Roman Domination In Complementary Prisms, 2017 East Tennessee State University

#### Roman Domination In Complementary Prisms, Alawi I. Alhashim

*Electronic Theses and Dissertations*

The complementary prism GG of a graph G is formed from the disjoint union of G and its complement G by adding the edges of a perfect match- ing between the corresponding vertices of G and G. A Roman dominating function on a graph G = (V,E) is a labeling f : V(G) → {0,1,2} such that every vertex with label 0 is adjacent to a vertex with label 2. The Roman domination number γR(G) of G is the minimum f(V ) = Σv∈V f(v) over all such functions of G. We study the Roman domination number ...

Session A-3: Three-Act Math Tasks, 2017 Illinois Mathematics and Science Academy

#### Session A-3: Three-Act Math Tasks, Lindsey Herlehy

*Professional Learning Day*

Participants will engage in a Three-Act Math task highlighting the application of properties of geometrical figures. Developed by Dan Meyer, an innovative and highly regarded mathematics instructor, Three-Act Math tasks utilize pedagogical skills that elicit student curiosity, collaboration and questioning. By posing a mathematical problem through active storytelling, this instructional approach redefines real-world mathematics and clarifies the role that a student plays in the learning process. Participants will be given multiple resources where they can access Three-Act Math tasks appropriate for upper elementary grades through Algebra and Geometry courses.

Lorentz Transformation From An Elementary Point Of View, 2017 Laboratoire de Physique Th\'{e}orique, Universit\'{e} de Toulouse III \& CNRS

#### Lorentz Transformation From An Elementary Point Of View, Arkadiusz Jadczyk, Jerzy Szulga

*Electronic Journal of Linear Algebra*

Elementary methods are used to examine some nontrivial mathematical issues underpinning the Lorentz transformation. Its eigen-system is characterized through the exponential of a $G$-skew symmetric matrix, underlining its unconnectedness at one of its extremes (the hyper-singular case). A different yet equivalent angle is presented through Pauli coding which reveals the connection between the hyper-singular case and the shear map.

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.

The Forbidden Number Of A Knot, 2017 Loyola Marymount University

#### The Forbidden Number Of A Knot, Alissa S. Crans, Blake Mellor, Sandy Ganzell

*Blake Mellor*

Every classical or virtual knot is equivalent to the unknot via a sequence of extended Reidemeister moves and the so-called forbidden moves. The minimum number of forbidden moves necessary to unknot a given knot is an invariant we call the forbid- den number. We relate the forbidden number to several known invariants, and calculate bounds for some classes of virtual knots.

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.

Tree Diagrams For String Links Ii: Determining Chord Diagrams, 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.

A Few Weight Systems Arising From Intersection Graphs, 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 ...

Counting Links In Complete Graphs, 2017 Loyola Marymount University

#### Counting Links In Complete Graphs, Thomas Fleming, Blake Mellor

*Blake Mellor*

We find the minimal number of non-trivial links in an embedding of any complete kk-partite graph on 7 vertices (including K_{7}, which has at least 21 non-trivial links). We give either exact values or upper and lower bounds for the minimal number of non-trivial links for all complete kk-partite graphs on 8 vertices. We also look at larger complete bipartite graphs, and state a conjecture relating minimal linking embeddings with minimal book embeddings.

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 ...

Drawing A Triangle On The Thurston Model Of Hyperbolic Space, 2017 Loyola Marymount University

#### Drawing A Triangle On The Thurston Model Of Hyperbolic Space, Curtis D. Bennett, Blake Mellor, Patrick D. Shanahan

*Blake Mellor*

In looking at a common physical model of the hyperbolic plane, the authors encountered surprising difficulties in drawing a large triangle. Understanding these difficulties leads to an intriguing exploration of the geometry of the Thurston model of the hyperbolic plane. In this exploration we encounter topics ranging from combinatorics and Pick’s Theorem to differential geometry and the Gauss-Bonnet Theorem.