Anthropocene Blues by John Lane

Review of John Lane\u27s Anthropocene Blue

Geodesics and compression bodies

We consider hyperbolic structures on the compression body C with genus 2 positive boundary and genus 1 negative boundary. Note that C deformation retracts to the union of the torus boundary and a single arc with its endpoints on the torus. We call this arc the core tunnel of C. We conjecture that, in any geometrically finite structure on C, the core tunnel is isotopic to a geodesic. By considering Ford domains, we show this conjecture holds for many geometrically finite structures. Additionally, we give an algorithm to compute the Ford domain of such a manifold, and a procedure which has been implemented to visualize many of these Ford domains. Our computer implementation gives further evidence for the conjecture.Comment: 31 pages, 11 figures. V2 contains minor changes. To appear in Experimental Mathematic

Links with no exceptional surgeries

We show that if a knot admits a prime, twist-reduced diagram with at least 4 twist regions and at least 6 crossings per twist region, then every non-trivial Dehn filling of that knot is hyperbolike. A similar statement holds for links. We prove this using two arguments, one geometric and one combinatorial. The combinatorial argument further implies that every link with at least 2 twist regions and at least 6 crossings per twist region is hyperbolic and gives a lower bound for the genus of a link.Comment: 28 pages, 15 figures. Minor rewording and organizational changes; also added theorem giving a lower bound on the genus of these link

Comparison of Various Mean Field Formulations for Retrieving Refractive Indices of Aerosol Particles Containing Inclusions

Application of effective medium approximation (EMA) methods to two-component systems are presented. Systems studied are composed of water, sulfate, soot, and dust as these are commonly encountered atmospheric aerosol components. Atmospheric models often employ EMAs to include internally mixed aerosols without the computational burden of exact theory. In the current work, several types of mixing rules (Maxwell-Garnet, Bruggeman, and coherent potential approximation) have been applied to various two-component internally mixed particles at 550 nm using volume fractions of the minor component below 0.1. As expected, results show that the formulations tested produce very similar effective refractive indices indicating that electric field interactions between inclusions is negligible at the tested volume fractions. This indicates that the differences in component refractive index has only a minor effect on the validity of the EMA at the tested volume fractions. In all cases considered, the linear average of the refractive indices of the two components provides an upper limit for the EMAs

Treewidth, crushing, and hyperbolic volume

We prove that there exists a universal constant $c$ such that any closed hyperbolic 3-manifold admits a triangulation of treewidth at most $c$ times its volume. The converse is not true: we show there exists a sequence of hyperbolic 3-manifolds of bounded treewidth but volume approaching infinity. Along the way, we prove that crushing a normal surface in a triangulation does not increase the carving-width, and hence crushing any number of normal surfaces in a triangulation affects treewidth by at most a constant multiple.Comment: 20 pages, 12 figures. V2: Section 4 has been rewritten, as the former argument (in V1) used a construction that relied on a wrong theorem. Section 5.1 has also been adjusted to the new construction. Various other arguments have been clarifie

Explicit Dehn filling and Heegaard splittings

We prove an explicit, quantitative criterion that ensures the Heegaard surfaces in Dehn fillings behave "as expected." Given a cusped hyperbolic manifold X, and a Dehn filling whose meridian and longitude curves are longer than 2pi(2g-1), we show that every genus g Heegaard splitting of the filled manifold is isotopic to a splitting of the original manifold X. The analogous statement holds for fillings of multiple boundary tori. This gives an effective version of a theorem of Moriah-Rubinstein and Rieck-Sedgwick.Comment: 17 pages. v3 contains minor revisions and cleaner arguments, incorporating referee comments. To appear in Communications in Analysis and Geometr