Geometry and Topology Commons^™

The Topology Of Surface Mediatrices, James Bernhard, J. J. P. Veerman

Mathematics and Statistics Faculty Publications and Presentations

Given a pair of distinct points p and q in a metric space with distance d, the mediatrix is the set of points x such that d(x,p)=d(x,q). In this paper, we examine the topological structure of mediatrices in connected, compact, closed 2-manifolds whose distance function is inherited from a Riemannian metric. We determine that such mediatrices are, up to homeomorphism, finite, closed simplicial 1-complexes with an even number of incipient edges emanating from each vertex. Using this and results from [J.J.P. Veerman, J. Bernhard, Minimally separating sets, mediatrices and Brillouin spaces, Topology Appl., in press], we give the classification …

Centers And Shore Points In Λ-Dendroids, Van C. Nall

Department of Math & Statistics Faculty Publications

A dendroid is the disjoint union of the set of centers and the set of shore points. We show this is also true for λ-dendroids and use this fact to show that the finite union of shore continua in a λ-dendroid is a shore set.

Connecting Polygonizations Via Stretches And Twangs, Mirela Damian, Robin Flatland, Joseph O'Rourke, Suneeta Ramswami

Computer Science: Faculty Publications

We show that the space of polygonizations of a fixed planar point set S of n points is connected by O(n² ) “moves” between simple polygons. Each move is composed of a sequence of atomic moves called “stretches” and "twangs". These atomic moves walk between weakly simple "polygonal wraps" of S. These moves show promise to serve as a basis for generating random polygons.

The Center Of Some Braid Groups And The Farrell Cohomology Of Certain Pure Mapping Class Groups, Craig A. Jensen, Yu Qing Chen, Henry H. Glover

Mathematics Faculty Publications

In this paper we first show that many braid groups of low genus surfaces have their centers as direct factors. We then give a description of centralizers and normalizers of prime order elements in pure mapping class groups of surfaces with spherical quotients using automorphism groups of fundamental groups of the quotient surfaces. As an application, we use these to show that the primary part of the Farrell cohomology groups of certain mapping class groups are elementary abelian groups. At the end we compute the primary part of the Farrell cohomology of a few pure mapping class groups.

The Euler Characteristic Of The Whitehead Automorphism Group Of A Free Product, Craig A. Jensen, Jon Mccammond, John Meier

Mathematics Faculty Publications

A combinatorial summation identity over the lattice of labelled hypertrees is established that allows one to gain concrete information on the Euler characteristics of various automorphism groups of free products of groups. In particular, we establish formulae for the Euler characteristics of: the group of Whitehead automorphisms...

Boundary Slopes Of 2-Bridge Links Determine The Crossing Number, Jim Hoste, Patrick D. Shanahan

Mathematics, Statistics and Data Science Faculty Works

A diagonal surface in a link exterior M is a properly embedded, incompressible, boundary incompressible surface which furthermore has the same number of boundary components and the same slope on each component of the boundary of M. We derive a formula for the boundary slope of a diagonal surface in the exterior of a 2-bridge link which is analogous to the formula for the boundary slope of a 2-bridge knot found by Hatcher and Thurston. Using this formula we show that the diameter of a 2-bridge link, that is, the difference between the smallest and largest finite slopes of …

Virtual Spatial Graphs, Thomas Fleming, Blake Mellor

Mathematics, Statistics and Data Science Faculty Works

Two natural generalizations of knot theory are t he study of spatially embedded graphs, and Kauffman's theory of virtual knots. In this paper we combine these approaches to begin the study of virtual spat ial graphs.

