrics is trivial by explicit methods. Our constructions are derived from scalar-flat K{\"a}hler ALE surfaces discovered by Lock-Viaclovsky. Next, we study psc ... More

(0,2) Dualities and the 4-SimplexMay 13 2019We propose that a simple, Lagrangian 2d $\mathcal{N}=(0, 2)$ duality interface between the 3d $\mathcal{N}=2$ XYZ model and 3d $\mathcal{N}=2$ SQED can be associated to the simplest triangulated 4-manifold: the 4-simplex. We then begin to flesh out a ... More

Bieberbach groups and flat manifolds with finite abelian holonomy from Artin braid groupsMay 13 2019Let $n\geq 3$. In this paper we show that for any finite abelian subgroup $G$ of $S_n$ the crystallographic group $B_n/[P_n,P_n]$ has Bieberbach subgroups $\Gamma_{G}$ with holonomy group $G$. Using this approach we obtain an explicit description of the ... More

Volumes of random 3-manifoldsMay 13 2019We prove a law of large numbers for the volumes of families of random hyperbolic mapping tori and Heegaard splittings providing a sharp answer to a conjecture of Dunfield and Thurston.

Alexander polynomials of simple-ribbon knotsMay 13 2019In a previous paper, we introduced special types of fusions, so called simple-ribbon fusions on links. A knot obtained from the trivial knot by a finite sequence of simple-ribbon fusions is called a simple-ribbon knot. Every ribbon knot with <10 crossings ... More

Persistently Foliar Composite KnotsMay 13 2019A knot $\kappa$ in $S^3$ is persistently foliar if, for each boundary slope, there is a co-oriented taut foliation meeting the boundary of the knot complement transversely in a foliation by curves of that slope. For rational slopes, these foliations may ... More

A hyperbolic counterpart to Rokhlin's cobordism theoremMay 12 2019The purpose of the present note is to prove existence of super-exponentially many compact orientable hyperbolic arithmetic $n$-manifolds that are geometric boundaries of compact orientable hyperbolic $(n+1)$-manifolds, for each $2 \leq n \leq 8$, thereby ... More

L-space surgeries on 2-component L-space linksMay 12 2019In this paper, we analyze L-space surgeries on two component L-space links. We show that if one surgery coefficient is negative for the L-space surgery, then the corresponding link component is an unknot. If the link admits very negative (i.e. $d_{1}, ... More

Limited Resource Optimal Distribution Algorithm Based on Game Iteration MethodMay 11 2019The article provides a solution algorithm for the linear programming problem (LPP) with the latter being presented as an antagonistic matrix game so the game's further solution is based on the iterative method. The algorithm is presented as a computer ... More

Effective counting of simple closed geodesics on hyperbolic surfacesMay 11 2019We prove a quantitative estimate, with a power saving error term, for the number of simple closed geodesics of length at most $L$ on a closed hyperbolic surface of genus $g$. The proof relies on the exponential mixing rate for the Teichm\"{u}ller geodesic ... More

Systole on locally symmetric spacesMay 10 2019Here we survey on the growth of systoles of arithmetic locally symmetric spaces under the congruence covering and give simple proofs for the best possible constants of Gromov for several important classes of symmetric spaces.

Asymptotics for the number of Simple $(4a+1)$-Knots of Genus 1May 10 2019We investigate the asymptotics of the total number of simple $4a+1$-knots with Alexander polynomial of the form $mt^2 +(1-2m) t + m$ for some $m \in [-X, X]$. Using Kearton and Levine's classification of simple knots, we give equivalent algebraic and ... More

Average Weights and Power in Weighted Voting GamesMay 10 2019We investigate a class of weighted voting games for which weights are randomly distributed over the unit simplex. We provide close-formed formulae for the expectation and density of the distribution of weight of the $k$-th largest player under the uniform ... More

