Networks, (K)nots, Nucleotides, and Nanostructures

Designing self-assembling DNA nanostructures often requires the identification of a route for a scaffolding strand of DNA through the target structure. When the target structure is modeled as a graph, these scaffolding routes correspond to Eulerian circuits subject to turning restrictions imposed by physical constraints on the strands of DNA. Existence of such Eulerian circuits is an NP-hard problem, which can be approached by adapting solutions to a version of the Traveling Salesperson Problem. However, the author and collaborators have demonstrated that even Eulerian circuits obeying these turning restrictions are not necessarily feasible as scaffolding routes by giving examples of nontrivially knotted circuits which cannot be traced by the unknotted scaffolding strand. Often, targets of DNA nanostructure self-assembly are modeled as graphs embedded on surfaces in space. In this case, Eulerian circuits obeying the turning restrictions correspond to A-trails, circuits which turn immediately left or right at each vertex. In any graph embedded on the sphere, all A-trails are unknotted regardless of the embedding of the sphere in space. We show that this does not hold in general for graphs on the torus. However, we show this property does hold for checkerboard-colorable graphs on the torus, that is, those graphs whose faces can be properly 2-colored, and provide a partial converse to this result. As a consequence, we characterize (with one exceptional family) regular triangulations of the torus containing unknotted A-trails. By developing a theory of sums of A-trails, we lift constructions from the torus to arbitrary n-tori, and by generalizing our work on A-trails to smooth circuit decompositions, we construct all torus links and certain sums of torus links from circuit decompositions of rectangular torus grids. Graphs embedded on surfaces are equivalent to ribbon graphs, which are particularly well-suited to modeling DNA nanostructures, as their boundary components correspond to strands of DNA and their twisted ribbons correspond to double-helices. Every ribbon graph has a corresponding delta-matroid, a combinatorial object encoding the structure of the ribbon-graph\u27s spanning quasi-trees (substructures having exactly one boundary component). We show that interlacement with respect to quasi-trees can be generalized to delta-matroids, and use the resulting structure on delta-matroids to provide feasible-set expansions for a family of delta-matroid polynomials, both recovering well-known expansions of this type (such as the spanning-tree expansion of the Tutte polynnomial) as well as providing several previously unknown expansions. Among these are expansions for the transition polynomial, a version of which has been used to study DNA nanostructure self-assembly, and the interlace polynomial, which solves a problem in DNA recombination

A computational approach for finding 6-List-critical graphs on the Torus

La coloració de grafs dibuixats a superfícies és un àrea antiga i molt estudiada de la teoria de grafs. Thomassen va demostrar que hi ha un nombre finit de grafs 6-crítics a qualsevol superfície fixa i va proporcionar el conjunt explícit dels grafs 6-crítics al torus. Després, Postle va demostrar que hi ha un nombre finit de grafs 6-llista-crítics a qualsevol superfície fixa. Amb l'objectiu de trobar el conjunt de grafs 6-llista-crítics al torus, desenvolupem i implementem tècniques algorítmiques per la cerca per ordinador de grafs crítics en diferents situacions de coloració per llistes.La coloración de grafos dibujados en superficies es un área antigua y muy estudiada de la teoría de grafos. Thomassen demostró que hay un número finito de grafos 6-críticos en cualquier superficie fija y proporcionó el conjunto explícito de los grafos 6-críticos en el toro. Después, Postle demostró que hay un número finito de grafos 6-lista-críticos en cualquier superficie fija. Con el objetivo de encontrar el conjunto de grafos 6-lista-críticos en el toro, desarrollamos e implementamos técnicas algorítmicas para la búsqueda por ordenador de grafos críticos en diferentes situaciones de coloración por listas.Coloring graphs embedded on surfaces is an old and well-studied area of graph theory. Thomassen proved that there are finitely many 6-critical graphs on any fixed surface and provided the explicit set of 6-critical graphs on the torus. Later, Postle proved that there are finitely many 6-list-critical graphs on any fixed surface. With the goal of finding the set of 6-list-critical graphs on the torus, we develop and implement algorithmic techniques for computer search of critical graphs in different list-coloring settings.Outgoin

Distributed branch points and the shape of elastic surfaces with constant negative curvature

We develop a theory for distributed branch points and investigate their role in determining the shape and influencing the mechanics of thin hyperbolic objects. We show that branch points are the natural topological defects in hyperbolic sheets, they carry a topological index which gives them a degree of robustness, and they can influence the overall morphology of a hyperbolic surface without concentrating energy. We develop a discrete differential geometric (DDG) approach to study the deformations of hyperbolic objects with distributed branch points. We present evidence that the maximum curvature of surfaces with geodesic radius $R$ containing branch points grow sub-exponentially, $O(e^{c\sqrt{R}})$ in contrast to the exponential growth $O(e^{c' R})$ for surfaces without branch points. We argue that, to optimize norms of the curvature, i.e. the bending energy, distributed branch points are energetically preferred in sufficiently large pseudospherical surfaces. Further, they are distributed so that they lead to fractal-like recursive buckling patterns.Comment: 59 pages, 20 figures. Major revisions including new proofs with weakened hypotheses, expanded discussion and additional references. Some images are not at their original resolution to keep them at a reasonable size. Comments are very welcome and much appreciate