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Non left-orderable surgeries on L-space twisted torus knotsMar 17 2019We show that if $K$ is an L-space twisted torus knot $T^{l,m}_{p,pk \pm 1}$ with $p \ge 2$, $k \ge 1$, $m \ge 1$ and $1 \le l \le p-1$, then the fundamental group of the $3$-manifold obtained by $\frac{r}{s}$-surgery along $K$ is not left-orderable whenever ... More
Generic-case complexity of Whitehead's algorithm, revisitedMar 17 2019In \cite{KSS06} it was shown that with respect to the simple non-backtracking random walk on the free group $F_N=F(a_1,\dots,a_N)$ the Whitehead algorithm has strongly linear time generic-case complexity and that "generic" elements of $F_N$ are "strictly ... More
Biquandle Module Invariants of Oriented Surface-LinksMar 16 2019We define invariants of oriented surface-links by enhancing the biquandle counting invariant using \textit{biquandle modules}, algebraic structures defined in terms of biquandle actions on commutative rings analogous to Alexander biquandles. We show that ... More
On solvability of certain equations of arbitrary length over torsion-free groupsMar 15 2019Let $G$ be a non-trivial torsion free group and $s(t)=g_{1}t^{\epsilon_{1}}g_{2}t^{\epsilon_{2}} \cdots g_{n}t^{\epsilon_{n}}=1 \; (g_{i} \in G,\ \epsilon_i=\pm 1)$ be an equation over $G$ containing no blocks of the form $t^{-1}g_{i}t^{-1}, \; g_{i} ... More
A differential form approach to the genus of Open Riemann surfacesMar 14 2019We will show that any open Riemann surface $M$ of finite genus is biholomorphic to an open set of a compact Riemann surface. Moreover, we will introduce a quotient space of forms in $M$ that determines if $M$ has finite genus and also the minimal genus ... More
Lifting coarse homotopiesMar 14 2019We prove a general lifting lemma for coarse homotopies with respect to a certain class of bornologous surjective maps. This class is wide enough to include quotients by group actions satisfying a coarse discontinuity condition, which allows us to obtain ... More
The Knot Invariant $Υ$ Using Grid HomologiesMar 14 2019According to the idea of Ozsv\'ath, Stipsicz and Szab\'o, we define the knot invariant $\Upsilon$ without the holomorphic theory, using constructions from grid homology. We develop a homology theory using grid diagrams, and show that $\Upsilon$, as introduced ... More
Knot Floer homology and strongly homotopy-ribbon concordancesMar 14 2019We prove that the map on knot Floer homology induced by a strongly homotopy-ribbon concordance is injective.
Torsion in thin regions of Khovanov homologyMar 13 2019In the integral Khovanov homology of links, the presence of odd torsion is rare. Homologically thin links, that is links whose Khovanov homology is supported on two adjacent diagonals, are known to only contain $\mathbb{Z}_2$ torsion. In this paper, we ... More
Strands algebras and Ozsváth-Szabó's Kauffman-states functorMar 13 2019We define new differential graded algebras A(n,k,S) in the framework of Lipshitz-Ozsv\'ath-Thurston's and Zarev's strands algebras from bordered Floer homology. The algebras A(n,k,S) are meant to be strands models for Ozsv\'ath-Szab\'o's algebras B(n,k,S); ... More
Generators, relations, and homology for Ozsváth-Szabó's Kauffman-states algebrasMar 13 2019We give a generators-and-relations description of differential graded algebras recently introduced by Ozsv\'ath and Szab\'o for the computation of knot Floer homology. We also compute the homology of these algebras and determine when they are formal.
