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Articles 31 - 38 of 38
Full-Text Articles in Geometry and Topology
Tree Diagrams For String Links Ii: Determining Chord Diagrams, Blake Mellor
Tree Diagrams For String Links Ii: Determining Chord Diagrams, Blake Mellor
Mathematics, Statistics and Data Science Faculty Works
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).
A Few Weight Systems Arising From Intersection Graphs, Blake Mellor
A Few Weight Systems Arising From Intersection Graphs, Blake Mellor
Mathematics, Statistics and Data Science Faculty Works
No abstract provided.
A Geometric Interpretation Of Milnor's Triple Invariants, Blake Mellor, Paul Melvin
A Geometric Interpretation Of Milnor's Triple Invariants, Blake Mellor, Paul Melvin
Mathematics, Statistics and Data Science Faculty Works
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.
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
Mathematics, Statistics and Data Science Faculty Works
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.
Finite Type Link Concordance Invariants, Blake Mellor
Finite Type Link Concordance Invariants, Blake Mellor
Mathematics, Statistics and Data Science Faculty Works
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 Ii: Milnor's Invariants, Blake Mellor
Finite Type Link Homotopy Invariants Ii: Milnor's Invariants, Blake Mellor
Mathematics, Statistics and Data Science Faculty Works
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.
The Intersection Graph Conjecture For Loop Diagrams, Blake Mellor
The Intersection Graph Conjecture For Loop Diagrams, Blake Mellor
Mathematics, Statistics and Data Science Faculty Works
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."
Finite Type Link Homotopy Invariants, Blake Mellor
Finite Type Link Homotopy Invariants, Blake Mellor
Mathematics, Statistics and Data Science Faculty Works
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.