Simple braids

Abstract

We study a subset of square free positive braids and we give a few algebraic characterizations of them and one geometric characterization: the set of positive braids whose closures are unlinks. We describe canonical forms of these braids and of their conjugacy classes.

Appendix

In this section we consider only positive braids: we compute the left least common multiple $(\mathrm{lcm}_{\,\mathrm L})$ of a generator $x_{i}$ and of the very simple braid, U(a, b) and D(c, d) respectively. The simplest case appears in Garside [11].

Lemma 7.1

(Garside) Suppose that .

• (a) If $|i-j|\geqslant 2$, then .

• (b) If $i+1=j$, then $x_{i}x_{i+1}x_{i}=x_{i+1}x_{i}x_{i+1}\,|_{\mathrm{L}}\,\beta$.

Lemma 7.2

• (a) If $x_{i}x_{i+1},x_{i+2}\in \mathrm{Div}_{\mathrm{L}}(\beta )$, then $x_{i}x_{i+1}(x_{i+2}x_{i+1})=x_{i+2}(x_{i} x_{i+1}x_{i+2})\,|_{\mathrm{L}}\,\beta$.

• (b) if $x_{i+1}x_{i+2},x_{i}\in \mathrm{Div}_{\mathrm{L}}(\beta )$, then $x_{i}(x_{i+1} x_{i}x_{i+2}x_{i+1})=x_{i+1}x_{i+2}(x_{i}x_{i+1} x_{i+2})\,|_{\mathrm{L}}\,\beta$.

• (c) if $x_{i+2}x_{i+1},x_{i}\in \mathrm{Div}_{\mathrm{L}}(\beta )$, then $x_{i}(x_{i+2} x_{i+1}x_{i})=x_{i+2}x_{i+1}x_{i}(x_{i+1})\,|_{\mathrm{L}}\,\beta$.

• (d) if $x_{i+1}x_{i},x_{i+2}\in \mathrm{Div}_{\mathrm{L}}\beta$, then $x_{i+1}x_{i}(x_{i +2}x_{i+1}x_{i})=x_{i+2}(x_{i+1}x_{i}x_{i+2} x_{i+1})\,|_{\mathrm{L}}\,\beta$.

Proof

(a): Garside Lemma (a) implies that , therefore $x_{i+2}\,|_{\mathrm{L}}\,x_{i+1} \beta '$, and case (b) of Garside Lemma implies that .

(b): Garside Lemma (b) implies $x_{i+1}x_{i+2}\beta ' =\beta =x_{i+1}x_{i}x_{i+1}\beta ^{'''}$, therefore $x_{i}x_{i+1}$ and $x_{i+2}$ are left divisors of $x_{i+2}\beta '$; case (a) of this lemma gives the result.

(c) and (d) can be checked in a similar way. $\square$

Using Lemmas 7.1 and 7.2 one can start an induction to prove the next results (or one can find a proof in [1]).

Proposition 7.3

Suppose that $x_{i},U(a,b)\in \mathrm{Div}_{\mathrm{L}}(\beta )$, $a+1\leqslant b$. We have the following implications:

• (a) if $i\notin \mathrm{e}\text {-}\mathrm{supp} U(a,b)$, then ;

• (b) if $i=a-1$, then ;

• (c) if $i=a$, then $\mathrm{lcm}_{\,\mathrm L}(x_{a},U(a,b))=U(a,b)$;

• (d) if $i\in [a+1,b]$, then ;

• (e) if $i=b+1$, then .

Proposition 7.4

Suppose that $x_{i},D(c,d)\in \mathrm{Div}_{\mathrm{L}} (\beta )\ (c\geqslant d+1)$. We have the following implications:

• (a) if $i\notin \mathrm{e}\text {-}\mathrm{supp}D(c,d)$ then ;

• (b) if $i=d-1$, then ;

• (c) if $i\in [d,c-1]$, then ;

• (d) if $i=c$, then $\mathrm{lcm}_{\,\mathrm L}(x_{c},D(c,d))=D(c,d)$;

• (e) if $i=c+1$, then .

Proposition 7.5

Given and a cycle $\gamma$, , we have the following implications:

• (a) if $x_{b-1}\,|_{\mathrm{L}}\,\gamma \beta$, then $x_{b-1}\,|_{\mathrm{L}}\,\beta$;

• (b) if $x_{e+1}\,|_{\mathrm{L}}\,\gamma \beta$, then $x_{e+1}\,|_{\mathrm{L}}\,\beta$.

Proof

Induction on the length of $\gamma$ and Garside Lemma 7.1 give the result. $\square$