Exotic Statistics For Strings In 4d Bf Theory, John C. Baez, Derek K. Wise, Alissa S. Crans

Mathematics, Statistics and Data Science Faculty Works

After a review of exotic statistics for point particles in 3d BF theory, and especially 3d quantum gravity, we show that string-like defects in 4d BF theory obey exotic statistics governed by the 'loop braid group'. This group has a set of generators that switch two strings just as one would normally switch point particles, but also a set of generators that switch two strings by passing one through the other. The first set generates a copy of the symmetric group, while the second generates a copy of the braid group. Thanks to recent work of Xiao-Song Lin, we can …

Intrinsic Linking And Knotting In Virtual Spatial Graphs, Thomas Fleming, Blake Mellor

Mathematics, Statistics and Data Science Faculty Works

We introduce a notion of intrinsic linking and knotting for virtual spatial graphs. Our theory gives two filtrations of the set of all graphs, allowing us to measure, in a sense, how intrinsically linked or knotted a graph is; we show that these filtrations are descending and nonterminating. We also provide several examples of intrinsically virtually linked and knotted graphs. As a byproduct, we introduce the virtual unknotting number of a knot, and show that any knot with nontrivial Jones polynomial has virtual unknotting number at least 2.

Derived Categories And The Analytic Approach To General Reciprocity Laws. Part Ii, Michael Berg

Mathematics, Statistics and Data Science Faculty Works

Building on the topological foundations constructed in Part I, we now go on to address the homological algebra preparatory to the projected final arithmetical phase of our attack on the analytic proof of general reciprocity for a number field. In the present work, we develop two algebraic frameworks corresponding to two interpretations of Kubota's n-Hilbert reciprocity formalism, presented in a quasi-dualized topological form in Part I, delineating two sheaf-theoretic routes toward resolving the aforementioned (open) problem. The first approach centers on factoring sheaf morphisms eventually to yield a splitting homomorphism for Kubota's n-fold cover of the adelized special linear group …

Clarifications Of Rule 2 In Teaching Geometric Dimensioning And Tolerancing, Cheng Lin, Alok Verma

Engineering Technology Faculty Publications

Geometric dimensioning and tolerancing is a symbolic language used on engineering drawings and computer generated three-dimensional solid models for explicitly describing nominal geometry and its allowable variation. Application cases using the concept of Rule 2 in the Geometric Dimensioning and Tolerancing (GD&T) are presented. The rule affects all fourteen geometric characteristics. Depending on the nature and location where each feature control frame is specified, interpretation on the applicability of Rule 2 is quite inconsistent. This paper focuses on identifying the characteristics of a feature control frame to remove this inconsistency. A table is created to clarify the confusions for students …

The Local Gromov–Witten Invariants Of Configurations Of Rational Curves, Dagan Karp, Chiu-Chu Melissa Liu, Marcos Mariño

All HMC Faculty Publications and Research

We compute the local Gromov–Witten invariants of certain configurations of rational curves in a Calabi–Yau threefold. These configurations are connected subcurves of the “minimal trivalent configuration”, which is a particular tree of ℙ¹’s with specified formal neighborhood. We show that these local invariants are equal to certain global or ordinary Gromov–Witten invariants of a blowup of ℙ³ at points, and we compute these ordinary invariants using the geometry of the Cremona transform. We also realize the configurations in question as formal toric schemes and compute their formal Gromov–Witten invariants using the mathematical and physical theories of the …

Applications Of Computational Homology, Christopher Aaron Johnson

Theses, Dissertations and Capstones

Homology is a field of topology that classifies objects based on the number of n- dimensional holes (cuts, tunnels, voids, etc.) they possess. The number of its real life ap- plications is quickly growing, which requires development of modern computational meth- ods. In my thesis, I will present methods of calculation, algorithms, and implementations of simplicial homology, alpha shapes, and persistent homology.

