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A Nonexpert Introduction To Rational Homotopy Theory, Nicholas Boschert 2017 University of Colorado, Boulder

A Nonexpert Introduction To Rational Homotopy Theory, Nicholas Boschert

Mathematics Undergraduate Contributions

This article attempts to render the rational homotopy theory of Sullivan and Quillen more comprehensible to non-experts in algebraic topology by expounding on many of the results and proofs in a more detailed and elementary way.


Conditions For Obtaining Nontrivial Knots From Collections Of Vectors, Joseph Borgatti 2017 Merrimack College

Conditions For Obtaining Nontrivial Knots From Collections Of Vectors, Joseph Borgatti

Honors Senior Capstone Projects

We explore under what conditions one can obtain a nontrivial knot, given a collection of n vectors. First, we show how to get a crossing from any 3 vectors equal in magnitude, by arbitrarily picking 2 vectors and identifying the sufficient and necessary criteria for picking a third vector that will guarantee a crossing when the vectors are reordered. We also show that it’s always possible for a set of vectors to be reordered to form the unknot, if they sum to ~0 when added together. Our main results are restricted to sets of n vectors that, when reordered ...


Intrinsic Linking And Knotting Of Graphs In Arbitrary 3–Manifolds, Erica Flapan, Hugh Howards, Don Lawrence, Blake Mellor 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, Blake Mellor 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, Blake Mellor, Dylan Thurston 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, Blake Mellor 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, Erica Flapan, Blake Mellor, Ramin Naimi 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, Blake Mellor 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, Erica Flapan, Nicole Lehle, Blake Mellor, Matt Pittluck, Xan Vongsathorn 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, Blake Mellor 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, Eric Flapan, Blake Mellor, Ramin Naimi 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, Blake Mellor, Paul Melvin 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, Blake Mellor 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, Blake Mellor 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, Terry Kong, Alec Lewald, Blake Mellor, Vadim Pigrish 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, Thomas Fleming, Blake Mellor 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, Blake Mellor 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.


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


Links With Finite N-Quandles, Jim Hoste, Patrick D. Shanahan 2017 Pitzer College

Links With Finite N-Quandles, Jim Hoste, Patrick D. Shanahan

Patrick Shanahan

We prove a conjecture of Przytycki which asserts that the n-quandle of a link L in the 3-sphere is finite if and only if the fundamental group of the n-fold cyclic branched cover of the 3-sphere, branched over L, is finite.


Derived Categories And The Analytic Approach To General Reciprocity Laws. Part Ii, Michael Berg 2017 Loyola Marymount University

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

Michael Berg

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 ...


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