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- Spatial graphs (2)
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Articles 1 - 19 of 19
Full-Text Articles in Physical Sciences and Mathematics
Intrinsic Linking And Knotting Of Graphs In Arbitrary 3–Manifolds, Erica Flapan, Hugh Howards, Don Lawrence, Blake Mellor
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
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, Alissa S. Crans, Blake Mellor, Sandy Ganzell
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, Blake Mellor, Dylan Thurston
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
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
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
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
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
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
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
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
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
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
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 of …
Counting Links In Complete Graphs, Thomas Fleming, Blake Mellor
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 K7, 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, Thomas Fleming, Blake Mellor
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, Curtis D. Bennett, Blake Mellor, Patrick D. Shanahan
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.
Alexander And Writhe Polynomials For Virtual Knots, Blake Mellor
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
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.