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Gaiane Panina

Publications and source records attributed to Gaiane Panina.

At least 37 records · Page 2Linked to original sources
arXiv2020

Diagonal complexes for surfaces of finite type and surfaces with involution

Two related constructions are studied: (1) The diagonal complex $\mathcal{D}$ and its barycentric subdivision $\mathcal{BD}$ related to a \textit{punctured} oriented surface $F$ equipped with a number of labeled marked points. (2) The symmetric diagonal complex $\mathcal{D}^{inv}$ and its barycentric subdivision $\mathcal{BD}^{inv}$ related to a symmetric (=with an involution) oriented surface $F$ equipped with a number of (symmetrically placed) labeled marked points. Eliminating a puncture gives rise to a bundle whose fibers are homeomorphic to a surgery of the surface $F$. The bundle can be viewed as the "universal curve with holes". The symmetric complex is shown to be homotopy equivalent to the complex of a punctured surface obtained by a surgery of the initial symmetric surface.

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arXiv2020

Splitting necklaces, with constraints

We prove several versions of N. Alon's "necklace-splitting theorem", subject to additional constraints, as illustrated by the following results. (1) The "almost equicardinal necklace-splitting theorem" claims that, without increasing the number of cuts, one guarantees the existence of a fair splitting such that each thief is allocated (approximately) one and the same number of pieces of the necklace (including "degenerate pieces" if they exist), provided the number of thieves $r=p^ν$ is a prime power. (2) The "binary splitting theorem" claims that if $r=2^d$ and the thieves are associated with the vertices of a $d$-cube then, without increasing the number of cuts, one can guarantee the existence of a fair splitting such that adjacent pieces are allocated to thieves that share an edge of the cube. This result provides a positive answer to the "binary splitting necklace conjecture" of Asada at al. (Conjecture 2.11 in [7]) in the case $r=2^d$. (3) An interesting variation arises when the thieves have their own individual preferences. We prove several "envy-free fair necklace-splitting theorems" of various level of generality. By specialization we obtain numerous corollaries, among them envy-free versions of (a) "almost equicardinal splitting theorem", (b) "necklace-splitting theorem for $r$-unavoidable preferences", (c) "envy-free binary splitting theorem", etc. As a corollary we also obtain a recent result of Avvakumov and Karasev [1] about envy-free divisions where players may prefer an empty part of the necklace.

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arXiv2020

Equilibrium stressability of multidimensional frameworks

We prove an equilibrium stressability criterium for trivalent multidimensional tensegrities. The criterium appears in different languages: (1) in terms of stress monodromies, (2) in terms of surgeries, (3) in terms of exact discrete 1-forms, and (4) in Cayley algebra terms.

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arXiv2020

Generalized chessboard complexes and discrete Morse theory

Chessboard complexes and their generalizations, as objects, and Discrete Morse theory, as a tool, are presented as a unifying theme linking different areas of geometry, topology, algebra and combinatorics. Edmonds and Fulkerson bottleneck (minmax) theorem is proved and interpreted as a result about a critical point of a discrete Morse function on the Bier sphere of an associated simplicial complex $K$. We illustrate the use of "standard discrete Morse functions" on generalized chessboard complexes by proving a connectivity result for chessboard complexes with multiplicities. Applications include new Tverberg-Van Kampen-Flores type results for $j$-wise disjoint partitions of a simplex.

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arXiv2020

Colored Tverberg problem, extensions and new results

We prove a "multiple colored Tverberg theorem" and a "balanced colored Tverberg theorem", by applying different methods, tools and ideas. The proof of the first theorem uses multiple chessboard complexes (as configuration spaces) and Eilenberg-Krasnoselskii theory of degrees of equivariant maps for non-free actions. The proof of the second result relies on high connectivity of the configuration space, established by discrete Morse theory.

