arXiv research

Combinatorial definition of bordered Floer theory for torus-boundary manifolds.

New definition of regular points for PL functions on manifolds.

We provide an intergral lift of the combinatorial definition of Heegaard Floer homology for nice diagrams, and show that the proof of independence using convenient diagrams adapts to this setting.

Defines annulus complex of handlebodies and proves its connectivity.

Formula derived for a magnetic line invariant.

This paper is a short introduction to the combinatorial version of tangle Floer homology defined in "Combinatorial tangle Floer homology". There are two equivalent definitions---one in terms of strand diagrams, and one in terms of bordered grid diagrams. We present both, discuss the correspondence, and carry out some e…

Paper introduces danceability index as a new bridge index definition.

The invariant $Θ$ is an invariant of rational homology 3-spheres $M$ equipped with a combing $X$ over the complement of a point. It is related to the Casson-Walker invariant $λ$ by the formula $Θ(M,X)=6λ(M)+p_1(X)/4$, where $p_1$ is an invariant of combings that is simply related to a Gompf invariant. In [arXiv:1209.32…

This note proves combinatorially that the intersection pairing on the middle dimensional compactly supported cohomology of a smooth toric hyperkaehler variety is always definite, providing a large number of non-trivial L^2 harmonic forms for toric hyperkaehler metrics on these varieties. This is motivated by a result o…

This paper shows how Hodge's theory of harmonic $p$-sets (a discrete version of his theory of harmonic forms) allows a new approach to be taken to the problem of providing a combinatorial definition of the Pontrjagin classes of a compact manifold. This approach is then related to the author's definition of flag vectors…

In a recent work [2] with Datta, we introduced the mu vector (with respect to a given field) of simplicial complexes and used it to study tightness and lower bounds. In this paper, we modify the definition of mu vectors. With the new definition, most results of [2] become correct without the hypothesis of 2-neighbourli…

Functors from web categories differ despite similar definitions.

Study lens spaces' definite fillings, classifying those with specific inequalities.

We call a Delta Diagram any diagram of a knot or link whose regions (including the unbounded one) have 3, 4, or 5 sides. We prove that any knot or link admits a delta diagram. We define and estimate combinatorial link invariants stemming from this definition.

New combinatorial approach to Goldman-Turaev Lie bialgebra using cyclic word partitions.

A new geometric definition of integration for differential forms.

We define a generalization of virtual links to arbitrary dimensions by extending the geometric definition due to Carter et al. We show that many homotopy type invariants for classical links extend to invariants of virtual links. We also define generalizations of virtual link diagrams and Gauss codes to represent virtua…

BIG Laplacians bridge combinatorial and Hodge Laplacians for discrete data.

In this paper we give a necessary combinatorial condition for a negative--definite plumbing tree to be suitable for rational blow--down, or to be the graph of a complex surface singularity which admits a rational homology disk smoothing. New examples of surface singularities with rational homology disk smoothings are a…

The basic theorems of vector calculus are illuminated when we replace the original 3 stooges of vector calculus: Grad, Div, and Curl, with combinatorial substitutes. In addition to providing simple proofs of Green's theorem and the equivalence of the integral and derivative definitions of curl, we also provide a brief …

Let E be a circle bundle over a Riemann surface that supports a contact structure transverse to the fibers. This paper presents a combinatorial definition of a differential graded algebra (DGA) that is an invariant of Legendrian knots in E. The invariant generalizes Chekanov's combinatorial DGA invariant of Legendrian …

A closed, orientable, splitting surface in an oriented $3$-manifold is a topologically minimal surface of index $n$ if its associated disk complex is $(n-2)$-connected but not $(n-1)$-connected. A critical surface is a topologically minimal surface of index $2$. In this paper, we use an equivalent combinatorial definit…

In 2003, Ozsváth and Szabó defined the concordance invariant $τ$ for knots in oriented 3-manifolds as part of the Heegaard Floer homology package. In 2011, Sarkar gave a combinatorial definition of $τ$ for knots in $S^3$ and a combinatorial proof that $τ$ gives a lower bound for the slice genus of a knot. Recently, Har…

We construct a non-normal affine monoid together with its modules associated with a negative definite plumbed $3$-manifold $M$. In terms of their structure, we describe the $H_1(M,\mathbb{Z})$-equivariant parts of the topological Poincaré series. In particular, we give combinatorial formulas for the Seiberg--Witten inv…

Improves scalability of Bayesian optimization for combinatorial spaces.

We discuss certain aspects of the combinatorial approach to the differential geometry of non-abelian gerbes, due to W. Messing and the author (arXiv:math.AG/0106083), and give a more direct derivation of the associated cocycle equations. This leads us to a more restrictive definition of the corresponding coboundary rel…

Coloring numbers are one of the simplest combinatorial invariants of knots and links to describe. And with Joyce's introduction of quandles, we can understand them more algebraically. But can we extend these invariants to tangles -- knots and links with free ends? Indeed we can, once we categorify. Starting from the de…

Simpler proof for non-basic sets in 2D.

