We start with some terminology from differential topology [1]. Let  be a circle and  ≥ 2 be an integer. An immersion  :  → R is a smooth function whose derivative never vanishes. An embedding  :  → R is an immersion that is oneto-one. It follows that () is a manifold but () need not be ( is only locally one-to-one, so consider the map that twists  into a figure eight). A knot is a s...

After proving a theorem about the general formulae for the signature of alternating knot and link families, we distinguished all families of knots obtained from generating alternating knots with at most 10 crossings and alternating links with at most 9 crossings, for which the unknotting (unlinking) number can be confirmed by using the general formulae for signatures.

A virtual string can be defined as a closed curve on a surface modulo certain equivalence relations. Turaev defined several invariants of virtual strings which we use to produce a table of virtual strings up to 4 crossings. We discuss progress in extending the tabulation to 5 crossings. We also provide a counter-example to a statement of Kadokami.

Variations in magnetic or electrostatic confinement fields give rise to trapping separatrices, and neoclassical transport theory analyzes effects from collision-induced separatrix crossings. Experiments on pure electron plasmas now quantitatively characterize a broad range of transport and wave damping effects due to "chaotic" separatrix crossings, which occur due to equilibrium plasma rotation...

Recently there has been growing interest among psychologists in human performance on the Euclidean Traveling Salesperson problem (E-TSP). A debate has been initiated on what strategy people use in solving visually presented E-TSP instances. The most prominent hypothesis is the convex-hull hypothesis, originally proposed by MacGregor and Ormerod (1996). We argue that, in the literature so far, t...