Asymptotics of multivariate sequences in the presence of a lacunaMay 10 2019We explain a discontinuous drop in the exponential growth rate for certain multivariate generating functions at a critical parameter value, in even dimensions $d \geq 4$. This result depends on computations in the homology of the algebraic variety where ... More

Parity in KnotoidsMay 10 2019This paper investigates the parity concept in knotoids in $S^2$ and in $\mathbb{R}^2$ in relation with virtual knots. We show that the virtual closure map is not surjective and give specific examples of virtual knots that are not in the image. We introduce ... More

Filtered instanton Floer homology and the homology cobordism groupMay 10 2019For any $s \in \mathbb{R}_{\leq 0} \cup \{-\infty\}$ and oriented homology $3$-sphere $Y$ , we introduce a homology cobordism invariant $r_s(Y )$ whose value is in $ \mathbb{R}_{>0} \cup \{\infty \}$. The values $\{r_s (Y )\}$ are contained in the critical ... More

On Geodesic Triangles in Hyperbolic PlaneMay 09 2019Let M be an orientable hyperbolic surface without boundary and let $\gamma$ be a closed geodesic in M. We prove that any side of any triangle formed by distinct lifts of $\gamma$ in H2 is shorter than $\gamma$.

Topological spines of 4-manifoldsMay 09 2019We show that infinitely many of the simply connected 4-manifolds constructed by Levine and Lidman that do not admit PL spines actually admit topological spines.

A Fox-Milnor Theorem for Knots in a Thickened SurfaceMay 09 2019A knot in a thickened surface $K$ is a smooth embedding $K:S^1 \rightarrow \Sigma \times [0,1]$, where $\Sigma$ is a closed, connected, orientable surface. There is a bijective correspondence between knots in $S^2 \times [0,1]$ and knots in $S^3$, so ... More

Writhe polynomials and shell moves for virtual knots and linksMay 09 2019The writhe polynomial is a fundamental invariant of an oriented virtual knot. We introduce a kind of local moves for oriented virtual knots called shell moves. The first aim of this paper is to prove that two oriented virtual knots have the same writhe ... More

Representations of braid groups and construction of projective surfacesMay 09 2019Braid groups are an important and flexible tool used in several areas of science, such as Knot Theory (Alexander's theorem), Mathematical Physics (Yang-Baxter's equation) and Algebraic Geometry (monodromy invariants). In this note we will focus on their ... More

Classical shadows of stated skein representations at roots of unityMay 09 2019We extend some results of Bonahon, Bullock, Turaev and Wong concerning the skein algebras of closed surfaces to L^e's stated skein algebra associated to open surfaces. We prove that the stated skein algebra with deforming parameter +1 embeds canonically ... More

Upper bounds on Renormalized Volume for Schottky groupsMay 08 2019In this article we show that for any given Riemann surface $\Sigma$ of genus $g$, we can bound (from above) the renormalized volume of a (hyperbolic) Schottky group with boundary at infinity conformal to $\Sigma$ in terms of the genus and the combined ... More

$k$-differentials on curves and rigid cycles in moduli spaceMay 08 2019For $g\geq2$, $j=1,\dots,g$ and $n\geq g+j$ we exhibit infinitely many new rigid and extremal effective codimension $j$ cycles in $\overline{\mathcal{M}}_{g,n}$ from the strata of quadratic differentials and projections of these strata under forgetful ... More

Surface braid groups, finite Heisenberg covers and double Kodaira fibrationsMay 08 2019We exhibit new examples of double Kodaira fibrations by using finite Galois covers of a product $\Sigma_b \times \Sigma_b$, where $\Sigma_b$ is a smooth projective curve of genus $b \geq 2$. Each cover is obtained by providing an explicit group epimorphism ... More

A diagrammatic presentation and its characterization of non-split compact surfaces in the 3-sphereMay 08 2019We give a presentation for a non-split compact surface embedded in the 3-sphere $S^3$ by using diagrams of spatial trivalent graphs equipped with signs and we define Reidemeister moves for such signed diagrams. We show that two diagrams of embedded surfaces ... More

Linear extensions of multiple conjugation quandles and MCQ Alexander pairsMay 07 2019A quandle is an algebra whose axioms are motivated from knot theory. A linear extension of a quandle can be described by using a pair of maps called an Alexander pair. In this paper, we show that a linear extension of a multiple conjugation quandle can ... More

Knots not concordant to L-space knotsMay 07 2019In this short note we use methods of Friedl, Livingston and Zentner to show that there are knots that are not algebraically concordant to a connected sum of positive and negative L-space knots.

