Handle slides for delta-matroids

Abstract

A classic exercise in the topology of surfaces is to show that, using handle slides, every disc-band surface, or 1-vertex ribbon graph, can be put in a canonical form consisting of the connected sum of orientable loops, and either non-orientable loops or pairs of interlaced orientable loops. Motivated by the principle that ribbon graph theory informs delta-matroid theory, we find the delta-matroid analogue of this surface classification. We show that, using a delta-matroid analogue of handle slides, every binary delta-matroid in which the empty set is feasible can be written in a canonical form consisting of the direct sum of the delta-matroids of orientable loops, and either non-orientable loops or pairs of interlaced orientable loops. Our delta-matroid results are compatible with the surface results in the sense that they are their ribbon graphic delta-matroidal analogues.Keywords: delta-matroid, disc-band surface, handle slide, ribbon graph 2010 MSC: 05B35, 05C10 Overview and backgroundMatroid theory is often thought of as a generalisation of graph theory. W. Tutte famously observed that, "If a theorem about graphs can be expressed in terms of edges and circuits alone it probably exemplifies a more general theorem about matroids" (see [14]). The merit of this point of view is that the more 'tactile' area of graph theory can serve as a guide for matroid theory, in the sense that results and properties for graphs can indicate what results and properties about matroids might hold. In [7] and [8], C. Chun et al. proposed that a similar relationship holds between topological graph theory and delta-matroid theory, writing "If a theorem about embedded graphs can be expressed in terms of its spanning quasi-trees then it probably exemplifies a more general theorem about delta-matroids". Taking advantage of this principle, here we use classical results from surface topology to guide us to a classification of binary delta-matroids.Informally, a ribbon graph is a "topological graph", whose vertices are discs and edges are ribbons, that arises from a regular neighbourhood of a graph in a surface. Formally, a ribbon graph G = (V, E) consists of a set of discs V whose elements are vertices, a set of discs E whose elements are edges, and is such that (i) the vertices and edges intersect in disjoint line segments; (ii) each such line segment lies on the boundary of precisely one vertex and precisely one edge; and (iii) every edge contains exactly two such line segments. We note that ribbon graphs describe exactly cellularly embedded graphs, and refer the reader to [10], or [12] where they are called reduced band

Lagrangian Subspaces, Delta-Matroids, and Four-Term Relations

“…In [10], the authors define the first and the second Vassiliev moves for binary delta-matroids B E . To define the second Vassiliev move, they use the recently introduced (see [13]) concept of handle sliding for deltamatroids. In [10], it is shown (see Proposition 4.10) that the action of the first and second Vassiliev moves on the space V E as defined by Kleptsyn-Smirnov coincides with the one defined by Zhukov and Lando for binary delta-matroids.…”

Delta-Matroids and Vassiliev Invariants

“…The definition of the 4-term relations requires the definition of two operations, namely, exchanging of handle ends (the first Vassiliev move) and handle sliding (the second Vassiliev move). The handle sliding for binary delta-matroids was defined in [16]. Below, we give the description of this operation, and define the operation of exchanging handle ends.…”

“…It was shown in [16] that for the delta-matroids of embedded graphs, the operation of handle sliding, when applied to two ribbons with neighboring ends, coincides with the handle sliding for embedded graphs. We prove a similar statement for the operation of exchanging handle ends.…”

“…Let D = (E; Φ) be a set system, a, b ∈ E be two different elements. It is proved in [16] that if D = (E(Γ); Φ(Γ)) is the delta-matroid of an embedded graph Γ and a, b are two ribbons in Γ with neighboring ends, then the delta-matroid of the ribbon graph Γ ab obtained from Γ by sliding the handle a over the handle b coincides with the delta-matroid D ab . However, if the ends of the ribbons a, b in Γ are not neighboring, then the handle sliding of the above definition can lead to a set system that is not isomorphic to the delta-matroid of any embedded graph.…”

“…However, if the ends of the ribbons a, b in Γ are not neighboring, then the handle sliding of the above definition can lead to a set system that is not isomorphic to the delta-matroid of any embedded graph. Moreover, the following example from [16] shows that a handle sliding applied to a delta-matroid can produce a set system that is not a delta-matroid. In the next section we prove a similar theorem for the other Vassiliev move, the first one.…”