The Alpha Shapes method represents a point cloud as the union of balls centered at each point, and based on these balls, a complex can be built and homology computed. If the balls are allowed to grow, one …

Geodesic On Surfaces Of Constant Gaussian Curvature, Veasna Chiek

Theses Digitization Project

The goal of the thesis is to study geodesics on surfaces of constant Gaussian curvature. The first three sections of the thesis is dedicated to the definitions and theorems necessary to study surfaces of constant Gaussian curvature. The fourth section contains examples of geodesics on these types of surfaces and discusses their properties. The thesis incorporates the use of Maple, a mathematics software package, in some of its calculations and graphs. The thesis' conclusion is that the Gaussian curvature is a surface invariant and the geodesics of these surfaces will be the so-called best paths.

Gauss-Bonnet Formula, Heather Ann Broersma

Theses Digitization Project

From fundamental forms to curvatures and geodesics, differential geometry has many special theorems and applications worth examining. Among these, the Gauss-Bonnet Theorem is one of the well-known theorems in classical differential geometry. It links geometrical and topological properties of a surface. The thesis introduced some basic concepts in differential geometry, explained them with examples, analyzed the Gauss-Bonnet Theorem and presented the proof of the theorem in greater detail. The thesis also considered applications of the Gauss-Bonnet theorem to some special surfaces.

Hausdorff Dimension, Loren Beth Nemeth

Theses Digitization Project

The purpose of this study was to define topological dimension and Hausdorff dimension, Namely metric space theory and measure theory. It was verified that in the sets of elementary geometry, the dimensions agree, while in the case of the fractals, the Hausdorff dimension is strictly larger than the topological dimension.

Intrinsic Linking And Knotting Of Graphs In Arbitrary 3–Manifolds, Erica Flapan, Hugh Howards, Don Lawrence, Blake Mellor

Pomona Faculty Publications and Research

We prove that a graph is intrinsically linked in an arbitrary 3–manifold M if and only if it is intrinsically linked in S³. Also, assuming the Poincaré Conjecture, we prove that a graph is intrinsically knotted in M if and only if it is intrinsically knotted in S³.

Notes For Mat 7500 – Winter '93, Revised Winter '06, David Handel

Mathematics Faculty Research Publications

These notes developed from a one semester course at Wayne State University, taught several times in the last three decades of the 1900s. The subject matter is analysis on manifolds, consisting of the theory of smooth manifolds, differential forms, integration of forms, the generalized Stokes' Theorem, de Rham cohomology, and some related topics. The course is intended for first or second year graduate students in Mathematics with a background in Advanced Calculus, General Topology, linear algebra (including quotient spaces), and a little elementary group theory (including some familiarity with the symmetric groups). Given the above background, the notes are self-contained. …

The Integral Cohomology Of The Group Of Loops, Craig Jensen, Jon Mccammond, John Meier

Mathematics Faculty Publications

No abstract provided.

Intrinsic Linking And Knotting Of Graphs In Arbitrary 3–Manifolds, Erica Flapan, Hugh Howards, Don Lawrence, Blake Mellor

Mathematics, Statistics and Data Science Faculty Works

We prove that a graph is intrinsically linked in an arbitrary 3–manifold MM if and only if it is intrinsically linked in S³. Also, assuming the Poincaré Conjecture, we prove that a graph is intrinsically knotted in M if and only if it is intrinsically knotted in S³.

Intersection Graphs For String Links, Blake Mellor

Mathematics, Statistics and Data Science Faculty Works

We extend the notion of intersection graphs for knots in the theory of finite type invariants to string links. We use our definition to develop weight systems for string links via the adjacency matrix of the intersection graph, and show that these weight systems are related to the weight systems induced by the Conway and Homfly polynomials.

Chord Diagrams And Gauss Codes For Graphs, Thomas Fleming, Blake Mellor

Mathematics, Statistics and Data Science Faculty Works

Chord diagrams on circles and their intersection graphs (also known as circle graphs) have been intensively studied, and have many applications to the study of knots and knot invariants, among others. However, chord diagrams on more general graphs have not been studied, and are potentially equally valuable in the study of spatial graphs. We will define chord diagrams for planar embeddings of planar graphs and their intersection graphs, and prove some basic results. Then, as an application, we will introduce Gauss codes for immersions of graphs in the plane and give algorithms to determine whether a particular crossing sequence is …

Tree Diagrams For String Links, Blake Mellor

Mathematics, Statistics and Data Science Faculty Works

In previous work, the author defined the intersection graph of a chord diagram associated with string links (as in the theory of finite type invariants). In this paper, we classify the trees which can be obtained as intersection graphs of string link diagrams.