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arXiv2020

Area-perimeter duality in polygon spaces

Two natural foliations, guided by area and perimeter, of the configurations spaces of planar polygons are considered and the topology of their leaves is investigated in some detail. In particular, the homology groups and the homotopy type of leaves are determined. The homology groups of the spaces of polygons with fixed area and perimeter are also determined. Besides, we extend the classical isoperimetric duality to all critical points. In conclusion a few general remarks on dual extremal problems in polygon spaces and beyond are given.

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arXiv2019

A universality theorem for stressable graphs in the plane

Universality theorems (in the sense of N. Mnëv) claim that the realization space of a combinatorial object (a point configuration, a hyperplane arrangement, a convex polytope, etc.) can be arbitrarily complicated. In the paper, we prove a universality theorem for a graph in the plane with a prescribed \textit{oriented matroid of stresses}, that is the collection of signs of all possible equilibrium stresses of the graph. This research is motivated by the Grassmanian stratification (Gelfand, Goresky, MacPherson, Serganova) by thin Schubert cells, and by a recent series of papers on stratifications of configuration spaces of tensegrities (Doray, Karpenkov, Schepers, Servatius).

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arXiv2019

Extremal areas of polygons with fixed perimeter

We consider the configuration space of planar $n$-gons with fixed perimeter, which is diffeomorphic to the complex projective space $\mathbb{C}P^{n-2}$. The oriented area function has the minimal number of critical points on the configuration space. We describe its critical points (these are regular stars) and compute their indices when they are Morse.

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arXiv2018

A Tverberg type theorem for collectively unavoidable complexes

We prove (Theorem 2.4) that the symmetrized deleted join $SymmDelJoin(\mathcal{K})$ of a "balanced family" $\mathcal{K} = \langle K_i\rangle_{i=1}^r$ of collectively $r$-unavoidable subcomplexes of $2^{[m]}$ is $(m-r-1)$-connected. As a consequence we obtain a Tverberg-Van Kampen-Flores type result (Theorem 3.2) which is more conceptual and more general then previously known results. Already the case $r=2$ of Theorem 3.2 seems to be new as an extension of the classical Van Kampen-Flores theorem. The main tool used in the paper is R. Forman's discrete Morse theory.

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arXiv2018

Compactifications of $M_{0,n}$ associated with Alexander self-dual complexes: Chow ring, $ψ$-classes and intersection numbers

An Alexander self-dual complex gives rise to a compactification of $M_{0,n}$, called ASD compactification, which is a smooth algebraic variety. ASD compactifications include (but are not exhausted by) the polygon spaces, or the moduli spaces of flexible polygons. We present an explicit description of the Chow rings of ASD compactifications. We study the analogues of Kontsevich tautological bundles, compute their Chern classes, compute top intersections of the Chern classes, and derive a recursion for the intersection numbers.

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arXiv2018

Diagonal complexes

Given an $n$-gon, the poset of all collections of pairwise non-crossing diagonals is isomorphic to the face poset of some convex polytope called \textit{associahedron}. We replace in this setting the $n$-gon (viewed as a disc with $n$ marked points on the boundary) with an arbitrary oriented surface with a number of labeled marked points ("vertices"). With appropriate definitions we arrive at cell complexes $\mathcal{D}$ (generalization of) and its barycentric subdivision $\mathcal{BD}$. The complex $\mathcal{D}$ generalizes the associahedron. If the surface is closed, the complex $\mathcal{D}$ (as well as $\mathcal{BD}$) is homotopy equivalent to the space of metric ribbon graphs $RG_{g,n}^{met}$, or, equivalently, to the decorated moduli space $\widetilde{\mathcal{M}}_{g,n}$. For bordered surfaces, we prove the following: (1) Contraction of a boundary edge does not change the homotopy type of the support of the complex. (2) Contraction of a boundary component to a new marked point yields a forgetful map between two diagonal complexes which is homotopy equivalent to the Kontsevich's tautological circle bundle. Thus, contraction of a boundary component gives a natural simplicial model for the tautological bundle. As an application, we compute the psi-class, that is, the first Chern class in combinatorial terms. The latter result is an application of the local combinatorial formula. (3) In the same way, contraction of several boundary components corresponds to Whitney sum of the tautological bundles.

