All roots of unity are detected by the A-polynomial

For an arbitrary positive integer n, we construct infinitely many one-cusped hyperbolic 3-manifolds where each manifold's A-polynomial detects every n-th root of unity. This answers a question of Cooper, Culler, Gillet, Long, and Shalen as to which roots of unity arise in this manner.Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol5/agt-5-11.abs.htm

Closed surfaces and character varieties

The powerful character variety techniques of Culler and Shalen can be used to find essential surfaces in knot manifolds. We show that module structures on the coordinate ring of the character variety can be used to identify detected boundary slopes as well as when closed surfaces are detected. This approach also yields new number theoretic invariants for the character varieties of knot manifolds.Comment: 28 pages, 1 figur

Mechanisms of segmentation in the American cockroach, periplaneta americana

A fully segmented body and jointed legs are defining characteristics of the Arthropoda (Insecta, Crustacea, Myriapoda, and Chelicerata). The underlying mechanisms involved in achieving these features are not well understood outside of the insect Drosophila melanogaster (fruit fly) – a long germ band organism where segmentation occurs all at once in a syncytial blastoderm. In the more common, ancestral mode of development, short germ band, new segments are added sequentially from the cellular environment of a posteriorly extending growth zone. Segmentation in these organisms may not always be comparable to the “Drosophila paradigm” and, therefore, require further analysis. My thesis will explore the conservation and divergence of the molecular mechanisms of segmentation in a phylogenetically basal, short germ band insect, Periplaneta americana (American cockroach). Presented over three results chapters, I will discuss aspects of cockroach segmentation processes, from the establishment of a posterior organiser and growth zone, to subsequent posterior growth and the formation of new segments. In particular, Chapter III describes how interactions between the Cad/Wnt-dependent posterior organiser and the Notch-segmentation clock control posterior growth and segmentation. Chapter IV encompasses the expression patterns and potential roles for Periplaneta homologues of the pair-rule genes: even-skipped, runt, pairberry, and sloppy-paired throughout embryogenesis, identifying deviations in function between anterior and posterior segmentation processes. New functions for the non-canonical, polycistronic small Open Reading Frame (smORF) gene tarsal-less in body patterning are discussed in Chapter V, along with the conserved roles for tarsal-less, nubbin, Notch, and Delta in leg and development. Elucidation of the networks involved in these processes will help establish putative ancestral gene functions allowing us to gain further insights into the evolution of insect (and arthropod) body segmentation and leg joint formation