A simple algorithm for finding short sigma-definite representatives

0.18 MB·English

by  Jean Fromentin

A SIMPLE ALGORITHM FOR FINDING SHORT SIGMA-DEFINITE
REPRESENTATIVES
1
JEAN FROMENTIN ANDLUIS PARIS
1
0
2

n
a Abstract. We describe a new algorithm which for each braid returns a quasi-geodesic σ-
J
definite word representative, defined as a braid word in which the generator σi with maximal
7 index i appears either only positively or only negatively.

]
R
Introduction
G
h. Since [4], we know that Artin’s braid groups Bn are left orderable, by an ordering that
enjoys many remarkable properties. This braid ordering is based on the property that every
t
a
nontrivial braid admits a σ-definite representative, defined to be a braid word in standard Artin
m
generators σ in which the generator σ with highest index i appears either with only positive
i i
[
exponentsorwithonlynegativeexponents. Inthepasttwodecades,manydifferentproofsofthis

1 result have been found, some of them based on algebra [3, 4, 5, 14], other on geometry [2, 8, 9].
v
Allthesemethodsturnouttobealgorithms. Butin thebestcases, startingwith abraidwordw
0
′
0 of length ℓ, they only prove the existence of a σ-definite word w equivalent to w with length
4 bounded by an exponential on ℓ. In [10], an algorithm returning a quasi-geodesic σ-definite
1
representative has been introduced. It is heavily based on technical properties of the so-called
.
1 rotating normal form on the Birman–Ko–Lee monoid. Quite effective, this algorithm remains
0
complicate.
1
1 The aim of this paper is to describe a simple algorithm returning a quasi-geodesic σ-definite
: representative. It is based either on the alternating normal form introduced in [6] or on the
v
i rotating normal form intoduced in [10, 11, 12]. The main advantage of this new algorithm is
X
that it can be describe with few technical results on these normal forms. Part of the algo-
r
rithm presented here uses some ideas from [10]. However, this new algorithm goes beyond the
a
simplification of the previous one, and the paper can be read independently from [10].
Thepaperisorganizedasfollows. InSection1,wegiveanoverviewonreversingprocessesand
give someelementary algorithm thatwillbeneededtodescribethemainalgorithm. InSection 2
we recall the definition of the Φ -splitting, that is a natural way to describe each braid of B+
n n
from a finite sequence of braid of B+ , and we give an algorithm to compute it. In Section 3,
n−1
we introduce two different algorithms that allow us to express a braid of B as a quotient of
n
braids lying in B+. In Section 4 we describe and prove the correctness of the main algorithm
n
Date: 14 d´ecembre 2010.
2010 Mathematics Subject Classification. 20F36,20M05,06F05.
Key words and phrases. Braid group, braid monoids, braid ordering, algorithm.
Both authors are partially supported by the Agence Nationale de la Recherche (projet Th´eorie de Garside,
ANR-08-BLAN-0269-03).
1
2 JEANFROMENTINANDLUISPARIS
in the context of the alternating normal form. Finally, in the last section, we investigate the
complexity of our algorithm in the context of the Birman–Ko–Lee monoid B+∗.
n
1. Reversing process
In this section, we recall how to perform elementary computations in a finitely generated
Garside monoid. The main tool is the reversing algorithm introduced in [7].
Assume that M is a Garside monoid. Then, we define two partial orderings on M. Given
′ ′ ′
elements β and β of M, we say that β left divides (resp. right divides) β , denoted by β ≺ β
′ ′ ′
(resp. β ≻ β ), if there exists γ in M such that βγ = β (resp. β = γβ ) is satisfy.
′
Theleft lcm of two elements β and β of M is the minimal element γ in M, with respect to ≺,
′ ′
satisfying β ≺ γ and β ≺ γ, and we denote it by β ∨L β . Of course, we define symmetrically
′ ′
the right lcm of β and β in M, which is denoted by β∨Rβ .
Definition 1.1. Let M be a monoid generated by a finite set S.
(i) A word on the alphabet S is called a positive S-word,
(ii) A word on the alphabet S ∪S−1 is called a S-word,
(iii) The element represented by an S-word w is denoted by w,
(iv) For w,w′ two S-words, we say that w is equivalent to w′, denoted by w ≡ w′, if w = w′
holds.
Let M be a Garside monoid generated by a finite set S. A left lcm selector on S in M is a
mapping fn : S ×S → S∗ such that, for all x,y in S, the words xfn(x,y) and yfn(y,x) both
L L L
represent x∨L y. We define also a right lcm selector on S in M to be a mapping fRn such that