The construction problem for Hodge numbers modulo an integerMar 13 2019For any integer $m\ge2$ and any dimension $n\ge1$, we show that any $n$-dimensional Hodge diamond with values in $\mathbb Z/m\mathbb Z$ is attained by the Hodge numbers of an $n$-dimensional smooth complex projective variety. As a corollary, there are ... More
The colored Jones polynomial and Kontsevich-Zagier series for double twist knots, IIMar 12 2019Let $K_{(m,p)}$ denote the family of double twist knots where $2m-1$ and $2p$ are non-zero integers denoting the number of half-twists in each region. Using a result of Takata, we prove a formula for the colored Jones polynomial of $K_{(-m,-p)}$ and $K_{(-m,p)}$. ... More
Volume versus rank of lattices in Lie groupsMar 12 2019We prove that the rank (that is, the minimal size of a generating set) of lattices in a general connected Lie group is bounded by the co-volume of the projection of the lattice to the semi-simple part of the group. This is a generalization of a result ... More
Smoothly non-isotopic Lagrangian disk fillings of Legendrian knotsMar 12 2019In this paper, we construct the first families of distinct Lagrangian ribbon disks in the standard symplectic 4-ball which have the same boundary Legendrian knots, and are not smoothly isotopic or have non-homeomorphic exteriors.
Directed diagrammatic reducibilityMar 11 2019We introduce the notion of directed diagrammatic reducibility which is a relative version of diagrammatic reducibility. Directed diagrammatic reducibility has strong group theoretic and topological consequences. A multi-relator version of the Freiheitssatz ... More
Pay to change lanes: A cooperative lane-changing strategy for connected/automated drivingMar 11 2019This paper proposes a cooperative lane changing strategy using a transferable utility games framework. This allows vehicles to engage in transactions where gaps in traffic are created in exchange for monetary compensation. We formulate gains in travel ... More
When can a link be obtained from another using crossing exchanges and smoothings?Mar 11 2019Mar 13 2019Let $L$ be a fixed link. Given a link diagram $D$, is there a sequence of crossing exchanges and smoothings on $D$ that yields a diagram of $L$? We approach this problem from the computational complexity point of view. It follows from work by Endo, Itoh, ... More
When can a link be obtained from another using crossing exchanges and smoothings?Mar 11 2019Let L be a fixed link. Given a link diagram D, is there a sequence of crossing exchanges and smoothings on D that yields a diagram of L? We approach this problem from the computational complexity point of view. It follows from work by Endo, Itoh, and ... More
The Levine-Tristram signature: a surveyMar 11 2019The Levine-Tristram signature associates to each oriented link $L$ in $S^3$ a function $\sigma_L \colon S^1 \to \mathbb{Z}.$ This invariant can be defined in a variety of ways, and its numerous applications include the study of unlinking numbers and link ... More
The Large genus asymptotic expansion of Masur-Veech volumesMar 11 2019We study the asymptotic behavior of Masur-Veech volumes as the genus goes to infinity. We show the existence of a complete asymptotic expansion of these volumes that depends only on the genus and the number of singularities. The computation of the first ... More
Exhausting Curve Complexes by Finite Superrigid Sets on Nonorientable SurfacesMar 11 2019Let $N$ be a compact, connected, nonorientable surface of genus $g$ with $n$ boundary components. Let $\mathcal{C}(N)$ be the curve complex of $N$. We prove that if $(g, n) \neq (1,2)$ and $g + n \neq 4$, then there is an exhaustion of $\mathcal{C}(N)$ ... More
A note on the $Θ$-invariant of 3-manifoldsMar 11 2019In this note, we revisit the $\Theta$-invariant as defined by R. Bott and the first author. The $\Theta$-invariant is an invariant of rational homology 3-spheres with acyclic orthogonal local systems, which is a generalization of the 2-loop term of the ... More