String$\mathbf{^c}$ Structures and Modular InvariantsMay 06 2019In this paper, we study some algebraic topology aspects of String$^c$ structures, more precisely, from the aspect of Whitehead tower and the aspect of the loop group of $Spin^c(n)$. We also extend the generalized Witten genus constructed for the first ... More

Modularity and value distribution of quantum invariants of hyperbolic knotsMay 06 2019We obtain an exact modularity relation for the $q$-Pochhammer symbol. Using this formula, we show that Zagier's modularity conjecture holds for hyperbolic knots $K\neq 7_2$ with at most seven crossings. For $K=4_1$, we also prove a complementary reciprocity ... More

Topological manifold bundles and the $A$-theory assembly mapMay 06 2019We give a new proof of an index theorem for fiber bundles of compact $topological$ manifolds due to Dwyer, Weiss, and Williams, which asserts that the parametrized $A$-theory characteristic of such a fiber bundle factors canonically through the assembly ... More

Well-quasi-order of plane minors and an application to link diagramsMay 06 2019A plane graph $H$ is a {\em plane minor} of a plane graph $G$ if there is a sequence of vertex and edge deletions, and edge contractions performed on the plane, that takes $G$ to $H$. Motivated by knot theory problems, it has been asked if the plane minor ... More

Connected sums of almost complex manifolds, products of rational homology spheres, and the twisted spin^c Dirac operatorMay 05 2019We record an answer to the question "In which dimensions is the connected sum of two closed almost complex manifolds necessarily an almost complex manifold?". In the process of doing so, we are naturally led to ask "For which values of l is the connected ... More

A tour through Mirzakhani's work on moduli spaces of Riemann surfacesMay 05 2019We survey Mirzakhani's work relating to Riemann surfaces, which spans about 20 papers. We target the discussion at a broad audience of non-experts.

On Legendrian products and twist spunsMay 04 2019The Legendrian product of two Legendrian knots, as defined by Lambert-Cole, is a Legendrian torus. We show that this Legendrian torus is a twist spun whenever one of the Legendrian knot components is sufficiently large. We then study examples of Legendrian ... More

On Goussarov-Polyak-Viro Conjecture of knots with degree threeMay 04 2019A knot invariant ordered by filtered finite dimensional vector spaces is called finite type. It has been conjectured that every finite type invariant of classical knots could be extended to a finite type invariant of long virtual knots (Goussarov-Polyak-Viro ... More

Dynamics on the Morse BoundaryMay 04 2019Let $X$ be a proper geodesic metric space and let $G$ be a group of isometries of $X$ which acts geometrically. Cordes constructed the Morse boundary of $X$ which generalizes the contracting boundary for CAT(0) spaces and the visual boundary for hyperbolic ... More

A partial order on multibranched surfaces in 3-manifoldsMay 03 2019In this paper, we introduce a partial order on neighborhood equivalence classes of essential maximally spread multibranched surfaces embedded in a 3-manifold. We show that if an essential maximally spread multibranched surface is atoroidal and acylindrical, ... More

On the barycentric extensionMay 02 2019In this paper, we will study the Douady-Earle / barycentric extension of maps on $S^{n-1}$. We will show the extension is uniformly Lipschitz if the map is quasiregular. In particular, we will show the barycentric extension for a rational map $f$ on $S^2$ ... More

