Unprovability results involving braids

Carlucci, Lorenzo; Dehornoy, Patrick; Weiermann, Andreas

doi:10.1112/plms/pdq016

Cited by 12 publications

(15 citation statements)

References 38 publications

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“…It is likely that a similar result involving B + ∞ and IΣ 2 could be established, but this was not made in [13].…”

Section: Proof Of Proposition 216 (Sketch)mentioning

confidence: 96%

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Laver’s results and low-dimensional topology

Dehornoy

2015

Arch. Math. Logic

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Abstract. In connection with his interest in selfdistributive algebra, Richard Laver established two deep results with potential applications in low-dimensional topology, namely the existence of what is now known as the Laver tables and the well-foundedness of the standard ordering of positive braids. Here we present these results and discuss the way they could be used in topological applications.Richard Laver established two remarkable results that might lead to significant applications in low-dimensional topology, namely the existence of a series of finite structures satisfying the left-selfdistributive law, now known as the Laver tables, and the well-foundedness of the standard ordering of Artin's positive braids. In this text, we shall explain the precise meaning of these results and discuss their (past or future) applications in topology. In one word, the current situation is that, although the depth of Laver's results is not questionable, few topological applications have been found. However, the example of braid groups orderability shows that, once initial obstructions are solved, topological applications of algebraic results involving selfdistributivity can be found; the situation with Laver tables is presumably similar, and the only reason explaining why so few applications are known is that no serious attempt has been made so far, mainly because the results themselves remain widely unknown in the topology community.Therefore this text is more a program than a report on existing results. Our aim is to provide a self-contained and accessible introduction to the subject, hopefully helping the algebraic and topological communities to better communicate. Most of the results mentioned below have already appeared in literature, a number of them even belonging to the folklore of their domain (whereas ignored outside of it). However, at least the observations about cocycles for Laver tables mentioned in Subsection 1.3 (and established in another paper) are new.Very naturally, the text comprises two sections, one devoted to Laver tables, and one devoted to the well-foundedness of the braid ordering. It should be noted that the above two topics (Laver tables, well-foundedness of the braid ordering) do not exhaust Laver's contributions to selfdistributive algebra and, from there, to potential topological applications: in particular, Laver constructed powerful tools for investigating free LD-structures, leading to applications of their own [68,69,70]. However, connections with topology are less obvious in these cases and we shall not develop them here (see the other articles in this volume).

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“…It is likely that a similar result involving B + ∞ and IΣ 2 could be established, but this was not made in [13].…”

Section: Proof Of Proposition 216 (Sketch)mentioning

confidence: 96%

“…Here we shall restrict to the case of 3-strand braids and refer to [13] for details and extensions. In order to construct a long sequence of braids, we start with an arbitrary braid in B + 3 and then repeat some transformation until, if ever, the trivial braid is obtained.…”

Section: 3mentioning

confidence: 99%

Laver’s results and low-dimensional topology

Dehornoy

2015

Arch. Math. Logic

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“…Proposition 6.7 shows that, for each d, the rank of ∆ d We refer to [7] for partial results about Question 6.8, and to [8] for further applications, consisting of unprovability statements involving braids.…”

Section: Definition 54 For N 3 and W In B +mentioning

confidence: 99%

Alternating normal forms for braids and locally Garside monoids

Dehornoy

2008

Journal of Pure and Applied Algebra

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Communicated by C. Kassel MSC: 20F36 20M05 06F05 a b s t r a c tWe describe new types of normal forms for braid monoids, Artin-Tits monoids, and, more generally, for all monoids in which divisibility has some convenient lattice properties ("locally Garside monoids"). We show that, in the case of braids, one of these normal forms coincides with the normal form introduced by Burckel and deduce that the latter can be computed easily. This approach leads to a new, simple description for the standard order ("Dehornoy order") of B n in terms of that of B n−1 , and to a quadratic upper bound for the complexity of this order.The first aim of this paper is to improve our understanding of well-order of positive braids and of the Burckel normal form of [5,6], which after more than ten years remain mysterious objects. This aim is achieved, at least partially, by giving a new, alternative definition for the Burckel normal form that makes it natural and easily computable. This new description is direct, involving right divisors only, while Burckel's original approach resorts to iterating some tricky reduction procedure. It turns out that the construction we describe below relies on a very general scheme for which many monoids are eligible, and we hope for further applications beyond the case of braids.After the seminal work of Garside [22], braid monoids are known to be equipped with a normal form, namely the so-called greedy normal form of [4,1,19,32], which gives for each element of the monoid a distinguished representative word. This normal form, which also exists in spherical Artin-Tits monoids and in Garside monoids that generalize them, is excellent both in theory and in practice as it provides a bi-automatic structure and it is easily computable [20,9,13].In this paper we proceed to construct a new type of normal form for braid monoids and their generalizations. Our construction keeps one of the ingredients of the (right) greedy normal form, namely considering the maximal right divisor that lies in some subset A, but, instead of taking for A the set of so-called simple elements, i.e., the divisors of the Garside element ∆, we choose A to be some standard parabolic submonoid M I of M, i.e., the monoid generated by some subset I of the standard generating set S. When I is a proper subset of S, the submonoid M I is a proper subset of M, and the construction stops after one step. However, by considering two parabolic submonoids M I , M J which together generate M, we can obtain a well-defined, unique decomposition consisting of alternating factors in M I and M J , as in the case of an amalgamated product. By considering convenient families of submonoids, we can iterate the process and obtain a unique normal form for each element of M. When it exists, typically in all Artin-Tits monoids, such a normal form is exactly as easy to compute as the greedy normal form, and, as the greedy form, it solves the word problem in quadratic time.The above construction is quite general, as it only requires the ground monoid M to be what is ...

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“…Laver while a graduate student affirmed this, soon after he had established Fraïssé's Conjecture. 13 Notably, Keith Milliken [97], in his UCLA thesis with Laver on the committee, applied LP d to derive a "pigeonhole principle", actually a partition theorem in terms of "strongly embedded trees" rather than perfect subtrees.…”

Section: Products Of Infinitely Many Treesmentioning

confidence: 99%

“…This would stand as a remarkable fact that would frame the emerging order theory of braid groups, and with a context set, e.g. Carlucci et al [13] investigated unprovability results according to the order type of long descending sequences in the Dehornoy order.…”

Section: Algebra Of Embeddingsmentioning

confidence: 99%