# Geometry and Topology Commons™

## All Articles in Geometry and Topology

487 full-text articles. Page 1 of 16.

Intrinsic Linking And Knotting Of Graphs In Arbitrary 3–Manifolds, 2017 Loyola Marymount University

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

##### Blake Mellor

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.

Tree Diagrams For String Links, 2017 Loyola Marymount University

#### Tree Diagrams For String Links, Blake Mellor

##### Blake Mellor

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.

On The Existence Of Finite Type Link Homotopy Invariants, 2017 Loyola Marymount University

#### On The Existence Of Finite Type Link Homotopy Invariants, Blake Mellor, Dylan Thurston

##### Blake Mellor

We show that for links with at most 5 components, the only finite type homotopy invariants are products of the linking numbers. In contrast, we show that for links with at least 9 components, there must exist finite type homotopy invariants which are not products of the linking numbers. This corrects previous errors of the first author.

2017 Loyola Marymount University

#### Tree Diagrams For String Links Ii: Determining Chord Diagrams, Blake Mellor

##### Blake Mellor

In previous work, we defined the intersection graph of a chord diagram associated with a string link (as in the theory of finite type invariants). In this paper, we look at the case when this graph is a tree, and we show that in many cases these trees determine the chord diagram (modulo the usual 1-term and 4-term relations).

Spatial Graphs With Local Knots, 2017 Loyola Marymount University

#### Spatial Graphs With Local Knots, Erica Flapan, Blake Mellor, Ramin Naimi

##### Blake Mellor

It is shown that for any locally knotted edge of a 3-connected graph in S3, there is a ball that contains all of the local knots of that edge and is unique up to an isotopy setwise fixing the graph. This result is applied to the study of topological symmetry groups of graphs embedded in S3.

The Intersection Graph Conjecture For Loop Diagrams, 2017 Loyola Marymount University

#### The Intersection Graph Conjecture For Loop Diagrams, Blake Mellor

##### Blake Mellor

Vassiliev invariants can be studied by studying the spaces of chord diagrams associated with singular knots. To these chord diagrams are associated the intersection graphs of the chords. We extend results of Chmutov, Duzhin and Lando to show that these graphs determine the chord diagram if the graph has at most one loop. We also compute the size of the subalgebra generated by these "loop diagrams."

Symmetries Of Embedded Complete Bipartite Graphs, 2017 Loyola Marymount University

#### Symmetries Of Embedded Complete Bipartite Graphs, Erica Flapan, Nicole Lehle, Blake Mellor, Matt Pittluck, Xan Vongsathorn

##### Blake Mellor

We characterize which automorphisms of an arbitrary complete bipartite graph Kn,m can be induced by a homeomorphism of some embedding of the graph in S3.

2017 Loyola Marymount University

#### A Few Weight Systems Arising From Intersection Graphs, Blake Mellor

##### Blake Mellor

No abstract provided.

Complete Graphs Whose Topological Symmetry Groups Are Polyhedral, 2017 Loyola Marymount University

#### Complete Graphs Whose Topological Symmetry Groups Are Polyhedral, Eric Flapan, Blake Mellor, Ramin Naimi

##### Blake Mellor

We determine for which m the complete graph Km has an embedding in S3 whose topological symmetry group is isomorphic to one of the polyhedral groups A4, A5 or S4.

A Geometric Interpretation Of Milnor's Triple Invariants, 2017 Loyola Marymount University

#### A Geometric Interpretation Of Milnor's Triple Invariants, Blake Mellor, Paul Melvin

##### Blake Mellor

Milnor's triple linking numbers of a link in the 3-sphere are interpreted geometrically in terms of the pattern of intersections of the Seifert surfaces of the components of the link. This generalizes the well known formula as an algebraic count of triple points when the pairwise linking numbers vanish.

Finite Type Link Concordance Invariants, 2017 Loyola Marymount University

#### Finite Type Link Concordance Invariants, Blake Mellor

##### Blake Mellor

This paper is a generalization of the author's previous work on link homotopy to link concordance. We show that the only real-valued finite type link concordance invariants are the linking numbers of the components.

Finite Type Link Homotopy Invariants, 2017 Loyola Marymount University

#### Finite Type Link Homotopy Invariants, Blake Mellor

##### Blake Mellor

Bar-Natan used Chinese characters to show that finite type invariants classify string links up to homotopy. In this paper, I construct the correct spaces of chord diagrams and Chinese characters for links up to homotopy. I use these spaces to show that the only rational finite type invariants of link homotopy are the pairwise linking numbers of the components.