Instantons, Bar-Natan homology, and some concordance invariants of knotsMay 02 2019May 05 2019A spectral sequence is established, from Bar-Natan's variant of Khovanov homology to a deformation of instanton homology for knots and links. This spectral sequence arises as a specialization of a spectral sequence from a characteristic-2 version of $F_5$ ... More

Instantons, Bar-Natan homology, and some concordance invariants of knotsMay 02 2019A spectral sequence is established, from Bar-Natan's variant of Khovanov homology to a deformation of instanton homology for knots and links. This spectral sequence arises as a specialization of a spectral sequence from a characteristic-2 version of $F_5$ ... More

Shadows of acyclic 4-manifolds with sphere boundaryMay 02 2019In terms of Turaev's shadows, we provide a sufficient condition for a compact, smooth, acyclic 4-manifold with boundary the 3-sphere to be diffeomorphic to the standard 4-ball. As a consequence, we prove that if a compact, smooth, acyclic 4-manifold with ... More

Distances between surfaces in 4-manifoldsMay 02 2019If $\Sigma$ and $\Sigma'$ are homotopic embedded surfaces in a $4$-manifold then they may be related by a regular homotopy (at the expense of introducing double points) or by a sequence of stabilisations and destabilisations (at the expense of adding ... More

Static Pricing: Universal Guarantees for Reusable ResourcesMay 02 2019We consider a fundamental pricing model in which a fixed number of units of a reusable resource are used to serve customers. Customers arrive to the system according to a stochastic process and upon arrival decide to purchase or not the service depending ... More

Static Pricing: Universal Guarantees for Reusable ResourcesMay 02 2019May 13 2019We consider a fundamental pricing model in which a fixed number of units of a reusable resource are used to serve customers. Customers arrive to the system according to a stochastic process and upon arrival decide whether or not to purchase the service, ... More

Some Structure Properties of Finite Normal-Form GamesMay 02 2019Game theory provides a mathematical framework for analysing strategic situations involving at least two players. Normal-form games model situations where the players simultaneously pick their moves. In this thesis we explore the strategic structure of ... More

Higher Arity Self-Distributive Operations in Cascades and their CohomologyMay 01 2019We investigate constructions and relations of higher arity self-distributive operations and their cohomology. We study the categories of mutually distributive structures both in the binary and ternary settings and their connections through functors. This ... More

Can tangle calculus be applicable to hyperpolynomials?May 01 2019We make a new attempt at the recently suggested program to express knot polynomials through topological vertices, which can be considered as a possible approach to the tangle calculus: we discuss the Macdonald deformation of the relation between the convolution ... More

The Complexity of POMDPs with Long-run Average ObjectivesApr 30 2019We study the problem of approximation of optimal values in partially-observable Markov decision processes (POMDPs) with long-run average objectives. POMDPs are a standard model for dynamic systems with probabilistic and nondeterministic behavior in uncertain ... More

On the regularity of critical points for O'Hara's knot energies: From smoothness to analyticityApr 30 2019We prove the analyticity of smooth critical points for O'Hara's knot energies $\mathcal{E}^{\alpha,p}$, with $p=1$ and $2<\alpha< 3$, subject to a fixed length constraint. This implies, together with the main result in \cite{BR13}, that bounded energy ... More

Matrix Group Integrals, Surfaces, and Mapping Class Groups II: $\mathrm{O}\left(n\right)$ and $\mathrm{Sp}\left(n\right)$Apr 30 2019Let $w$ be a word in the free group on $r$ generators. The expected value of the trace of the word in $r$ independent Haar elements of $\mathrm{O}(n)$ gives a function ${\cal T}r_{w}^{\mathrm{O}}(n)$ of $n$. We show that ${\cal T}r_{w}^{\mathrm{O}}(n)$ ... More

Doubly slice odd pretzel knotsApr 29 2019We prove that an odd pretzel knot is doubly slice if it has $2n+1$ twist parameters consisting of $n+1$ copies of $a$ and $n$ copies of $-a$ for some odd integer $a$. Combined with the work of Issa and McCoy, it follows that these are the only doubly ... More