Linked Exact Triples Of Triangulated Categories And A Calculus Of T-Structures, Michael Berg

Mathematics, Statistics and Data Science Faculty Works

We introduce a new formalism of exact triples of triangulated categories arranged in certain types of diagrams. We prove that these arrangements are well-behaved relative to the process of gluing and ungluing t-structures defined on the indicated categories and we connect our con. structs to· a problem (from number theory) involving derived categories. We also briefly address a possible connection with a result of R. Thomason.

A Brief Study Of Real-Valued Continuous Functions On Various Spaces, Dusty Ross

Honors Program Theses

The intent of this study is to find sufficient criteria on a space X in order to bound the cardinality of the real-valued continuous functions on X by 2w. The desired result is known for X separable and for X first-countable, Hausdorff, and either Lindelof or ccc, but these are all very strong properties on a space. It is the goal of this study to find properties that are weaker, yet sufficient in bounding .the number of real-valued continuous functions on a space by the size of the continuum.

Wallpaper Groups, Cassondra Fett

Honors Capstones

Capstone submitted as a graduation requirement for the BSU Honors Program.

Neutrosophic Methods In General Relativity, Florentin Smarandache, Dmitri Rabounski, Larissa Borissova

Branch Mathematics and Statistics Faculty and Staff Publications

In this work the authors apply concepts of Neutrosophic Logic to the General Theory of Relativity to obtain a generalisation of Einstein’s fourdimensional pseudo-Riemannian differentiable manifold in terms of Smarandache Geometry (Smarandache manifolds), by which new classes of relativistic particles and non-quantum teleportation are developed. Fundamental features of Neutrosophic Logic are its denial of the Law of Excluded Middle, and open (or estimated) levels of truth, falsity and indeterminancy. Both Neutrosophic Logic and Smarandache Geometry were invented some years ago by one of the authors (F. Smarandache). The application of these purely mathematical theories to General Relativity reveals hitherto unknown …

A Constructive Proof Of Ky Fan's Generalization Of Tucker's Lemma, Timothy Prescott '02, Francis E. Su

All HMC Faculty Publications and Research

We present a proof of Ky Fan's combinatorial lemma on labellings of triangulated spheres that differs from earlier proofs in that it is constructive. We slightly generalize the hypotheses of Fan's lemma to allow for triangulations of Sⁿ that contain a flag of hemispheres. As a consequence, we can obtain a constructive proof of Tucker's lemma that holds for a more general class of triangulations than the usual version.

Differentials In The Homological Homotopy Fixed Point Spectral Sequence, Robert R. Bruner, John Rognes

Mathematics Faculty Research Publications

We analyze in homological terms the homotopy fixed point spectrum of a T–equivariant commutative S–algebra R. There is a homological homotopy fixed point spectral sequence with E^2_(s,t) = H^(−s)_(gp) (��;H_t(R;��_p)), converging conditionally to the continuous homology H^c_(s+t)(R^(h��);��_p) of the homotopy fixed point spectrum. We show that there are Dyer–Lashof operations β^ϵQ^i acting on this algebra spectral sequence, and that its differentials are completely determined by those originating on the vertical axis. More surprisingly, we show that for each class x in the E^(2r)–term of the spectral sequence there are 2r other classes in the E^(2r)–term (obtained mostly by Dyer–Lashof operations …

Orthogonal Curvilinear Coordinates, Lindsey J. Bromenshenkel

Honors Capstones

Capstone submitted as a graduation requirement for the BSU Honors Program.