Promoting circular-orderability to left-orderabilityMar 11 2019Motivated by recent activity in low-dimensional topology, we provide a new criterion for left-orderability of a group under the assumption that the group is circularly-orderable: A group $G$ is left-orderable if and only if $G \times \mathbb{Z}/n\mathbb{Z}$ ... More
Dilogarithm identities for solutions to Pell's equation in terms of continued fraction convergentsMar 10 2019In this note, we derive a number of formulae for the dilogarithm using hyperbolic geometry. One such formula shows that solutions to the Pell equation $x^2-ny^2 =\pm 1$ satisfy an infinite dilogarithm formula in terms of their continued fraction expansion. ... More
Reversible Quaternionic Hyperbolic IsometriesMar 10 2019Let $G$ be a group. An element $g$ in $G$ is called reversible if it is conjugate to $g^{-1}$ within $G$, and called strongly reversible if it is conjugate to its inverse by an order two element of $G$. Let $\textbf{H}_{\mathbb H}^n$ be the $n$-dimensional ... More
Oriented Local Moves and Divisibility of the Jones PolynomialMar 10 2019For any virtual link $L = S \cup T$ that may be decomposed into a pair of oriented $n$-tangles $S$ and $T$, an oriented local move of type $T \mapsto T'$ is a replacement of $T$ with the $n$-tangle $T'$ in a way that preserves the orientation of $L$. ... More
On the top homology group of Johnson kernelMar 09 2019The action of the mapping class group $\mathrm{Mod}_g$ of an oriented surface $\Sigma_g$ on the lower central series of $\pi_1(\Sigma_g)$ defines the descending filtration in $\mathrm{Mod}_g$ called the Johnson filtration. The first two terms of it are ... More
Generalized Chillingworth Classes on Subsurface Torelli GroupsMar 09 2019The contraction of the image of the Johnson homomorphism is called the Chillingworth class. In this paper, we derive a combinatorial description of the Chillingworth class for Putman's subsurface Torelli groups. We also prove the naturality and uniqueness ... More
Properly discontinuous actions versus uniform embeddingsMar 08 2019Whenever a finitely generated group $G$ acts properly discontinuously by isometries on a metric space $X$, there is an induced uniform embedding (a Lipschitz and uniformly proper map) $\rho: G \rightarrow X$ given by mapping $G$ to an orbit. We study ... More
The $\mathbb{R}$ invariant solutions to the Kapustin Witten equations on $(0,\infty) \times \mathbb{R}^2 \times \mathbb{R}$ with generalized Nahm pole asymptoticsMar 08 2019This paper supplies a new characterization of the Kapustin Witten equation solutions on $(0,\infty) \times \mathbb{R}^2 \times \mathbb{R}$ that play a key role in Edward Witten's program to obtain the Jones polynomial knot invariants using solutions to ... More
Golden ratio on nonorientable surfacesMar 08 2019On each nonorientable surface of odd genus $g \geq 5$, we give a mapping class whose dilatation on an invariant subsurface is the golden ratio.
On large orientation-reversing finite group-actions on 3-manifolds and equivariant Heegaard decompositionsMar 08 2019We consider finite group-actions on closed, orientable and nonorientable 3-manifolds; such a finite group-action leaves invariant the two handlebodies of a Heegaard splitting of M of some genus g. The maximal possible order of a finite group-action of ... More
Price of Anarchy in Stochastic Atomic Congestion Games with Affine CostsMar 08 2019We consider an atomic congestion game with stochastic demand in which each player participates in the game with probability $p$, and incurs no cost with probability $1-p$. We assume that $p$ is common knowledge among all players and that players are independent. ... More
Cyclic coverings of virtual link diagramsMar 08 2019A virtual link diagram is called mod $m$ almost classical if it admits an Alexander numbering valued in integers modulo $m$, and a virtual link is called mod $m$ almost classical if it has a mod $m$ almost classical diagram as a representative. In this ... More