The positive scalar curvature cobordism categoryApr 29 2019We prove that many spaces of positive scalar curvature metrics have the homotopy type of infinite loop spaces. Our result in particular applies to the path component of the round metric inside $\mathcal{R}^+ (S^d)$ if $d \geq 6$. To achieve that goal, ... More

Hierarchies and semistability of relatively hyperbolic groupsApr 29 2019A finitely presented group is semistable if all proper rays in the Cayley 2-complex are properly homotopic. A long standing open question asks whether all finitely presented groups are semistable. In this article, we prove semistability of groups that ... More

Branch Stabilisation for the Components of Hurwitz Moduli Spaces of Galois CoversApr 29 2019We consider components of Hurwitz moduli space of G-Galois covers and set up a powerful algebraic framework to study the set of corresponding equivalence classes of monodromy maps. Within that we study geometric stabilisation by various G-covers branched ... More

Small exotic 4-manifolds from lines and quadrics in $\mathbb{CP}^{2}$Apr 29 2019We construct potentially new manifolds homeomorphic but not diffeomorphic to $\mathbb{CP}^{2} \# 8 \overline{\mathbb{CP}^{2}}$ and $\mathbb{CP}^{2} \# 9 \overline{\mathbb{CP}^{2}}$ via rational blowdown surgery along certain $4$-valent plumbing graphs. ... More

When Sally Found Harry: A Stochastic Search GameApr 29 2019Harry hides on an edge of a graph and does not move from there. Sally, starting from a known origin, tries to find him as soon as she can. Harry's goal is to be found as late as possible. At any given time, each edge of the graph is either active or inactive, ... More

Compact hyperbolic manifolds without spin structuresApr 29 2019Apr 30 2019We exhibit the first examples of compact orientable hyperbolic manifolds that do not have any spin structure. We show that such manifolds exist in all dimensions $n \geq 4$. The core of the argument is the construction of a compact orientable hyperbolic ... More

Compact hyperbolic manifolds without a spin structureApr 29 2019We exhibit the first examples of compact orientable hyperbolic manifolds that do not have any spin structure. We show that such manifolds exist in all dimensions $n \geq 4$. The core of the argument is the construction of a compact orientable hyperbolic ... More

Definition of the cord algebra of knots using Morse TheoryApr 29 2019We redefine the cord algebra, which was introduced by Lenhard Ng as a topological knot invariant, in terms of Morse Theory. The determination of the cord algebra of the unknot and of the righthanded trefoil are given. We proove that the cord algebra in ... More

On proper branched coverings and a question of VuorinenApr 29 2019We study global injectivity of proper branched coverings defined on the Euclidean $n$-ball in the case when the branch set is compact. In particular we show that such mappings are homeomorphisms when $n=3$ or when the branch set is empty. This proves ... More

Space of minimal discs and its compactificationApr 29 2019We investigate the class of geodesic metric discs satisfying a uniform quadratic isoperimetric inequality and uniform bounds on the length of the boundary circle. We show that the closure of this class as a subset of Gromov-Hausdorff space is intimately ... More

Controlled surgery and $\mathbb{L}$-homologyApr 29 2019This paper presents an alternative approach to controlled surgery obstructions. The obstruction for a degree one normal map $(f,b): M^n \rightarrow X^n$ with control map $q: X^n \rightarrow B$ to complete controlled surgery is an element $\sigma^c (f, ... More

Homotopy versus isotopy: spheres with duals in 4--manifoldsApr 28 2019David Gabai recently proved a smooth 4-dimensional ``Light Bulb Theorem'' in the absence of involutions in the fundamental group. More precisely, he showed that homotopy implies isotopy for embedded 2-spheres which have a common geometric dual. We extend ... More

Subgroup distortion of 3-manifold groupsApr 28 2019May 06 2019In this paper, we compute the subgroup distortion of all finitely generated subgroups of all finitely generated 3-manifold groups, and the subgroup distortion in this case can only be linear, quadratic, exponential and double exponential. It turns out ... More

