Critical Points Of Index Research Articles

Additive energy functions have predictable landscape topologies.

Recent work [Mirth et al., J. Chem. Phys. 154, 114114 (2021)] has demonstrated that sublevelset persistent homology provides a compact representation of the complex features of an energy landscape in 3 N-dimensions. This includes information about all transition paths between local minima (connected by critical points of index ≥1) and allows for differentiation of energy landscapes that may appear similar when considering only the lowest energy pathways (as tracked by other representations, such as disconnectivity graphs, using index 1 critical points). Using the additive nature of the conformational potential energy landscape of n-alkanes, it became apparent that some topological features-such as the number of sublevelset persistence bars-could be proven. This work expands the notion of predictable energy landscape topology to any additive intramolecular energy function on a product space, including the number of sublevelset persistent bars as well as the birth and death times of these topological features. This amounts to a rigorous methodology to predict the relative energies of all topological features of the conformational energy landscape in 3N dimensions (without the need for dimensionality reduction). This approach is demonstrated for branched alkanes of varying complexity and connectivity patterns. More generally, this result explains how the sublevelset persistent homology of an additive energy landscape can be computed from the individual terms comprising that landscape.

Hyperbolic surfaces with sublinearly many systoles that fill

For any \varepsilon > 0 , we construct a closed hyperbolic surface of genus g=g(\varepsilon) with a set of at most \varepsilon g systoles that fill, meaning that each component of the complement of their union is contractible. This surface is also a critical point of index at most \varepsilon g for the systole function, disproving the lower bound of 2g-1 posited by Schmutz Schaller.

A distance for circular Heegaard splittings

For a knot [Formula: see text], its exterior [Formula: see text] has a singular foliation by Seifert surfaces of [Formula: see text] derived from a circle-valued Morse function [Formula: see text]. When [Formula: see text] is self-indexing and has no critical points of index 0 or 3, the regular levels that separate the index-1 and index-2 critical points decompose [Formula: see text] into a pair of compression bodies. We call such a decomposition a circular Heegaard splitting of [Formula: see text]. We define the notion of circular distance (similar to Hempel distance) for this class of Heegaard splitting and show that it can be bounded under certain circumstances. Specifically, if the circular distance of a circular Heegaard splitting is too large: (1) [Formula: see text] cannot contain low-genus incompressible surfaces, and (2) a minimal-genus Seifert surface for [Formula: see text] is unique up to isotopy.

On a banded link presentation of knotted surfaces

We will discuss a method for visual presentation of knotted surfaces in the four space, by examining a number and a position of its Morse’s critical points. Using this method, we will investigate surface-knot with one critical point of index [Formula: see text]. Then we show infinitely many mutually distinct surface-knots that have an embedding with two critical points of index [Formula: see text]. Next we define a long flat form of a banded link for any surface-knot and show diagrammatically a long flat form of [Formula: see text]-twist-spun [Formula: see text]-torus knots.

Surface reconstruction based on a dynamical system

We present an efficient algorithm that computes a manifold triangular mesh from a set of unorganized sample points in . The algorithm builds on the observation made by several researchers that the Gabriel graph of the sample points provides a good surface description. However, this surface description is only one-dimensional. We associate the edges of the Gabriel graph with index 1 critical points of a dynamical system induced by the sample points. Exploiting also the information contained in the critical points of index 2 provides a two-dimensional surface description which can be easily turned into a manifold.

Controlled connectivity of closed 1–forms

We discuss controlled connectivity properties of closed 1-forms and their cohomology classes and relate them to the simple homotopy type of the Novikov complex. The degree of controlled connectivity of a closed 1-form depends only on positive multiples of its cohomology class and is related to the Bieri-Neumann-Strebel-Renz invariant. It is also related to the Morse theory of closed 1-forms. Given a controlled 0-connected cohomology class on a manifold M with n = dim M > 4 we can realize it by a closed 1-form which is Morse without critical points of index 0, 1, n-1 and n. If n = dim M > 5 and the cohomology class is controlled 1-connected we can approximately realize any chain complex D_* with the simple homotopy type of the Novikov complex and with D_i=0 for i < 2 and i > n-2 as the Novikov complex of a closed 1-form. This reduces the problem of finding a closed 1-form with a minimal number of critical points to a purely algebraic problem.

An analogue of Morse theory for planar linear networks and the generalized Steiner problem

A study is made of the generalized Steiner problem: the problem of finding all the locally minimal networks spanning a given boundary set (terminal set). It is proposed to solve this problem by using an analogue of Morse theory developed here for planar linear networks. The space of all planar linear networks spanning a given boundary set is constructed. The concept of a critical point and its index is defined for the length function of a planar linear network. It is shown that locally minimal networks are local minima of on and are critical points of index 1. The theorem is proved that the sum of the indices of all the critical points is equal to . This theorem is used to find estimates for the number of locally minimal networks spanning a given boundary set.

Morse index and causal continuity. A criterion for topology change in quantum gravity

Studies in 1 + 1 dimensions suggest that causally discontinuous topology-changing spacetimes are suppressed in quantum gravity. Borde and Sorkin have conjectured that causal discontinuities are associated precisely with index 1 or n - 1 Morse points in topology-changing spacetimes built from Morse functions. We establish a weaker form of this conjecture. Namely, if a Morse function f on a compact cobordism has critical points of index 1 or n - 1, then all the Morse geometries associated with f are causally discontinuous, while if f has no critical points of index 1 or n - 1, then there exist associated Morse geometries which are causally continuous.

On Gradient Dynamical Systems

We consider in this paper a Co vector field X on a Co compact manifold Mn (&M, the boundary of M, may be empty or not) satisfying the following conditions: (1) At each singular point /8 of X, there is a cell neighborhood N and a Co function f on N such that X is the gradient of f on N in some riemannian structure on N. Furthermore /8 is a non-degenerate critical point of f. Let ,81 , m denote these singularities. (2) If x e &M, X at x is transversal (not tangent) to SM. Hence X is not zero on SM. (3) If x e M let p,(x) denote the orbit of X (solution curve) through x satisfying p0(x) = x. Then for each x e M, the limit set of p,(x) as t +-~ oo is contained in the union of the /3i. (4) The stable and unstable manifolds of the /3i have normal intersection with each other. This has the following meaning. The stable manifold Wj* of /3i is the set of all x e M such that limits ...p,(x) = /i. The unstable manifold Wi of 8i is the set of all x e M such that limit,,-,. t(x) = /i. It follows from conditions (1), (2) and a local theorem in [1, p. 330], that if /3i is a critical point of index X, then Wi is the image of a 1-1, Co map pi: U-s M, where Uc Rn A has the property if x e U, tx e U, 0 ? t ? 1 and pi has rank n X everywhere (see [4] for more details). A similar statement holds for Wi* with the U c RA. Now for x e Wi (or Wi*) let Wi2, (or We*) be the tangent space of Wi (or Wi*) at x. Then for each i, j, if x e Wf nWj*, condition (4) means that