A concordance analogue of the $4$-dimensional light bulb theoremMar 07 2019We prove a concordance analogue of Gabai's $4$-dimensional light bulb theorem. That is, we show that when $R$ and $R'$ are homotopically (smoothly) embedded $2$-spheres in a $4$-manifold $X^4$ where $\pi_1(X^4)$ has no $2$-torsion and one of $R$ or $R'$ ... More
Unchaining surgery and topology of symplectic 4-manifoldsMar 07 2019Mar 12 2019We study a symplectic surgery operation we call unchaining, which effectively reduces the second Betti number and the symplectic Kodaira dimension at the same time. Using unchaining, we give novel constructions of symplectic Calabi-Yau surfaces from complex ... More
Unchaining surgery and topology of symplectic 4-manifoldsMar 07 2019We study a symplectic surgery operation we call, which effectively reduces the second Betti number and the symplectic Kodaira dimension at the same time. Using unchaining, we give novel constructions of symplectic Calabi-Yau surfaces from complex surfaces ... More
Orbit closures of Zariski dense subgroups in homogeneous spacesMar 07 2019We present a new proof of Benoist-Quint's finite or dense dichotomy for orbits of Zariski dense subgroups acting on the quotient space of SO(d,1) by a cocompact lattice. Our proof is topological. We use ideas from the study of dynamics of unipotent flows ... More
Cross ratios on ${\rm CAT(0)}$ cube complexes and marked length-spectrum rigidityMar 06 2019We show that group actions on irreducible ${\rm CAT(0)}$ cube complexes with no free faces are uniquely determined by their $\ell^1$ length function. Actions are allowed to be non-proper and non-cocompact, as long as they are essential and have no finite ... More
Multi-tribracketsMar 05 2019We introduce multi-tribrackets, algebraic structures for region coloring of diagrams of knots and links with different operations at different kinds of crossings. In particular we consider the case of component multi-tribrackets which have different tribracket ... More
The knots that lie above all shadowsMar 05 2019We show that for each even integer $m\ge 2$, every reduced shadow with sufficiently many crossings is a shadow of a torus knot T(2,m+1), or of a twist knot $T_m$, or of a connected sum of $m$ trefoil knots.
Isotopy and equivalence of closed surface braidsMar 05 2019The closure of a braid in a closed orientable surface $\Sigma$ is a link in $\Sigma\times S^1$. We classify such closed surface braids up to equivalence and isotopy (with a small indeterminacy for isotopy of closed sphere braids), in terms of the algebra ... More
Deformations of smooth functions on $2$-torusMar 05 2019Let $f $ be a Morse function on a smooth compact surface $M$ and $\mathcal{S}'(f)$ be a group of $f$-preserving diffeomorphisms of $M$ which are isotopic to the identity map. Let also $G(f)$ be a group of automorphisms of the graph of $f$ induced by elements ... More
A diagrammatic approach to the AJ ConjectureMar 05 2019The AJ Conjecture relates a quantum invariant, a minimal order recursion for the colored Jones polynomial of a knot (known as the $\hat{A}$ polynomial), with a classical invariant, namely the defining polynomial $A$ of the $\psl$ character variety of ... More
On the Bauer-Furuta and Seiberg-Witten invariants of families of $4$-manifoldsMar 05 2019We show how the families Seiberg-Witten invariants of a family of smooth $4$-manifolds can be recovered from the families Bauer-Furuta invariant via a cohomological formula. We use this formula to deduce several properties of the families Seiberg-Witten ... More
Convergence of gradient descent-ascent analyzed as a Newtonian dynamical system with dissipationMar 05 2019A dynamical system is defined in terms of the gradient of a payoff function. Dynamical variables are of two types, ascent and descent. The ascent variables move in the direction of the gradient, while the descent variables move in the opposite direction. ... More
Khovanov homology and ribbon concordanceMar 04 2019We show that a ribbon concordance between two links induces an injective map on Khovanov homology.
