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Laver’s results and low-dimensional topology. (English) Zbl 1342.57003

The subject of the paper are two important results of Richard Laver: (1) his study of free left distributive systems with one generator, including what is now known as Laver tables, and (2) well-foundedness of the braid ordering.
The aim is to publicize Laver’s results in a form accessible to anybody working in any of the areas that meet in Laver’s work (low dimensional topology, set theory, non-associative algebra and algebra of braids), but also to propose new applications of Laver’s results in topology. Therefore, part (1) includes also introductions to braid and knot coloring, rack (co)homology and set-theoretical solutions to the Yang-Baxter equation, and in each case, there is a dicussion of a potential use of Laver tables in these problems. Part (2) introduces Dehornoy’s ordering of the braid group, presents the main idea behind Laver’s proof of its well-foundedness on positive braids, and culminates with a handful of results on the order type of various subsets of the braid group, with applications to unprovability of certain braid properties in certain fragments of arithmetic. In the end, a potential application of the ordering to solving the conjugacy and Markov equivalence problems in braid groups is proposed.
The paper is written with a great care and is a pleasure to read. The research program outlined in the paper is worth attention.

MSC:

57M25 Knots and links in the \(3\)-sphere (MSC2010)
20N02 Sets with a single binary operation (groupoids)
20F36 Braid groups; Artin groups
20F10 Word problems, other decision problems, connections with logic and automata (group-theoretic aspects)
03F30 First-order arithmetic and fragments
03E55 Large cardinals
16T25 Yang-Baxter equations
57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
18G60 Other (co)homology theories (MSC2010)
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