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

Perturbative renormalization of lattice N=4 super Yang-Mills theory

We consider N=4 super Yang-Mills theory on a four-dimensional lattice. The lattice formulation under consideration retains one exact supersymmetry at non-zero lattice spacing. We show that this feature combined with gauge invariance and the large point group symmetry of the lattice theory ensures that the only counterterms that appear at any order in perturbation theory correspond to renormalizations of existing terms in the bare lattice action. In particular we find that no mass terms are generated at any finite order of perturbation theory. We calculate these renormalizations by examining the fermion and auxiliary boson self energies at one loop and find that they all exhibit a common logarithmic divergence which can be absorbed by a single wavefunction renormalization. This finding implies that at one loop only a fine tuning of the finite parts is required to regain full supersymmetry in the continuum limit.Comment: v2. Minor corrections, references adde

SMF-1, SMF-2 and SMF-3 DMT1 Orthologues Regulate and Are Regulated Differentially by Manganese Levels in C. elegans

Manganese (Mn) is an essential metal that can exert toxic effects at high concentrations, eventually leading to Parkinsonism. A major transporter of Mn in mammals is the divalent-metal transporter (DMT1). We characterize here DMT1-like proteins in the nematode C. elegans, which regulate and are regulated by Mn and iron (Fe) content. We identified three new DMT1-like genes in C. elegans: smf-1, smf-2 and smf-3. All three can functionally substitute for loss of their yeast orthologues in S. cerevisiae. In the worm, deletion of smf-1 or smf-3 led to an increased Mn tolerance, while loss of smf-2 led to increased Mn sensitivity. smf mRNA levels measured by QRT-PCR were up-regulated upon low Mn and down-regulated upon high Mn exposures. Translational GFP-fusions revealed that SMF-1 and SMF-3 strongly localize to partially overlapping apical regions of the gut epithelium, suggesting a differential role for SMF-1 and SMF-3 in Mn nutritional intake. Conversely, SMF-2 was detected in the marginal pharyngeal epithelium, possibly involved in metal-sensing. Analysis of metal content upon Mn exposure in smf mutants revealed that SMF-3 is required for normal Mn uptake, while smf-1 was dispensable. Higher smf-2 mRNA levels correlated with higher Fe content, supporting a role for SMF-2 in Fe uptake. In smf-1 and smf-3 but not in smf-2 mutants, increased Mn exposure led to decreased Fe levels, suggesting that both metals compete for transport by SMF-2. Finally, SMF-3 was post-translationally and reversibly down-regulated following Mn-exposure. In sum, we unraveled a complex interplay of transcriptional and post-translational regulations of 3 DMT1-like transporters in two adjacent tissues, which regulate metal-content in C. elegans

A characterization of intrinsic reciprocity

Reciprocity, Extended preferences, Game theory,