An Adaptive Grid Algorithm for Computing the Homology Group of Semialgebraic SetMar 04 2019Looking for an efficient algorithm for the computation of the homology groups of an algebraic set or even a semi-algebraic set is an important problem in the effective real algebraic geometry. Recently, Peter Burgisser, Felipe Cucker and Pierre Lairez ... More
Fibers of maps to totally nonnegative spacesMar 04 2019This paper undertakes a study of the structure of the fibers of a family of maps $f_{(i_1,\dots ,i_d)}$ arising from representation theory, motivated both by connections to Lusztig's theory of canonical bases and also by the fact that these fibers encode ... More
Embedding lens spaces in definite 4-manifoldsMar 04 2019Every lens space has a locally flat embedding in a connected sum of 8 copies of the complex projective plane and a smooth embedding in n copies of the complex projective plane for some positive integer n. We show that there is no n such that every lens ... More
A graphical categorification of the two-variable Chebyshev polynomials of the second kindMar 04 2019We show that the $A_2$ clasps in the Karoubi envelope of $A_2$ spider satisfy the recursive formula of the two-variable Chebyshev polynomials of the second kind associated with a root system of type $A_2$. The $A_2$ spider is a diagrammatic description ... More
Augmentations and link group representationsMar 03 2019We construct the augmentation representation. It is a representation of the fundamental group of the link complement associated to an augmentation of the framed cord algebra. This construction connects representations of two link invariants of different ... More
Uniform exponential mixing for congruence covers of convex cocompact hyperbolic manifoldsMar 03 2019Let $\Gamma$ be a Zariski dense convex cocompact subgroup contained in an arithmetic lattice of $\operatorname{SO}(n, 1)^{\circ}$. We prove uniform exponential mixing of the geodesic flow for congruence covers of the hyperbolic manifold $\Gamma \backslash ... More
Approximation of Metric Spaces by Reeb Graphs: Cycle Rank of a Reeb Graph, the Co-rank of the Fundamental Group, and Large Components of Level Sets on Riemannian ManifoldsMar 02 2019For a connected locally path-connected topological space $X$ and a continuous function $f$ on it such that its Reeb graph $R_f$ is a finite topological graph, we show that the cycle rank of $R_f$, i.e., the first Betti number $b_1(R_f)$, in computational ... More
Tolling for Constraint Satisfaction in Markov Decision Process Congestion GamesMar 02 2019Markov decision process (MDP) congestion game is an extension of classic congestion games, where a continuous population of selfish agents solves Markov decision processes with congestion: the payoff of a strategy decreases as more population uses it. ... More
Computability Aspects of Differential Games in Euclidian SpacesMar 02 2019We study computability-theoretic aspects of differential games. Our focus is on pursuit and evasion games played in Euclidean spaces in the tradition of Rado's "Lion versus Man" game. In some ways, these games can be viewed as continuous versions of reachability ... More
Acylindrical Hyperbolicity of SubgroupsMar 02 2019Mar 15 2019Suppose $G$ is a finitely generated group and $H$ is a finitely generated subgroup of $G$. Let $\partial_{M}G$ denote the Morse boundary of $G$ with Cashen-Mackay topology. In this article we show that if the limit set $\Lambda(H)$ of $H$ in $\partial_{M}G$ ... More
Acylindrical Hyperbolicity of SubgroupsMar 02 2019Suppose $G$ is a finitely generated group and $H$ is a finitely generated subgroup of $G$. Let $\partial_{M}G$ denote the Morse boundary of $G$ with Cashen-Mackay topology. In this article we show that if the limit set $\Lambda(H)$ of $H$ in $\partial_{M}G$ ... More
Extension of KNTZ trick to non-rectangular representationsMar 01 2019We claim that the recently discovered universal-matrix precursor for the $F$ functions, which define the differential expansion of colored polynomials for twist and double braid knots, can be extended from rectangular to non-rectangular representations. ... More
Geometric simplicial embeddings of arc-type graphsFeb 28 2019In this paper, we investigate a family of graphs associated to collections of arcs on surfaces. These {\it multiarc graphs} naturally interpolate between arc graphs and flip graphs, both well studied objects in low dimensional geometry and topology. We ... More
Artin's braids, Braids for three space, and groups $Γ_{n}^{4}$ and $G_{n}^{k}$Feb 28 2019Mar 08 2019We construct a group $\Gamma_{n}^{4}$ corresponding to the motion of points in $\mathbb{R}^{3}$ from the point of view of Delaunay triangulations. We study homomorphisms from pure braids on $n$ strands to the product of copies of $\Gamma_{n}^{4}$. We ... More
