Enumerating fundamental normal surfaces: Algorithms, experiments and invariants

Computational knot theory and 3-manifold topology have seen significant breakthroughs in recent years, despite the fact that many key algorithms have complexity bounds that are exponential or greater. In this setting, experimentation is essential for understanding the limits of practicality, as well as for gauging the relative merits of competing algorithms. In this paper we focus on normal surface theory, a key tool that appears throughout low-dimensional topology. Stepping beyond the well-studied problem of computing vertex normal surfaces (essentially extreme rays of a polyhedral cone), we turn our attention to the more complex task of computing fundamental normal surfaces (essentially an integral basis for such a cone). We develop, implement and experimentally compare a primal and a dual algorithm, both of which combine domain-specific techniques with classical Hilbert basis algorithms. Our experiments indicate that we can solve extremely large problems that were once though intractable. As a practical application of our techniques, we fill gaps from the KnotInfo database by computing 398 previously-unknown crosscap numbers of knots.Comment: 17 pages, 5 figures; v2: Stronger experimental focus, restrict attention to primal & dual algorithms only, larger and more detailed experiments, more new crosscap number

The complexity of the normal surface solution space

Normal surface theory is a central tool in algorithmic three-dimensional topology, and the enumeration of vertex normal surfaces is the computational bottleneck in many important algorithms. However, it is not well understood how the number of such surfaces grows in relation to the size of the underlying triangulation. Here we address this problem in both theory and practice. In theory, we tighten the exponential upper bound substantially; furthermore, we construct pathological triangulations that prove an exponential bound to be unavoidable. In practice, we undertake a comprehensive analysis of millions of triangulations and find that in general the number of vertex normal surfaces is remarkably small, with strong evidence that our pathological triangulations may in fact be the worst case scenarios. This analysis is the first of its kind, and the striking behaviour that we observe has important implications for the feasibility of topological algorithms in three dimensions.Comment: Extended abstract (i.e., conference-style), 14 pages, 8 figures, 2 tables; v2: added minor clarification

Investigating the Role of FGF8 Signaling in Neurogenesis of the Developing Zebrafish Eye

In the embryonic zebrafish, the fibroblast growth factor 8a (FGF8) signaling network is essential for proper development and maintenance of retinal ganglion cells (RGCs) as well as the hyaloid vasculature, the vessels that supply the eye with nutrients during development. Disruption of FGF8 signaling via knock down of FGF8 or pharmacologic inhibition of FGF receptors (FGFRs) results in extensive abnormalities throughout the developing eye. Our preliminary data indicated that in developing zebrafish, mRNA expression of fgf8a is present exclusively in the RGCs, while the fibroblast growth factor receptor 1 (fgfr1b) is expressed exclusively in the area of the hyaloid vasculature. These results led us to hypothesize that FGF8 signals from the RGCs to the vasculature of the developing eye, and that this signaling network is essential for proper eye development. In order to test this hypothesis, we demonstrated the ability to detect downstream phosphorylation events in response to acute FGF8 stimulation in cells that expressed FGFR1 using Western blot and immunofluorescence (IF). Next, we established a zebrafish eye explant culture system to treat the cells of the developing zebrafish eye in vitro. Using transgenic zebrafish lines expressing green fluorescent protein (GFP) tags in either the differentiating RGCs or the vascular cells of the eye, we attempted to identify the specific cells capable of responding to FGF8. Our data indicate that recombinant FGF8 is capable of activating detectable intracellular signaling cascades, such as ERK phosphorylation, in cultured endothelial cells. Furthermore, FGF8 is capable of inducing signaling in some of the cells from the developing zebrafish eye, but not in the RGCs. These findings support our proposed model in which FGF8 signals from the RGCs to the hyaloid vasculature, resulting in the activation of signaling pathways that are necessary for proper development of the hyaloid vasculature and RGCs

A duplicate pair in the SnapPea census

We identify a duplicate pair in the well-known Callahan-Hildebrand-Weeks census of cusped finite-volume hyperbolic 3-manifolds. Specifically, the six-tetrahedron non-orientable manifolds x101 and x103 are homeomorphic.Comment: 5 pages, 3 figures; v2: minor edits. To appear in Experimental Mathematic

Increasing rainwater yield in water sensitive cities using short-term rainfall forecasts

Rainwater harvesting performance is examined in Brisbane and Sydney for three rainwater tank configurations comprising: 1) A conventional 5kL tank; 2) A 5kL tank with a fixed leaking compartment for baseflow (240 L/d); and 3) As with leaking but including a variable diversion compartment (480 L/day) controlled by short-term rainfall forecasts. This concept is referred to as adaptive rainwater diversion (ARD). Result show the ARD system achieves superior emulation of pre-urban runoff frequencies, runoff volumes and baseflows, while maintaining the household rainwater supply to within 90% of conventional systems, and also while producing an additional water resources of up to 60kL/hh/y. With these outcomes, the ARD system could potentially create an avenue to approach water sensitive cities in Eastern Australia

Enumeration of non-orientable 3-manifolds using face pairing graphs and union-find

Drawing together techniques from combinatorics and computer science, we improve the census algorithm for enumerating closed minimal P^2-irreducible 3-manifold triangulations. In particular, new constraints are proven for face pairing graphs, and pruning techniques are improved using a modification of the union-find algorithm. Using these results we catalogue all 136 closed non-orientable P^2-irreducible 3-manifolds that can be formed from at most ten tetrahedra.Comment: 37 pages, 34 figure