The Khovanov homology of alternating virtual linksApr 28 2019In this paper, we study the Khovanov homology of an alternating virtual link $L$ and show that it is supported on $g+2$ diagonal lines, where $g$ equals the virtual genus of $L$. Specifically, we show that $Kh^{i,j}(L)$ is supported on the lines $j=2i-\sigma_{\xi}+2k-1$ ... More

Verified computations for closed hyperbolic 3-manifoldsApr 27 2019Extending methods first used by Casson, we show how to verify a hyperbolic structure on a finite triangulation of a closed 3-manifold using interval arithmetic methods. A key ingredient is a new theoretical result (akin to a theorem by Neumann-Zagier ... More

Knot reversal and mutation act non-trivially on the concordance group of topologically slice knotsApr 26 2019We construct a topologically slice knot that is not smoothly concordant to its reverse. Moreover, we show that string reversal induces a nontrivial involution on the quotient of the concordance group of topologically slice knots modulo the subgroup generated ... More

On the Burau representation for $n=4$Apr 26 2019The problem of faithfulness of the (reduced) Burau representation for $n =4$ is known to be equivalent to the problem of whether certain two matrices A and B generate a free group of rank two. In [Ber-Tra] we gave a simple proof that $(A^3, B^3)$ is a ... More

Cable knots are not thinApr 25 2019We prove that the $(p,q)$-cable of a non-trivial knot is not Floer homologically thin. Using this and a theorem of Ian Zemke in \cite{zemke}, we find a larger class of satellite knots, containing non-cable knots as well, which are not Floer homologically ... More

Cusp excursion in hyperbolic manifolds and singularity of harmonic measureApr 25 2019We generalize the notion of cusp excursion of geodesic rays by introducing for any $k \geq 1$ the $k^{th}$ excursion in the cusps of a hyperbolic $N$-manifold of finite volume. We show that on one hand, this excursion is at most linear for geodesics that ... More

Prediction with Expert Advice: a PDE PerspectiveApr 25 2019This work addresses a classic problem of online prediction with expert advice. We assume an adversarial opponent, and we consider both the finite-horizon and random-stopping versions of this zero-sum, two-person game. Focusing on an appropriate continuum ... More

Hyperbolization of infinite-type 3-manifoldsApr 25 2019We study the class $\mathcal M^B$ of 3-manifolds $M$ that have a compact exhaustion $M=\cup_{i\in\mathbb N} M_i$ satisfying: each $M_i$ is hyperbolizable with incompressible boundary and each component of $\partial M_i$ has genus at most $g= g(M)$. For ... More

Qualitative counting closed geodesicsApr 25 2019We investigate the geometry of word metrics on fundamental groups of manifolds associated with the generating sets consisting of elements represented by closed geodesics. We ask whether the diameter of such a metric is finite or infinite. The first answer ... More

A new proof of Bowers-Stephenson conjectureApr 25 2019Inversive distance circle packing on surfaces was introduced by Bowers-Stephenson as a generalization of Thurston's circle packing and conjectured to be rigid. The infinitesimal and global rigidity of circle packing with nonnegative inversive distance ... More

Thurston's sphere packing on 3-dimensional manifolds, IApr 25 2019Thurston's sphere packing on 3-dimensional manifolds is a generalization of Thusrton's circle packing on surfaces, allowing adjacent spheres to intersect with non-obtuse angles. In this paper, we prove that the discrete Laplacian for a large class of ... More

Volume and Homology for Hyperbolic 3-OrbifoldsApr 24 2019Let ${\mathfrak M}$ be a closed, orientable, hyperbolic 3-orbifold such that $\pi_1({\mathfrak M})$ contains no hyperbolic triangle group. We show that strict upper bounds of 0.07625, 0.1525 and 0.22875 for ${\rm vol}\ {\mathfrak M}$ imply respective ... More