Artin's braids, Braids for three space, and groups $Γ_{n}^{4}$ and $G_{n}^{k}$Feb 28 2019Mar 01 2019We construct a group $\Gamma_{n}^{4}$ corresponding to the motion of points in $\mathbb{R}^{3}$ from the point of view of Delaunay triangulations. We study homomorphisms from pure braids on $n$ strands to the product of copies of $\Gamma_{n}^{4}$. We ... More
Artin's braids, Braids for three space, and groups $Γ_{n}^{4}$ and $G_{n}^{k}$Feb 28 2019We construct a group $\Gamma_{n}^{4}$ corresponding to the motion of points in $\mathbb{R}^{3}$ from the point of view of Delaunay triangulations. We study homomorphisms from pure braids on $n$ strands to the product of copies of $\Gamma_{n}^{4}$. We ... More
Bounded cohomology of transformation groupsFeb 28 2019Let $M$ be a complete connected Riemannian manifold of finite volume. In this paper we present a new method of constructing classes in bounded cohomology of transformation groups such as $Homeo_0(M,\mu)$, $Diff_0(M,vol)$ and $Symp_0(M,\omega)$ (in case ... More
Milnor invariants, $2n$-moves and $V^{n}$-moves for welded string linksFeb 28 2019In a previous paper, the authors proved that Milnor link-homotopy invariants modulo $n$ classify classical string links up to $2n$-move and link-homotopy. As analogues to the welded case, in terms of Milnor invariants, we give here two classifications ... More
The Shape of Thurston's Master TeapotFeb 27 2019We establish basic geometric and topological properties of Thurston's Master Teapot and the Thurston set for superattracting unimodal self-maps of intervals. In particular, the Master Teapot is connected, contains the unit cylinder, and its intersection ... More
The augmented deformation space of rational mapsFeb 27 2019The Epstein deformation space parameterizes marked rational maps with prescribed combinatorial and dynamical structure. For the family of quadratic rational maps with a periodic critical cycle of order 4 and an extra critical point not lying in this cycle, ... More
Multivariate Alexander quandles, II. The involutory medial quandle of a linkFeb 27 2019Mar 11 2019Joyce showed that for a classical knot $K$, the order of the involutory medial quandle is $| \det K |$. Generalizing Joyce's result, we show that for a classical link $L$ of $\mu \geq 1$ components, the order of the involutory medial quandle is $\mu | ... More
Multivariate Alexander quandles, II. The involutory medial quandle of a linkFeb 27 2019Joyce showed that for a classical knot $K$, the order of the involutory medial quandle is $|\det K|$. Generalizing Joyce's result, we show that for a classical link $L$ of $\mu \geq 1$ components, the order of the involutory medial quandle is $\mu |\det ... More
A cobordism category attached to Khovanov-Rozansky link homologies based on operadsFeb 27 2019We consider colored operads and their actions on categories. As a special example we construct a cobordism category with a colored operad action arising from oriented planar arc diagrams. This is used to construct an invariant of oriented tangle diagrams ... More
A cobordism category attached to Khovanov-Rozansky link homologies based on operadsFeb 27 2019Mar 15 2019We consider colored operads and their actions on categories. As a special example we construct a cobordism category with a colored operad action arising from oriented planar arc diagrams. This is used to construct an invariant of oriented tangle diagrams ... More
Weighted games of best choiceFeb 26 2019The game of best choice (also known as the secretary problem) is a model for sequential decision making with a long history and many variations. The classical setup assumes that the sequence of candidate rankings are uniformly distributed. Given a statistic ... More
Mapping class groups of highly connected $(4k+2)$-manifoldsFeb 26 2019We compute the mapping class group of the manifolds $\sharp^g(S^{2k+1}\times S^{2k+1})$ for $k>0$ in terms of the automorphism group of the middle homology and the group of homotopy $(4k+3)$-spheres. We furthermore identify its Torelli subgroup, determine ... More
Alternative versions of the Johnson homomorphisms and the LMO functorFeb 26 2019Let $\Sigma$ be a compact connected oriented surface with one boundary component and let $\mathcal{M}$ denote the mapping class group of $\Sigma$. By considering the action of $\mathcal{M}$ on the fundamental group of $\Sigma$ it is possible to define ... More
On the classifying spaces of cobordisms of singular mapsFeb 26 2019The classifying spaces of cobordisms of singular maps have two fairly different constructions. We expose a homotopy theoretical connection between them. As a corollary we show that the classifying spaces in some cases have a simple product structure.
