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Periodic Prime Knots And Toplogically Transitive Flows On 3-Manifolds, William Basener, Michael C. Sullivan
Periodic Prime Knots And Toplogically Transitive Flows On 3-Manifolds, William Basener, Michael C. Sullivan
Articles and Preprints
Suppose that φ is a nonsingular (fixed point free) C1 flow on a smooth closed 3-dimensional manifold M with H2(M)=0. Suppose that φ has a dense orbit. We show that there exists an open dense set N ⊆ M such that any knotted periodic orbit which intersects N is a nontrivial prime knot.
Visually Building Smale Flows In S3, Michael C. Sullivan
Visually Building Smale Flows In S3, Michael C. Sullivan
Articles and Preprints
A Smale flow is a structurally stable flow with one dimensional invariant sets. We use information from homology and template theory to construct, visualize and in some cases, classify, Smale flows in the 3-sphere.
Positive Braids With A Half Twist Are Prime, Michael C. Sullivan
Positive Braids With A Half Twist Are Prime, Michael C. Sullivan
Articles and Preprints
We shall prove that a knot which can be represented by a positive braid with a half twist is prime. This is done by associating to each such braid a smooth branched 2-manifold with boundary and studying its intersection with a would-be cutting sphere.
The Prime Decomposition Of Knotted Periodic Orbits In Dynamical Systems, Michael C. Sullivan
The Prime Decomposition Of Knotted Periodic Orbits In Dynamical Systems, Michael C. Sullivan
Articles and Preprints
Templates are used to capture the knotting and linking patterns of periodic orbits of positive entropy flows in 3 dimensions. Here, we study the properties of various templates, especially whether or not there is a bound on the number of prime factors of the knot types of the periodic orbits. We will also see that determining whether two templates are different is highly nontrivial.