Hard unknots and collapsing tangles (Q2884338)

From the text: `We envision this paper as useful for research knot theorists, for students beginning in knot theory and for scientists such as biologists who would like an entrance into knot theory that leads to connections with other fields.' `The main theme of this paper [is] that, given two fractions \(p/q\) and \(r/s\) such that \(|ps-qr|=1\), we can construct an unknot diagram \(N([p/q]-[r/s])\) from the rational tangles \([p/q]\) and \([r/s]\).'NEWLINENEWLINEFor a reader unfamiliar with the field, this paper will provide an introduction to knot theory and, in particular, to the study of rational links and rational tangles.NEWLINENEWLINEThe content of this paper relates to the question of finding hard link diagrams. Here a link diagram \(D\) is said to be hard if it does not admit any simplifying Reidemeister moves of type I or II, or any Reidemeister moves of type III. Thus any Reidemeister move on \(D\) must increase the number of crossings in the diagram. In particular, the authors study such diagrams given as the numerator closure of the sum of two (usually rational) tangles. Fractions describing the resulting diagrams are calculated by multiplying certain \(2\times 2\) matrices.NEWLINENEWLINEThe results are then related to processive DNA recombination.NEWLINENEWLINEFor the entire collection see [Zbl 1237.57001].