In this survey, we study three different notions of curvature that are defined on graphs, namely, combinatorial curvature, Bakry-Émery curvature, and Ollivier's Ricci curvature. For each curvature notion, the definition and its motivation from Riemannian geometry will be explained. Moreover, we bring together some glob…

This paper introduces an inner product on chain complexes of finite simplicial complexes that is well-adapted to the harmonic study of subdivisions. Its definition utilizes a decomposition of the chain spaces that suggests a sequence of subdivision invariants which we show do not all vanish for non-trivial subdivisions…

The abstract extends Reidemeister theorem to 3-manifolds using diagrams of links and bands.

We suggest a new definition for discrete minimal surfaces in terms of sphere packings with orthogonally intersecting circles. These discrete minimal surfaces can be constructed from Schramm's circle patterns. We present a variational principle which allows us to construct discrete analogues of some classical minimal su…

We extend Turaev's definition of torsion invariants of 3-dimensional manifolds equipped with non-singular vector fields, by allowing (suitable) tangency circles to the boundary, and manifolds with non-zero Euler characteristic. We show that these invariants apply in particular to (the exterior of) Legendrian links in c…

Homotopy classes of nanowords and nanophrases are combinatorial generalizations of virtual knots and links. Goussarov, Polyak and Viro defined finite type invariants for virtual knots and links via semi-virtual crossings. We extend their definition to nanowords and nanophrases. We study finite type invariants of low de…

In this paper, we generalize our results in \cite{GX3} to triangulated surfaces in hyperbolic background geometry, which means that all triangles can be embedded in the standard hyperbolic space. We introduce a new discrete Gaussian curvature by dividing the classical discrete Gauss curvature by an area element, which …

We focus on kernel methods for set-valued inputs and their application to Bayesian set optimization, notably combinatorial optimization. We investigate two classes of set kernels that both rely on Reproducing Kernel Hilbert Space embeddings, namely the ``Double Sum'' (DS) kernels recently considered in Bayesian set opt…

This survey paper begins with the description of the duality between arc systems and ribbon graphs embedded in a punctured surface. Then we explain how to cellularize the moduli space of curves in two different ways: using Jenkins-Strebel differentials and using hyperbolic geometry. We also briefly discuss how these tw…

Motivated by problems in search and detection we present a solution to a Combinatorial Multi-Armed Bandit (CMAB) problem with both heavy-tailed reward distributions and a new class of feedback, filtered semibandit feedback. In a CMAB problem an agent pulls a combination of arms from a set $\{1,...,k\}$ in each round, g…

Grid diagrams encode useful geometric information about knots in S^3. In particular, they can be used to combinatorially define the knot Floer homology of a knot K in S^3, and they have a straightforward connection to Legendrian representatives of K in (S^3, ξ_\st), where ξ_\st is the standard, tight contact structure.…

Classifies tilings of hyperbolic plane by regular polygons.

In the present paper we extend the definition of slice-torus invariant to links. We prove a few properties of the newly-defined slice-torus link invariants: the behaviour under crossing change, a slice genus bound, an obstruction to strong sliceness, and a combinatorial bound. Furthermore, we provide an application to …

We define a generalization of virtual links to arbitrary dimensions by extending the geometric definition due to Carter et al. We show that many homotopy type invariants for classical links extend to invariants of virtual links. We also define generalizations of virtual link diagrams and Gauss codes to represent virtua…

Ozsváth and Szabó gave a combinatorial description for the Heegaard Floer homology of boundaries of certain negative-definite plumbings. Némethi constructed a remarkable algorithm for executing these computations for almost-rational plumbings, and his work gives a formula computing the invariants for the Brieskorn homo…

We generalize Turaev's definition of torsion invariants of pairs $(M,ξ)$, where $M$ is a 3-dimensional manifold and $ξ$ is an Euler structure on $M$ (a non-singular vector field up to homotopy relative to the boundary of $M$ and local modifications in the interior of $M$). Namely, we allow $M$ to have arbitrary boundar…

Assume that $M(\mathcal{T})$ is a rational homology sphere plumbed 3-manifold associated with a connected negative definite graph $\mathcal{T}$. We consider the combinatorial multivariable Poincaré series associated with $\mathcal{T}$ and its counting functions, which encode rich topological information. Using the `per…

A function of several variables is called holonomic if, roughly speaking, it is determined from finitely many of its values via finitely many linear recursion relations with polynomial coefficients. Zeilberger was the first to notice that the abstract notion of holonomicity can be applied to verify, in a systematic and…

We present a short analytic proof of the equality between the analytic and combinatorial torsion. We use the same approach as in the proof given by Burghelea, Friedlander and Kappeler, but avoid using the difficult Mayer-Vietoris type formula for the determinants of elliptic operators. Instead, we provide a direct way …

The paper proves a conjecture linking two metrics on manifold cohomology.