Pseudo links and singular links in the solid torus (Q6585125)

A pseudo knot is a standard knot whose projections contain crossings with over/under information missing at some of the double points (these crossings are called precrossings). One reason for studying pseudo knots comes from Biology: there are DNA knots in which it is not possible to distinguish a positive crossing from a negative one, even studying them with electron microscopes.\N\NA singular knot is a knot with a finite number of rigid self-intersections (these crossings are called singular crossings).\N\NPseudo and singular links in \(S^3\) are closely related [\textit{V. G. Bardakov} et al., J. Knot Theory Ramifications 25, No. 9, Article ID 1641002, 13 p. (2016; Zbl 1396.20033)]. In the paper under review, this relation is extended to the case when these links are considered in the solid torus (classical knots and links in a solid torus \(ST\) can be seen as \textit{mixed} links in \(S^3\), with a fixed unknotted circle representing the complement of \(ST\) in \(S^3\) and a moving part representing the link in \(ST\)).\N\NThe paper starts by reviewing the foundations of the theory of pseudo links in \(S^3\). The diagrams of pseudo and singular links are the same, but a difference appears when dealing with the first Reidemeister move: it works if the loop involves a precrossing, but not for a singular crossing. As a consequence, it is proved that the corresponding singular and pseudo braid monoids are isomorphic. Alexander and Markov theorems are then reviewed, after recalling how the \(L\)-move (see for example [\textit{S. Lambropoulou}, J. Knot Theory Ramifications 16, No. 10, 1459--1468 (2007; Zbl 1152.57003)]) works for both singular and pseudo links.\N\NThe rest of the paper deals with the same topics, but considering pseudo and singular links in a solid torus: diagrams, moves, braid monoids and the corresponding Alexander and Markov theorems. A Jones polynomial is also defined in the case of pseudo links in a solid torus.\N\NIn [\textit{L. Paris} and \textit{L. Rabenda}, Ann. Inst. Fourier 58, No. 7, 2413--2443 (2008; Zbl 1171.57008)] the singular Hecke algebra of type \(A\) is introduced. Finally, in the paper under review, this definition is adapted to pseudo links, both in \(S^3\) and the solid torus, as a first step to the construction of Homflypt-type invariants for pseudo links in \(S^3\) and the solid torus, something that the author announces for a sequel paper.