Geometric automorphism groups of symplectic 4-manifoldsFeb 26 2019Let $M$ be a closed, oriented, smooth $4-$manifold with intersection form $\Gamma$, $A(\Gamma)$ the automorphism group of $\Gamma$ and $D(M)$ the subgroup induced by orientation-preserving diffeomorphisms of $M$. In this note we study the question when ... More
On spectra of Cayley graphs of the lamplighter group and their spectral measuresFeb 26 2019We show that the lamplighter group L has a system of generators for which the spectrum of the discrete Laplacian on the Cayley graph is a union of an interval and a countable set of isolated points accumulating to a point outside this interval. This is ... More
Minimal Entropy of $3$-manifoldsFeb 25 2019We compute the Minimal Entropy of every closed, orientable $3$-manifold, showing that its cube equals the sum of the cubes of the minimal entropies of each hyperbolic component arising from the $JSJ$ decomposition of each prime summand. As a consequence ... More
Impossible configurations for geodesics on negatively-curved surfacesFeb 24 2019Hass and Scott's example of a 4-valent graph on the 3-punctured sphere that cannot be realized by geodesics in any metric of negative curvature is generalized to impossible configurations filling surfaces of genus $n$ with $k$ punctures for any $n$ and ... More
Classification of virtual string links up to cobordismFeb 24 2019Cobordism of virtual string links on $n$ strands is a combinatorial generalization of link cobordism. There exists a bijection between virtual string links up to cobordisms and elements of the group $\mathbb{Z}^{n(n-1)}$. This paper also shows that virtual ... More
A generating polynomial for the two-bridge knot with Conway's notation C(n,r)Feb 24 2019We construct an integer polynomial whose coefficients enumerate the Kauffman states of the two-bridge knot with Conway's notation C(n,r).
Non-orientable Lagrangian surfaces in rational 4-manifoldsFeb 24 2019We show that for any nonzero class $A$ in $H_2(X; \mathbb{Z}_2)$ in a rational 4-manifold $X$, $A$ is represented by a nonorientable embedded Lagrangian surface L (for some symplectic structure) if and only if $P(A)\equiv (L) (mod\ 4)$; where $P(A)$ denotes ... More
On Reeb graphs induced from smooth functions on 3-dimensional closed orientable manifolds with finite singular valuesFeb 23 2019The Reeb graph of a function on a smooth manifold is the graph obtained as the space of all connected components of inverse images such that the set of all vertices coincides with the set of all connected components of inverse images including singular ... More
On complete hyperKähler manifolds with compact isometry groupFeb 23 2019The basic model of hyperK\"ahler manifold is the quaternionic number space $\HH^n$ with flat and symmetric metric. In this note we shall construct a one-parameter family of complete hyperK\"ahler metrics on $\HH^n$ whose isometry group is the compact ... More
Topology of complements to real affine space line arrangementsFeb 22 2019It is shown that the diffeomorphism type of the complement to a real space line arrangement in any dimensional affine ambient space is determined only by the number of lines and the data on multiple points.
