The Computational Complexity of Knot and Link Problems

We consider the problem of deciding whether a polygonal knot in 3-dimensional Euclidean space is unknotted, capable of being continuously deformed without self-intersection so that it lies in a plane. We show that this problem, {\sc unknotting problem} is in {\bf NP}. We also consider the problem, {\sc unknotting problem} of determining whether two or more such polygons can be split, or continuously deformed without self-intersection so that they occupy both sides of a plane without intersecting it. We show that it also is in NP. Finally, we show that the problem of determining the genus of a polygonal knot (a generalization of the problem of determining whether it is unknotted) is in {\bf PSPACE}. We also give exponential worst-case running time bounds for deterministic algorithms to solve each of these problems. These algorithms are based on the use of normal surfaces and decision procedures due to W. Haken, with recent extensions by W. Jaco and J. L. Tollefson.Comment: 32 pages, 1 figur

Finding the Right Plugin: Mosquitoes Have the Answer

The chemical complexity of the male ejaculate is truly extraordinary and every bit as remarkable as the most extravagant male plumage and courtship displays. This protein-rich seminal fluid delivers chemical messages that can enter the female brain and modify behaviour, stimulate muscle contractions in, and change the appearance of the female reproductive tract, and cause females to release reproductive hormones. In fact, these chemicals can collectively modify almost all aspects of female reproductive behaviour and physiology (e.g., [1]–[3]). Our understanding of the role of the non-sperm constituents of insect ejaculates has been hugely enriched by classic work in Drosophila melanogaster. In a series of papers dating from the 1980s, the Wolfner and Kubli research groups, and more recently the Swanson laboratory, have identified over 100 proteins and peptides synthesised in the male reproductive system and transferred to females during mating (e.g., [4]–[9]). Increasingly the functional significance of these substances is being shown, with many having significant effects on fitness [10]–[12]. A more recent focus has been on identifying seminal fluid proteins in species of agricultural and medical importance [13],[14], and excellent papers have recently been published on the identification of seminal fluid proteins in the mosquito vectors of dengue/yellow fever and of malaria [15],[16]