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In the mathematical area of braid theory, the Dehornoy order is a left-invariant total order on the braid group, found by Patrick Dehornoy[1][2]. Dehornoy's original discovery of the order on the braid group used huge cardinals, but there are now several more elementary constructions of it [3].

DefinitionEdit

Suppose that   are the usual generators of the braid group   on   strings. Define a  -positive word to be a braid that admits at least one expression in the elements   and their inverses, such that the word contains  , but does not contain   nor   for  .

The set   of positive elements in the Dehornoy order is defined to be the elements that can be written as a  -positive word for some  .

The set   satisfies  , the sets  ,  , and   are disjoint ("acyclicity property"), and the braid group is a union of  ,  , and   ("comparison property"). These properties imply that if we define " " to mean " " then we get a left-invariant total order on the braid group. For example,   because the braid word   is not  -positive, but, by the braid relations, it is equivalent to the  -positive word  , which lies in  .

HistoryEdit

Set theory introduces the hypothetical existence of various "hyper-infinity" notions such as large cardinals. In 1989, it was proved that one such notion, axiom  , implies the existence of an algebraic structure called an acyclic shelf which in turn implies the decidability of the word problem for the left selfdistributivity law  , a property that is a priori unconnected with large cardinals [4][5].

In 1992, Dehornoy produced an example of an acyclic shelf by introducing a certain groupoid   that captures the geometrical aspects of the   law. As a result, an acyclic shelf was constructed on the braid group  , which happens to be a quotient of  , and this implies the existence of the braid order directly [2]. Since the braid order appears precisely when the large cardinal assumption is eliminated, the link between the braid order and the acyclic shelf was only evident via the original problem from set theory [6].

PropertiesEdit

  • The existence of the order shows that every braid group   is an orderable group and that, consequently, the algebras   and   have no zero-divisor.
  • For  , the Dehornoy order is not invariant on the right : we have   and  . Anyway, no order of   with   may be invariant on both sides.
  • For  , the Dehornoy order is neither Archimedian, nor Conradian: there exist braids   satisfying   for every   (for instance,   and  ), and braids   greater than   satisfying   for every   (for instance,   and  ).
  • The Dehornoy order is a well-ordering when restricted to the posititive braid monoid   generated by   (Richard Laver [7]). The order type of the Dehornoy order restricted to   is the ordinal   (Serge Burckel [8]).
  • The Dehornoy order is also a well-ordering when restricted to the dual posititive braid monoid   generated by the elements   with  , and the order type of the Dehornoy order restricted to   is also   (Jean Fromentin [9]).
  • As a binary relation, the Dehornoy order is decidable. The best decision algorithm is based on Dynnikov's tropical formulas (Ivan Dynnikov [10], see Chapter XII of [3]); the resulting algorithm admits a uniform complexity  .

Connection with knot theoryEdit

  • Let   be Garside's fundamental half-turn braid. Every braid   lies in a unique interval  ; call the integer   the Dehornoy floor of  , denoted  . Then the link closure of braids with a large floor behave nicely, namely the properties of   can be read easily from  . Here are some examples.
  • If   holds, then   is prime, nonsplit, and nontrivial (Andrei Malyutin and Nikita Netstetaev [11]).
  • If   holds and   is a knot, then   is a toric knot if and only if   is periodic,   is a satellite knot if and only if   is reducible, and   is hyperbolic if and only if   is pseudo-Anosov (Tetsuya Ito [12]).

ReferencesEdit

  1. ^ Dehornoy, Patrick (1992), "Deux propriétés des groupes de tresses", Comptes Rendus de l'Académie des Sciences. Série I. Mathématique, 315 (6): 633–638, ISSN 0764-4442, MR 1183793
  2. ^ a b Dehornoy, Patrick (1994), "Braid groups and left distributive operations", Transactions of the American Mathematical Society, 345 (1): 115–150, doi:10.2307/2154598, MR 1214782
  3. ^ a b Dehornoy, Patrick; Dynnikov, Ivan; Rolfsen, Dale; Wiest, Bert (2008), Ordering braids, Mathematical Surveys and Monographs, 148, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-4431-1, MR 2463428
  4. ^ Dehornoy, Patrick (1989), "Sur la structure des gerbes libres", Comptes Rendus des Séances de l'Académie des Sciences. Série I. Mathématique, 309 (3): 143–148, MR 1005627
  5. ^ Laver, Richard (1992), "The left distributive law and the freeness of an algebra of elementary embeddings", Advances in Mathematics, 91 (2): 209–231, doi:10.1016/0001-8708(92)90016-E, MR 1149623
  6. ^ Dehornoy, Patrick (1996), "Another use of set theory", Bulletin of Symbolic Logic, 2 (4): 379–391, doi:10.2307/421170, MR 1321290
  7. ^ Laver, Richard (1996), "Braid group actions on left distributive structures, and well orderings in the braid groups", Journal of Pure and Applied Algebra, 108: 81–98, doi:10.1016/0022-4049(95)00147-6, MR 1382244
  8. ^ Burckel, Serge (1997), "The wellordering on positive braids", Journal of Pure and Applied Algebra, 120 (1): 1–17, doi:10.1016/S0022-4049(96)00072-2, MR 1466094
  9. ^ Fromentin, Jean (2011), "Every braid admits a short sigma-definite expression", Journal of the European Mathematical Society, 13 (6): 1591–1631, doi:10.4171/JEMS/289 | mr = 2835325
  10. ^ Dynnikov, Ivan (2002), "On a Yang-Baxter mapping and the Dehornoy ordering", Rossi\u{\i}skaya Akademiya Nauk. Moskovskoe Matematicheskoe Obshchestvo. Uspekhi Matematicheskikh Nauk, 57 (3): 151–152, doi:10.1070/RM2002v057n03ABEH000519, MR 1918864
  11. ^ Malyutin, Andrei; Netsvetaev, Nikita Yu. (2003), "Dehornoy order in the braid group and transformations of closed braids", Rossiĭskaya Akademiya Nauk. Algebra i Analiz, 15 (3): 170–187, doi:10.1090/S1061-0022-04-00816-7, MR 2052167
  12. ^ Ito, Tetsuya (2011), "Braid ordering and knot genus", Journal of Knot Theory and its Ramifications, 20 (9): 1311–1323, doi:10.1142/S0218216511009169, MR 2844810

Further readingEdit