Statistics of Square-tiled Surfaces: Symmetry and Short LoopsFeb 21 2019Square-tiled surfaces are a class of translation surfaces that are of particular interest in geometry and dynamics because, as covers of the square torus, they share some of its simplicity and structure. In this paper, we study counting problems that ... More
Currents, Systoles, and Compactifications of Character VarietiesFeb 20 2019We study the Thurston--Parreau boundary both of the Hitchin and of the maximal character varieties and determine therein an open set of discontinuity for the action of the mapping class group. This result is obtained as consequence of a canonical decomposition ... More
Cohomological invariants of representations of 3-manifold groupsFeb 20 2019Suppose $\Gamma$ is a discrete group, and $\alpha\in Z^3(B\Gamma;A)$, with $A$ an abelian group. Given a representation $\rho:\pi_1(M)\to\Gamma$, with $M$ a closed 3-manifold, put $F(M,\rho)=\langle(B\rho)^\ast[\alpha],[M]\rangle$, where $B\rho:M\to B\Gamma$ ... More
Width of codimension two knotsFeb 19 2019We extend the classical definition of {\it width} to higher dimensional, smooth codimension 2 knots and show in each dimension there are knots of arbitrarily large width.
A systematic classification of knotoids on the plane and on the sphereFeb 19 2019Feb 21 2019In this paper we generate and systematically classify all prime planar knotoids with up to 5 crossings. We also extend the existing list of knotoids in $S^2$ and add all knotoids with 6 crossings.
Arithmeticity of hyperbolic 3-manifolds containing infinitely many totally geodesic surfacesFeb 19 2019We prove that if a closed hyperbolic 3-manifold M contains infinitely many totally geodesic surfaces, then M is arithmetic.
On the 2-head of the colored Jones polynomial for pretzel knotsFeb 19 2019In this paper, we prove a formula for the 2-head of the colored Jones polynomial for an infinite family of pretzel knots. Following Hall, the proof utilizes skein-theoretic techniques and a careful examination of higher order stability properties for ... More
Taking-and-merging games as rewrite gamesFeb 19 2019This work contributes to the study of rewrite games where positions are words and the moves are local rewriting rules of the form u->v belonging to a finite set. We introduce and investigate taking-and-merging games where each rule is of the form a^k->epsilon. ... More
Chebotarev link is stably genericFeb 19 2019We discuss the relationship between two analogues in a 3-manifold of the set of prime ideals in a number field. We prove that if $(K_i)_{i\in \mathbb{N}_{>0}}$ is a sequence of knots obeying the Chebotarev law in the sense of Mazur and McMullen, then ... More
Lickorish type construction of manifolds over simple polytopesFeb 19 2019This paper is a survey on the Lickorish type construction of some kind of closed manifolds over simple convex polytopes. Inspired by Lickorish's theorem, we propose a method to describe certain families of manifolds over simple convex polytopes with torus ... More
Generating twist subgroup of mapping class group of non-orienatable surface by involutionsFeb 19 2019Let $N_{g}$ denote the closed non-orientable surface of genus $g$ and let ${\mathcal M} _g$ denote the mapping class group of $N_{g}$. Let ${\mathcal T} _g$ denote the twist subgroup of ${\mathcal M} _g$ which is the subgroup of ${\mathcal M} _g$ is generated ... More
Shake Slice and Shake Concordant LinksFeb 18 2019Mar 08 2019We can construct a 4-manifold by attaching 2-handles to a 4-ball with framing r along the components of a link in the boundary of the 4-ball. We define a link as r-shake slice if there exists embedded spheres that represent the generators of the second ... More
Shake Slice and Shake Concordant LinksFeb 18 2019We can construct a 4-manifold by attaching 2-handles to a 4-ball with framing r along the components of a link in the boundary of the 4-ball. We define a link as r-shake slice if there exists embedded spheres that represent the generators of the second ... More
Classifying spaces for projections of immersions with controlled singularitiesFeb 18 2019Feb 26 2019We give an explicit simple construction for classifying spaces of maps obtained as hyperplane projections of immersions. We prove structure theorems for these classifying spaces.