Dehornoy order

Suppose that ${\displaystyle \sigma _{1},\ldots ,\sigma _{n-1}}$ are the usual generators of the braid group ${\displaystyle B_{n}}$ on ${\displaystyle n}$ strings. Define a ${\displaystyle \sigma _{i}}$-positive word to be a braid that admits at least one expression in the elements ${\displaystyle \sigma _{1},\ldots ,\sigma _{n-1}}$ and their inverses, such that the word contains ${\displaystyle \sigma _{i}}$, but does not contain ${\displaystyle \sigma _{i}^{-1}}$ nor ${\displaystyle \sigma _{j}^{\pm 1}}$ for ${\displaystyle j<i}$.

The set ${\displaystyle P}$ of positive elements in the Dehornoy order is defined to be the elements that can be written as a ${\displaystyle \sigma _{i}}$-positive word for some ${\displaystyle i}$. We have:

• ${\displaystyle PP\subseteq P;}$
• ${\displaystyle P,\{1\}}$ and ${\displaystyle P^{-1}}$ are disjoint ("acyclicity property");
• the braid group is the union of ${\displaystyle P,\{1\}}$ and ${\displaystyle P^{-1}}$ ("comparison property").

These properties imply that if we define ${\displaystyle a<b}$ as ${\displaystyle a^{-1}b\in P}$ then we get a left-invariant total order on the braid group. For example, ${\displaystyle \sigma _{1}<\sigma _{2}\sigma _{1}}$ because the braid word ${\displaystyle \sigma _{1}^{-1}\sigma _{2}\sigma _{1}}$ is not ${\displaystyle \sigma _{1}}$-positive, but, by the braid relations, it is equivalent to the ${\displaystyle \sigma _{1}}$-positive word ${\displaystyle \sigma _{2}\sigma _{1}\sigma _{2}^{-1}}$, which lies in ${\displaystyle P}$.

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 ${\displaystyle I_{3}}$, implies the existence of an algebraic structure called an acyclic shelf which in turn implies the decidability of the word problem for the left self-distributivity law ${\displaystyle LD:x(yz)=(xy)(xz),}$ 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 ${\displaystyle {\mathcal {G}}_{LD}}$ that captures the geometrical aspects of the ${\displaystyle LD}$ law. As a result, an acyclic shelf was constructed on the braid group ${\displaystyle B_{\infty }}$, which happens to be a quotient of ${\displaystyle {\mathcal {G}}_{LD}}$, 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]

• The existence of the order shows that every braid group ${\displaystyle B_{n}}$ is an orderable group and that, consequently, the algebras ${\displaystyle \mathbb {Z} B_{n}}$ and ${\displaystyle \mathbb {C} B_{n}}$ have no zero-divisor.

• For ${\displaystyle n\geqslant 3}$, the Dehornoy order is not invariant on the right: we have ${\displaystyle \sigma _{2}<\sigma _{1}}$ and ${\displaystyle \sigma _{2}\sigma _{1}>\sigma _{1}^{2}}$. In fact no order of ${\displaystyle B_{n}}$ with ${\displaystyle n\geqslant 3}$ may be invariant on both sides.

• For ${\displaystyle n\geqslant 3}$, the Dehornoy order is neither Archimedean, nor Conradian: there exist braids ${\displaystyle \beta _{1},\beta _{2}}$ satisfying ${\displaystyle \beta _{1}^{p}<\beta _{2}}$ for every ${\displaystyle p}$ (for instance ${\displaystyle \beta _{1}=\sigma _{2}}$ and ${\displaystyle \beta _{2}=\sigma _{1}}$), and braids ${\displaystyle \beta _{1},\beta _{2}}$ greater than ${\displaystyle 1}$ satisfying ${\displaystyle \beta _{1}>\beta _{2}\beta _{1}^{p}}$ for every ${\displaystyle p}$ (for instance, ${\displaystyle \beta _{1}=\sigma _{2}^{-1}\sigma _{1}}$ and ${\displaystyle \beta _{2}=\sigma _{2}^{-2}\sigma _{1}}$).

• The Dehornoy order is a well-ordering when restricted to the positive braid monoid ${\displaystyle B_{n}^{+}}$ generated by ${\displaystyle \sigma _{1},\ldots ,\sigma _{n-1}}$ (Richard Laver [7]). The order type of the Dehornoy order restricted to ${\displaystyle B_{n}^{+}}$ is the ordinal ${\displaystyle \omega ^{\omega ^{n-2}}}$ (Serge Burckel[8]).

• The Dehornoy order is also a well-ordering when restricted to the dual posititive braid monoid ${\displaystyle B_{n}^{*+}}$ generated by the elements ${\displaystyle \sigma _{i}\dots \sigma _{j-1}\sigma _{j}\sigma _{j-1}^{-1}\dots \sigma _{i}^{-1}}$ with ${\displaystyle 1\leqslant i<j\leqslant n}$, and the order type of the Dehornoy order restricted to ${\displaystyle B_{n}^{*+}}$ is also ${\displaystyle \omega ^{\omega ^{n-2}}}$ (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 ${\displaystyle O(\ell ^{2})}$.

• Let ${\displaystyle \Delta _{n}}$ be Garside's fundamental half-turn braid. Every braid ${\displaystyle \beta }$ lies in a unique interval ${\displaystyle [\Delta _{n}^{2m},\Delta _{n}^{2m+2})}$; call the integer ${\displaystyle m}$ the Dehornoy floor of ${\displaystyle \beta }$, denoted ${\displaystyle \lfloor \beta \rfloor }$. Then the link closure of braids with a large floor behave nicely, namely the properties of ${\displaystyle {\widehat {\beta }}}$ can be read easily from ${\displaystyle \beta }$. Here are some examples.

• If ${\displaystyle \vert \lfloor \beta \rfloor \vert >1}$ then ${\displaystyle {\widehat {\beta }}}$ is prime, non-split, and non-trivial (Andrei Malyutin and Nikita Netstetaev[11]).

• If ${\displaystyle \vert \lfloor \beta \rfloor \vert >1}$ and ${\displaystyle {\widehat {\beta }}}$ is a knot, then ${\displaystyle {\widehat {\beta }}}$ is a toric knot if and only if ${\displaystyle \beta }$ is periodic, ${\displaystyle {\widehat {\beta }}}$ is a satellite knot if and only if ${\displaystyle \beta }$ is reducible, and ${\displaystyle {\widehat {\beta }}}$ is hyperbolic if and only if ${\displaystyle \beta }$ is pseudo-Anosov (Tetsuya Ito[12]).

• Kassel, Christian (2002), "L'ordre de Dehornoy sur les tresses", Astérisque (276): 7–28, ISSN 0303-1179, MR 1886754
• Dehornoy, Patrick (1997), "A fast method for comparing braids", Advances in Mathematics, 125 (2): 200–235, doi:10.1006/aima.1997.1605, MR 1434111