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arXiv2017

Intersection numbers of Chern classes of tautological line bundles on the moduli spaces of flexible polygons

Given a flexible $n$-gon with generic side lengths, the moduli space of its configurations in $\mathbb{R}^2$ as well as in $\mathbb{R}^3$ is a smooth manifold. It is equipped with $n$ \textit{tautological} line bundles whose definition is motivated by M. Kontsevich's tautological bundles over $\mathcal{M}_{0,n}$. We study their Euler classes, first Chern classes and intersection numbers, that is, top monomials in Chern (Euler) classes. The latter are interpreted geometrically as the signed numbers of some \textit{triangular configurations} of the flexible polygon.

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arXiv2017

Polygons with prescribed edge slopes: configuration space and extremal points of perimeter

We describe the configuration space $\mathbf{S}$ of polygons with prescribed edge slopes, and study the perimeter $\mathcal{P}$ as a Morse function on $\mathbf{S}$. We characterize critical points of $\mathcal{P}$ (these are \textit{tangential} polygons) and compute their Morse indices. This setup is motivated by a number of results about critical points and Morse indices of the oriented area function defined on the configuration space of polygons with prescribed edge lengths (flexible polygons). As a by-product, we present an independent computation of the Morse index of the area function (obtained earlier by G. Panina and A. Zhukova).

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arXiv2017

Alexander $r$-tuples and Bier complexes

Alexander $r$-tuples are introduced as a common generalization of pairs of Alexander dual complexes (Alexander $2$-tuples) and $r$-unavoidable complexes of Blagojević, Frick and Ziegler. The associated "Bier complexes" include both the Bier spheres and "optimal multiple chessboard complexes" as interesting, special cases. Our main result is Theorem 4.3 saying that (1) the $r$-fold deleted join of Alexander $r$-tuple is a pure complex homotopy equivalent to a wedge of spheres, and (2) the $r$-fold deleted join of a collectively unavoidable $r$-tuple is $(n-r-1)$-connected. We also give a complete classification (Theorem 5.1 and Corollary 5.2) of Alexander $r$-tuples and Bier complexes.

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arXiv2017

Moduli space of planar polygonal linkage: a combinatorial description

We explicitly describe a structure of a regular cell complex $K(L)$ on the moduli space $M(L)$ of a planar polygonal linkage $L$. The combinatorics is very much related (but not equal) to the combinatorics of the permutahedron. In particular, the cells of maximal dimension are labeled by elements of the symmetric group. For example, if the moduli space $M$ is a sphere, the complex $K$ is dual to the boundary complex of the permutahedron. The dual complex $K^*$ is patched of Cartesian products of permutohedra and carries a natural PL-structure. It can be explicitly realized as a polyhedron in the Euclidean space via a surgery on the permutohedron.

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arXiv2017

On the area of constrained polygonal linkages

We study configuration spaces of linkages whose underlying graph are polygons with diagonal constrains, or more general, partial two-trees. We show that (with an appropriate definition) the oriented area is a Bott-Morse function on the configuration space. Its critical points are described and Bott-Morse indices are computed. This paper is a generalization of analogous results for polygonal linkages (obtained earlier by G. Khimshiashvili, G. Panina, and A. Zhukova).

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arXiv2016

Oriented area is a perfect Morse function

We show that an appropriate generalization of the oriented area function is a perfect Morse function on the space of three-dimensional configurations of an equilateral polygonal linkage with odd number of edges. Therefore cyclic equilateral polygons (which appear as Morse points) are interpreted as independent generators of the homology groups of the (decorated) configuration space.

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