Singular crossings and Ozsváth-Szabó's Kauffman-states functorApr 24 2019Recently, Ozsv\'ath and Szab\'o introduced some algebraic constructions computing knot Floer homology in the spirit of bordered Floer homology, including a family of algebras B(n) and, for a generator of the braid group on n strands, a certain type of ... More

Enhanced IoV Security Network by using Blockchain Governance GameApr 24 2019This paper deals with the design of the secure network of the Enhanced Internet of Vehicles by using the Blockchain Governance Game. The BGG is the system model of the stochastic game to find best strategies towards preparation for preventing a network ... More

First cohomology of pure mapping class groups of big genus one and zero surfacesApr 23 2019We prove that the first integral cohomology of pure mapping class groups of infinite type genus one surfaces is trivial. For genus zero surfaces we prove that not every homomorphism to $\mathbb{Z}$ factors through a sphere with finitely many punctures. ... More

Matrix-Valued Mean-Field-Type Games: Risk-Sensitive, Adversarial, and Risk-Neutral Linear-Quadratic CaseApr 23 2019In this paper we study a class of matrix-valued linear-quadratic mean-field-type games for both the risk-neutral, risk-sensitive and robust cases. Non-cooperation, full cooperation and adversarial between teams are treated. We provide a semi-explicit ... More

On the quasi-isometric rigidity of graphs of surface groupsApr 23 2019Let $\Gamma$ be a word hyperbolic group with a cyclic JSJ decomposition that has only rigid vertex groups, which are all fundamental groups of closed surface groups. We show that any group $H$ quasi-isometric to $\Gamma$ is abstractly commensurable with ... More

The kernel of the monodromy of the universal family of degree $d$ smooth plane curvesApr 23 2019We consider the parameter space $\mathcal U_d$ of smooth plane curves of degree $d$. The universal smooth plane curve of degree $d$ is a fiber bundle $\cal E_d\to\cal U_d$ with fiber diffeomorphic to a surface $\Sigma_g$. This bundle gives rise to a monodromy ... More

Nimble evolution for pretzel Khovanov polynomialsApr 23 2019We conjecture explicit evolution formulas for Khovanov polynomials for pretzel knots in some regions in the windings space. Our description is exhaustive for genera 1 and 2. As previously observed, evolution at T != -1 is not fully smooth: it switches ... More

Ghrist Barcoded Video Frames. Application in Detecting Persistent Visual Scene Surface Shapes captured in VideosApr 23 2019Apr 24 2019This article introduces an application of Ghrist barcodes in the study of persistent Betti numbers derived from vortex nerve complexes found in triangulations of video frames. A Ghrist barcode is a topology of data pictograph useful in representing the ... More

Ghrist Barcoded Video Frames. Application in Detecting Persistent Visual Scene Surface Shapes captured in VideosApr 23 2019This article introduces an application of Ghrist barcodes in the study of persistent Betti numbers derived from vortex nerve complexes found in triangulations of video frames. A Ghrist barcode is a topology of data pictograph useful in representing the ... More

$A_\infty$-Minimal Model on Differential Graded AlgebrasApr 23 2019For a formal differential graded algebra, if extended by an odd degree element, we prove that the extended algebra has an $A_\infty$-minimal model with only $m_2$ and $m_3$ non-trivial. As an application, the $A_\infty$-algebras constructed by Tsai, Tseng ... More

Centers of subgroups of big mapping class groups and the Tits alternativeApr 22 2019In this note we show that many subgroups of mapping class groups of infinite-type surfaces without boundary have trivial centers, including all normal subgroups. Using similar techniques, we show that every nontrivial normal subgroup of a big mapping ... More

Random trees in the boundary of Outer spaceApr 22 2019We prove that for the harmonic measure associated to a random walk on Out$(F_r)$ satisfying some mild conditions, a typical tree in the boundary of Outer space is trivalent and nongeometric. This answers a question of M. Bestvina.