Papers by Nikolaos E Apostolakis
arXiv: Combinatorics, 2018
The set of factorizations of permutations in to $m$ transpositions of some symmetric group $\math... more The set of factorizations of permutations in to $m$ transpositions of some symmetric group $\mathcal{S}_n$ is naturally in bijection with the set of graphs of order $n$ and size $m$ with both edges and vertices labeled. We define a notion of duality (the \emph{mind-body duality}) for factorizations and such labeled graphs and interpret it in terms of Properly Embedded Graphs, a class of graphs embedded in a bounded compact oriented surface with all the vertices lying in the boundary, and show a close connection of this duality with the Hurwitz action of the Braid Group. Connections with the theory of Cellularly Embedded Graphs are highlighted and hints of possible applications are given. In this paper we focus on developing the necessary theory, leaving specific applications and further developments for future projects.
Using the theory developed in arXiv:1804.01214 we define an involutory duality for non-crossing t... more Using the theory developed in arXiv:1804.01214 we define an involutory duality for non-crossing trees and provide a bijection between the set of non-crossing trees with $n$ vertices and quadrangular dissections of a $2n$-gon by $n-1$ non-crossing diagonals that transforms that duality to reflection across an axis connecting the midpoints of two diametrically opposite sides of the $2n$-gon. We also show that this bijection fits well with well known bijections involving the set of ternary trees with $n-1$ internal vertices and the set of Flagged Perfectly Chain Decomposed Binary Ditrees. Further by analyzing the natural dihedral group action on the set of quadrangular dissections of a $2n$-gon we provide closed formulae for the number of quadrangular dissections up to rotations and up to rotations and reflections, the set of non-crossing trees up to rotations and up to rotations and reflections, the number of self-dual non-crossing trees, and the number of oriented and unoriented unla...
Proceedings of the London Mathematical Society, 2013
We provide a complete set of moves relating any two Lefschetz fibrations over the disc having as ... more We provide a complete set of moves relating any two Lefschetz fibrations over the disc having as their total space the same four-dimensional 2-handlebody up to 2-equivalence. As a consequence, we also obtain moves relating diffeomorphic three-dimensional open books, providing a different approach to an analogous previous result by Harer.
Journal of Knot Theory and Its Ramifications, 2007
We derive a formula expanding the bracket with respect to a natural deformation parameter. The ex... more We derive a formula expanding the bracket with respect to a natural deformation parameter. The expansion is in terms of a two-variable polynomial algebra of diagram resolutions generated by basic operations involving the Goldman bracket. A functorial characterization of this algebra is given. Differentiability properties of the star product underlying the Kauffman bracket are discussed.
Algebraic & Geometric Topology, 2003
We prove the existence of a finite set of moves sufficient to relate any two representations of t... more We prove the existence of a finite set of moves sufficient to relate any two representations of the same 3-manifold as a 4-fold simple branched covering of S 3. We also prove a stabilization result: after adding a fifth trivial sheet two local moves suffice. These results are analogous to results of Piergallini in degree 3 and can be viewed as a second step in a program to establish similar results for arbitrary degree coverings of S 3 .
A combinatorial presentation of closed orientable 3-manifolds as bi-tricolored links is given tog... more A combinatorial presentation of closed orientable 3-manifolds as bi-tricolored links is given together with two versions of a calculus via moves to manipulate bitricolored links without changing the represented manifold. That is, we provide a finite set of moves sufficient to relate any two manifestations of the same 3-manifold as a simple 4-sheeted branched covering of S 3 .
We derive a formula expanding the bracket with respect to a natural deformation parameter. The ex... more We derive a formula expanding the bracket with respect to a natural deformation parameter. The expansion is in terms of a two-variable polynomial algebra of diagram resolutions generated by basic operations involving the Goldman bracket. A functorial characterization of this algebra is given. Differentiability properties of the star product underlying the Kauffman bracket are discussed.
We prove the existence of a finite set of moves sufficient to relate any two representations of t... more We prove the existence of a finite set of moves sufficient to relate any two representations of the same 3–manifold as a 4–fold simple branched covering of S 3. We also prove a stabilization result: after adding a fifth trivial sheet two local moves suffice. These results are analogous to results of Piergallini in degree 3 and can be viewed as a second step in a program to establish similar results for arbitrary degree coverings of S 3 .
Proceedings of the London Mathematical Society, 2013
We provide a complete set of moves relating any two Lefschetz fibrations over the disk having as ... more We provide a complete set of moves relating any two Lefschetz fibrations over the disk having as their total space the same 4-dimensional 2-handlebody up to 2-equivalence. As a consequence, we also obtain moves relating diffeomorphic 3-dimensional open books, providing a different approach to an analogous previous result by Harer.
