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Full-Text Articles in Mathematics
Finite Type Link Homotopy Invariants Ii: Milnor's Invariants, Blake Mellor
Finite Type Link Homotopy Invariants Ii: Milnor's Invariants, Blake Mellor
Mathematics 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 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 Concordance Invariants, Blake Mellor
Finite Type Link Concordance Invariants, Blake Mellor
Mathematics 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.
Cyclic Dehn Surgery And The A-Polynomial, Patrick Shanahan
Cyclic Dehn Surgery And The A-Polynomial, Patrick Shanahan
Mathematics Faculty Works
We present a necessary condition for Dehn surgery on a knot in double-struck S sign3 to be cyclic which is based on the A-polynomial of the knot. The condition involves a width of the Newton polygon of the A-polynomial, and provides a simple method of computing a list of possible cyclic surgery slopes. The width produces a list of at most three slopes for a hyperbolic knot which contains no closed essential surface in its complement (in agreement with the Cyclic Surgery Theorem). We conclude with an application to cyclic surgeries along non-boundary slopes of hyperbolic mutant knots.