The paper under review mainly deals with classical and virtual knot
theory; it focuses on a diagrammatical move (so-called glide move, which locally
exchanges the relative
position, on a strand, of an over-crossing and an under-crossing) and how it can (or
cannot) be used to sort (by a so-called gliding process) all the crossings of a
tangle with no closed component, so that one meet first only
over-crossings and then only under-crossing when running any component
of the tangle (such a tangle diagram being called a OU-tangle by the
authors).
The paper take the form of a long, but segmented, discussion about
this gliding process, how it can be used to prove new or already known
results, how it can be computed and how it was already implicitely
present in the litterature. The paper is divided into eight sections
including the introduction and the acknowledgements. The
remaining six sections are as follows.
2) The first one presents the gliding process in a quite surprising
way. It is presented along a false fheorem (f for false)
which is proven by a froof. It is followed by a discussion explaining
why the proof is false: in general, the process does not converge; the rest of the
paper is devoted to several situations where it does. This curious way
to introduce their main tool may look disruptive, but at least it is
not misleading.
3) This section deals with classical
tangles. Characterized by an acyclic property (no cyclic path along the
diagram, where the path is allowed, at crossings, to jump from the over
to the under-strand), they prove that braids are exactly the subclass
of tangles for which the gliding process converges. This provides a
complete and computable invariant for braids, given in terms of reduced OU-diagrams.
4) This section deals with virtual tangles. The authors give here an
alternative proof of a theorem by Oleg Chterenthal which provides a
complete invariant of virtual braids. Rephrased in terms of OU-tangles, it says that
the gliding process converges for virtual braids into
a unique reduced OU-diagram, providing hence the complete invariant.
But contrary to the classical case, there exists virtual OU-tangles
which are not equivalent to a braid. This section is certainly the most
technical one, but the idea behind is actually quite simple: replace
a virtual OU-tangle T by couples (B,D) where B and D are respectively a virtual
braid and a virtual OU-diagram such that B.D=T, consider
the binary relation which simplifies (B,D) into (Bg,g^{-1}D) where
g is braid generator and the reduced representative of g^{-1}D has
fewer crossings than the one of D, and show that this relation is
Noetherian and satifies the Diamond Lemma. This ensures the existence
of a unique terminal element among the couples representing T.
5) This section make several comments on the previous one. It walks
from how it is related to Oleg Chterental's work to how all this
relates to Hopf algebras, by way of
how it can be used to recover the classical results of Section 2 and
how reduced OU-diagrams may embed to virtual tangles. It also
discusses the, so-called extraction, graphs associated to the above-mentionned
binary relation.
6) This sections provides Mathematica implementations leading to a few
intriguing pictures of extraction graphs.
7) The last section shortly travels through time: forward, with some
prospect for further developments; and then backward, by tracking down
earlier occurences of gliding and OU-ideas in the litterature.
The paper is written in a very non-academic style, but contains some
new and interesting results (such as a characterization of classical
braids among tangles, and a complete and computable
invariants for them) and a lot of very interesting
comments and ideas. It can be read as a pleasant journey in some
``Over then Under''-world. As such, I really recommend it for publication.