Colorings, Determinants And Alexander Polynomials For Spatial Graphs, 2017 Loyola Marymount University

#### Colorings, Determinants And Alexander Polynomials For Spatial Graphs, Terry Kong, Alec Lewald, Blake Mellor, Vadim Pigrish

##### Blake Mellor

A {\em balanced} spatial graph has an integer weight on each edge, so that the directed sum of the weights at each vertex is zero. We describe the Alexander module and polynomial for balanced spatial graphs (originally due to Kinoshita \cite{ki}), and examine their behavior under some common operations on the graph. We use the Alexander module to define the determinant and p-colorings of a balanced spatial graph, and provide examples. We show that the determinant of a spatial graph determines for which p the graph is p-colorable, and that a p-coloring of a graph corresponds to a representation ...

Chord Diagrams And Gauss Codes For Graphs, 2017 Loyola Marymount University

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

##### Blake Mellor

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 ...

Alexander And Writhe Polynomials For Virtual Knots, 2017 Loyola Marymount University

#### Alexander And Writhe Polynomials For Virtual Knots, Blake Mellor

##### Blake Mellor

We give a new interpretation of the Alexander polynomial Δ0 for virtual knots due to Sawollek and Silver and Williams, and use it to show that, for any virtual knot, Δ0 determines the writhe polynomial of Cheng and Gao (equivalently, Kauffman's affine index polynomial). We also use it to define a second-order writhe polynomial, and give some applications.

2017 Loyola Marymount University

#### Finite Type Link Homotopy Invariants Ii: Milnor's Invariants, Blake Mellor

##### Blake Mellor

We define a notion of finite type invariants for links with a fixed linking matrix. We show that Milnor's triple link homotopy invariant is a finite type invariant, of type 1, in this sense. We also generalize the approach to Milnor's higher order homotopy invariants and show that they are also, in a sense, of finite type. Finally, we compare our approach to another approach for defining finite type invariants within linking classes.

Long And Short-Range Air Navigation On Spherical Earth, 2017 AAR Aerospace Consulting, LLC

#### Long And Short-Range Air Navigation On Spherical Earth, Nihad E. Daidzic

##### International Journal of Aviation, Aeronautics, and Aerospace

Global range air navigation implies non-stop flight between any two airports on Earth. Such effort would require airplanes with the operational air range of at least 12,500 NM which is about 40-60% longer than anything existing in commercial air transport today. Air transportation economy requires flying shortest distance, which in the case of spherical Earth are Orthodrome arcs. Rhumb-line navigation has little practical use in long-range flights, but has been presented for historical reasons and for comparison. Database of about 50 major international airports from every corner of the world has been designed and used in testing and route ...

2016 University of Maine

#### Microstructural Analysis Of Thermoelastic Response, Nonlinear Creep, And Pervasive Cracking In Heterogeneous Materials, Alden C. Cook

##### Electronic Theses and Dissertations

This dissertation is concerned with the development of robust numerical solution procedures for the generalized micromechanical analysis of linear and nonlinear constitutive behavior in heterogeneous materials. Although the methods developed are applicable in many engineering, geological, and materials science fields, three main areas are explored in this work. First, a numerical methodology is presented for the thermomechanical analysis of heterogeneous materials with a special focus on real polycrystalline microstructures obtained using electron backscatter diffraction techniques. Asymptotic expansion homogenization and finite element analysis are employed for micromechanical analysis of polycrystalline materials. Effective thermoelastic properties of polycrystalline materials are determined and compared ...

Embedding Of Hypercube Graphs On Orientable Surfaces, 2016 University of Maine

#### Embedding Of Hypercube Graphs On Orientable Surfaces, Prateek Kunwar

##### Electronic Theses and Dissertations

The purpose of this thesis is to study hypercube graphs and their embeddings on orientable surfaces. We use rotation systems to represent these embeddings. We prove some results about the effect of adjacent switches in rotation system and create a rotation system called the ABC rotation system and prove general results about it. Using this rotation system, we give a general theorem about the minimal embedding of $Q_{n}$. We also look at some interesting types of maximal embedding of $Q_{n}$, such as the Eulerian walk embedding and the "big-face embedding". We prove a theorem that gives a recursively ...

Rainich Conditions In (2+1)-Dimensional Gravity, 2016 Selected Works

#### Rainich Conditions In (2+1)-Dimensional Gravity, Dionisios Krongos, Charles G. Torre

##### Charles G. Torre

In (3 + 1) spacetime dimensions the Rainich conditions are a set of equations expressed solely in terms of the metric tensor which are equivalent to the Einstein-Maxwell equations for non-null electromagnetic fields. Here we provide the analogous conditions for (2 + 1)-dimensional gravity coupled to electromagnetism. Both the non-null and null cases are treated. The construction of these conditions is based upon reducing the problem to that of gravity coupled to a scalar field, which we have treated elsewhere. These conditions can be easily extended to other theories of (2 + 1)-dimensional gravity. For example, we apply the geometrization conditions ...