Drafts by Nikolaos E Apostolakis
Using the theory of Properly Embedded Graphs developed in an earlier work we define an involutory... more Using the theory of Properly Embedded Graphs developed in an earlier work we define an involutory duality on the set labeled non-crossing trees that lifts the obvious duality in the set of unlabeled non-crossing trees. The set of non-crossing trees is a free ternary magma with one generator and this duality is an instance of a duality that is defined in any such magma. Any two free ternary magmas with one generator are isomorphic via a unique isomorphism that we call the structural bijection. Besides the set of non-crossing trees we also consider as free ternary magmas with one generator the set of ternary trees, the set of quadrangular dissections, and the set of flagged Perfectly Chain Decomposed Ditrees, and we give topological and/or combinatorial interpretations of the structural bijections between them. In particular the bijection from the set of quadrangular dissections to the set of non-crossing trees seems to be new. Further we give explicit formulas for the number of self-dual labeled and unlabeled non-crossing trees and the set of quadrangular dissections up to rotations and up to rotations and reflections.
Using the theory developed in [1] we define an involutory duality for non-crossing trees and prov... more Using the theory developed in [1] we define an involutory duality for non-crossing trees and provide a bijection between the set of non-crossing trees with n vertices and quadrangular dissec-tions of a 2n-gon by n − 1 non-crossing diagonals that transforms that duality to reflection across an axis connecting the midpoints of two diametrically opposite sides of the 2n-gon. We also show that this bijection fits well with well known bijections involving the set of ternary trees with n − 1 internal vertices and the set of Flagged Perfectly Chain Decomposed Binary Ditrees. Further by analyzing the natural dihedral group action on the set of quadrangular dissections of a 2n-gon we provide closed formulae for the number of quadrangular dissections up to rotations and up to rotations and reflections, the set of non-crossing trees up to rotations and up to rotations and reflections, the number of self-dual non-crossing trees, and the number of oriented and unori-ented unlabeled self-dual non-crossing trees. With the exception of the formula giving the number of unoriented unlabeled non-crossing trees, these formulae are new. 1. THE BIJECTIONS In [1] we introduced a notion of duality (called mind-body duality) for factorizations in a symmetric group S n , and interpreted it in terms of e-v-graphs (that is graphs with ordered edges and vertices) and pegs (that is graphs properly embedded in surfaces with boundary). In this paper we focus on vertex-labeled trees pegged on a disk, or as they are more commonly known, non-crossing trees. We start by fixing conventions and definitions and recalling some basic facts from [1], and refer the reader there for more details. By a non-crossing tree we mean a labeled tree t pegged in D 2 the 2-dimensional disk endowed with the counterclockwise orientation, and we denote the set of non-crossing trees with n vertices by N n. We assume that the vertices of t form the vertices of a regular n-gon and that the order induced by their labels is compatible with the cyclic order of the boundary circle induced by the orientation of the disk, and to be concrete for each n we fix the vertices of a regular n-gon with a standard labeling and we assume that all non-crossing trees have those vertices and that all edges are embedded as chords of the circle. We emphasize that the orientation of the disk is part of the definition and we denote by N ⊺ n the set of trees with n vertices pegged in D 2 ⊺ , the disk endowed with the clockwise orientation. We assume that the elements of N ⊺ n have the same vertices as the elements of N n but with their labels reflected across the diameter that passes through the vertex labeled 1. For a t ∈ N n we denote by t ⊺ the element of N ⊺ n that has the same underlying vertex labeled tree, see the left and middle of Figure 1 for an example. On the other hand the element of N n that is obtained from t by reflecting the edges of t across the diameter passing through 1 will be denoted ¯ t, in other words ¯ t has an edge (n + 2 − i, n + 2 − j) (addition is taken (mod n)) for every edge (i, j) of t. We will sometimes denote by s : N n
The set of factorizations of permutations in to m transpositions of some symmetric group Sn is na... more The set of factorizations of permutations in to m transpositions of some symmetric group Sn is naturally in bijection with the set of graphs of order n and size m with both edges and vertices labeled. We define a notion of duality (the mind-body duality) for factorizations and such labeled graphs and interpret it in terms of Properly Embedded Graphs, a class of graphs embedded in a bounded compact oriented surface with all the vertices lying in the boundary, and show a close connection of this duality with the Hurwitz action of the Braid Group. Connections with the theory of Cellularly Embedded Graphs are highlighted and hints of possible applications are given. In this paper we focus on developing the necessary theory, leaving specific applications and further developments for future projects.
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Papers by Nikolaos E Apostolakis
Drafts by Nikolaos E Apostolakis