Untangling Planar Curves

Any generic closed curve in the plane can be transformed into a simple closed curve by a finite sequence of local transformations called homotopy moves. We prove that simplifying a planar closed curve with n self-crossings requires Theta(n^{3/2}) homotopy moves in the worst case. Our algorithm improves the best previous upper bound O(n^2), which is already implicit in the classical work of Steinitz; the matching lower bound follows from the construction of closed curves with large defect, a topological invariant of generic closed curves introduced by Aicardi and Arnold. This lower bound also implies that Omega(n^{3/2}) degree-1 reductions, series-parallel reductions, and Delta-Y transformations are required to reduce any planar graph with treewidth Omega(sqrt{n}) to a single edge, matching known upper bounds for rectangular and cylindrical grid graphs. Finally, we prove that Omega(n^2) homotopy moves are required in the worst case to transform one non-contractible closed curve on the torus to another; this lower bound is tight if the curve is homotopic to a simple closed curve

Lower Bounds for Electrical Reduction on Surfaces

We strengthen the connections between electrical transformations and homotopy from the planar setting - observed and studied since Steinitz - to arbitrary surfaces with punctures. As a result, we improve our earlier lower bound on the number of electrical transformations required to reduce an n-vertex graph on surface in the worst case [SOCG 2016] in two different directions. Our previous Omega(n^{3/2}) lower bound applies only to facial electrical transformations on plane graphs with no terminals. First we provide a stronger Omega(n^2) lower bound when the planar graph has two or more terminals, which follows from a quadratic lower bound on the number of homotopy moves in the annulus. Our second result extends our earlier Omega(n^{3/2}) lower bound to the wider class of planar electrical transformations, which preserve the planarity of the graph but may delete cycles that are not faces of the given embedding. This new lower bound follow from the observation that the defect of the medial graph of a planar graph is the same for all its planar embeddings

Computing the Geometric Intersection Number of Curves

The geometric intersection number of a curve on a surface is the minimal number of self-intersections of any homotopic curve, i.e. of any curve obtained by continuous deformation. Given a curve c represented by a closed walk of length at most l on a combinatorial surface of complexity n we describe simple algorithms to (1) compute the geometric intersection number of c in O(n+ l^2) time, (2) construct a curve homotopic to c that realizes this geometric intersection number in O(n+l^4) time, (3) decide if the geometric intersection number of c is zero, i.e. if c is homotopic to a simple curve, in O(n+l log^2 l) time. To our knowledge, no exact complexity analysis had yet appeared on those problems. An optimistic analysis of the complexity of the published algorithms for problems (1) and (3) gives at best a O(n+g^2l^2) time complexity on a genus g surface without boundary. No polynomial time algorithm was known for problem (2). Interestingly, our solution to problem (3) is the first quasi-linear algorithm since the problem was raised by Poincare more than a century ago. Finally, we note that our algorithm for problem (1) extends to computing the geometric intersection number of two curves of length at most l in O(n+ l^2) time

The Unbearable Hardness of Unknotting

We prove that deciding if a diagram of the unknot can be untangled using at most k Reidemeister moves (where k is part of the input) is NP-hard. We also prove that several natural questions regarding links in the 3-sphere are NP-hard, including detecting whether a link contains a trivial sublink with n components, computing the unlinking number of a link, and computing a variety of link invariants related to four-dimensional topology (such as the 4-ball Euler characteristic, the slicing number, and the 4-dimensional clasp number)

Properly immersed curves in arbitrary surfaces via apparent contours on spines of traversing flows

Let S be a compact surface with boundary and F be the set of the orbits of a traversing flow on S. If the flow is generic, its orbit space is a spine G of S, namely G is a graph embedded in S and S is a regular neighbourhood of G. Moreover an extra structure on G turns it into a flow-spine, from which one can reconstruct S and F. In this paper we study properly immersed curves C in S. We do this by considering generic C's and their apparent contour relative to F, namely the set of points of G corresponding to orbits that either are tangent to C, or go through a self-intersection of C, or meet the boundary of C. We translate this apparent contour into a decoration of G that allows one to reconstruct C, and then we allow C to vary up to homotopy within a fixed generic F, and next also F to vary up to homotopy, and we identify a finite set of local moves on decorated graphs that translate these homotopies.Comment: 39 pages, 50 figure

From Curves to Words and Back Again: Geometric Computation of Minimum-Area Homotopy

Let $\gamma$ be a generic closed curve in the plane. Samuel Blank, in his 1967 Ph.D. thesis, determined if $\gamma$ is self-overlapping by geometrically constructing a combinatorial word from $\gamma$. More recently, Zipei Nie, in an unpublished manuscript, computed the minimum homotopy area of $\gamma$ by constructing a combinatorial word algebraically. We provide a unified framework for working with both words and determine the settings under which Blank's word and Nie's word are equivalent. Using this equivalence, we give a new geometric proof for the correctness of Nie's algorithm. Unlike previous work, our proof is constructive which allows us to naturally compute the actual homotopy that realizes the minimum area. Furthermore, we contribute to the theory of self-overlapping curves by providing the first polynomial-time algorithm to compute a self-overlapping decomposition of any closed curve $\gamma$ with minimum area.Comment: 27 pages, 16 figure

Structure and enumeration of K4-minor-free links and link diagrams

We study the class L of link-types that admit a K4-minor-free diagram, i.e., they can be projected on the plane so that the resulting graph does not contain any subdivision of K4. We prove that L is the closure of a subclass of torus links under the operation of connected sum. Using this structural result, we enumerate L and subclasses of it, with respect to the minimum number of crossings or edges in a projection of L' in L. Further, we obtain counting formulas and asymptotic estimates for the connected K4-minor-free link-diagrams, minimal K4-minor-free link-diagrams, and K4-minor-free diagrams of the unknot.Peer ReviewedPostprint (author's final draft

Transformaciones ∆ − Y en redes

La Combinatoria se compone de varias ramas que involucran el estudio de procesos finitos y estructuras discretas. Las gráficas son estructuras que constituyen el concepto central del estudio de una rama muy robusta de la Combinatoria. Así que resulta de gran interés investigar sus propiedades, invariantes y clasificaciones en familias que comparten propiedades. Un conjunto de resultados en teoría de graficas muestran que una clase de graficas pueden ser reducidas a una forma canónica mediante la aplicación de ciertas operaciones. Se puede demostrar, que la aplicación inversa de esas mismas operaciones puede generar todas las gráficas en una clase. Uno de los conjuntos más importantes de reducciones usadas en la teoría de graficas son las reducciones serie-paralelo, al aplicar una reducción de este tipo a una gráfica, se disminuye el número de sus aristas. Las operaciones que no alteran el número de aristas de la gráfica se llaman transformaciones, en particular y como centro de análisis en este trabajo, se estudian las transformaciones ∆ − Y