arXiv subjects

Nikos Apostolakis

Publications and source records attributed to Nikos Apostolakis.

6 recordsLinked to original sources

arXiv2020

Non-crossing trees, quadrangular dissections, ternary trees, and duality preserving bijections

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.

math.CO↗

arXiv2018

A duality for labeled graphs and factorizations with applications to graph embeddings and Hurwitz enumeration

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.

math.CO↗

arXiv2013

Lefschetz fibrations over the disc

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.

math.GT↗

arXiv2007

Kauffman state sums and bracket deformation

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.

math.GT↗

arXiv2004

On 4-fold covering moves

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.

math.GT↗

arXiv2001

On moves between branched coverings of S^3: The case of four sheets

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 bi-tricolored 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.

math.GT↗