fRn(x,y)y and fRn(y,x)x represent x∨Ry for all x,y in S.
Example 1.2. We recall that the positive braid monoid B+ is defined for n > 2 by the presen-
n
tation
σσ = σ σ for |i−j| > 2 +
i j j i
σ ,...,σ ; . (1)
1 n−1 σσ σ = σ σσ for |i−j| = 1
i j i j i j
(cid:28) (cid:29)
We put Σ = {σ ,...,σ }. Then the applications fn and fn defined on Σ ×Σ by
n 1 n−1 L R n n
σ for |i−j| > 2,
fn(σ,σ ) = fn(σ,σ ) = j
L i j R i j
(σjσi for |i−j| = 1.
are respectively left and right lcm selectors on Σ in B+.
n n
For the rest of this section, we fix a Garside monoid M, a finite generating set S of M, a left
lcm selector fn and a right lcm selector fn on S in M.
L R
Definition 1.3. Let w,w′ be S-words. We say that wy(1)w′ is true if w′ is obtained from w by
R
replacing a subword x−1y of w by fn(x,y)fn(y,x)−1. We say that w y w′ is true if there exists
L L R
a sequence w = w ,...,w =w′ of S-words such that w y(1)w holds for all i= 0,1,...,k−1.
0 k i R i+1
Symmetrically, we say that w y w′ is true, if w′ is obtained form w by repeatedly replacing
L
a subword xy−1 of w by the word fn(y,x)−1fn(x,y).
R R
Wenowintroducethenotionofrightreversingdiagrams. Assumethatw ,...,w isareversing
0 k
sequence, i.e., a sequence of S-words such that w y(1)w holds for each i. First we associate
i R i+1
with w apath labelled withits successive letters: weassociate to apositive letter x ahorizontal
0
A SIMPLE ALGORITHM FOR FINDING SHORT SIGMA-DEFINITE REPRESENTATIVES 3
right-oriented arrow labelled x, and to a negative letter x−1 a vertical down-oriented arrow
labelled x. Then we successively represent the S-words w ,...,w as follows: if w is obtained
1 k i+1
formw byreplacingthesubwordx−1y ofw byfn(x,y)fn(y,x)−1,thenwecomplete thepattern
i i L L
corresponding to x−1y using a right-oriented arrow labelled fn(x,y) and a down-oriented arrow
L
labelled fn(y,x) to obtain a square:
L
y y
x completed into x yR fLn(y,x)
fLn(x,y)
Symmetrically, we define a left reversing diagram, in which we complete the pattern cor-
responding to xy−1 using a right-oriented arrow labelled fn(x,y) and a down-oriented arrow
R
labelled fn(y,x):
R
fRn(x,y)
y completed into fRn(y,x) yL y
x x
Proposition 1.4. [7] For every S-word w, there exist unique positive S-words u and v such that
w y uv−1 holds. Moreover the words u and v are obtained from w in time O(pos(w)·neg(w)),
R
where pos(w) is the number of positive letters occurring in w and neg(w) is the number of
negative letters occurring in w. A similar result occurs for y .
L
Letw beaS-word. AsthereexistuniquepositiveS-wordsu,v suchthatw y uv−1 holds,we
R
say that u is the right numerator of w, denoted by N (w), and that v is the right denominator
R
of w, denoted by D (w). Symmetrically, we define left numerator and left denominator of w
R
respectively denoted by N (w) and D (w). An immediate consequence of [7] is:
L L
Proposition 1.5. For all positive S-words u,v:
(i) u≺ v holds if and only if D (u−1v) is the empty word ε,
R
(ii) u ≻ v holds if and only if D (uv−1) is the empty word ε.
L
′ ′
Let β,β be two elements of M. The left gcd of β and β is the maximal element γ with
′
respect to ≺ such that γ ≺ β and γ ≺ β holds.
Proposition 1.6. [7, Proposition 7.7] Let u,v be positive S-words. Then the left gcd of u and
v is the element represented by
N (uD (N (u−1v)D (u−1v)−1)−1). (2)
L L R R
See Figure 1 for a description of (2) in terms of reversing diagrams.
Thereversingprocesstakesawordoninputandreturnsaword. Inordertosimplifynotations
we shall use divisor and gcd symbols on words.
Notation 1.7. Let u,v be positive S-words.
(i) If u≺ v holds, we denote by u\v the word N (u−1v).
R
(ii) If u≻ v holds, we denote by u/v the word N (uv−1).
L
(iii) We denote by u∧ v the word of (2).
L
4 JEANFROMENTINANDLUISPARIS
v
NL(uu′′−1)
y NL(u′v′−1)
u L yL v′=DR(u−1v)
u′′=DL(u′v′−1)
u′=NR(u−1v)
Figure 1. Reversingdiagramcorrespondingtothecomputationoftheleftgcdofuandv.
Firstly, we right reverse u−1v to obtain u′v′−1. Secondly we left reverse u′v′−1 to com-
pute D (u′v′−1), denoted by u′′. Finally we left reverse uu′′−1 to compute N (uu′′−1),
L L
which represents u∧ v.
L
With these notations, the element u\v is equal to u−1v, the element u/v is equal to uv−1 and
the element u∧ v is the left gcd of u and v.
L
Inthesequelwe willconsider two Garsidemonoids naturally generated by afiniteset, namely
the positive braid monoid B+ generated by Σ and the dual braid monoid B+∗ generated by A
n n n n
(see Section 5). From now on, we will not specify the lcm selectors for left and right reversing
operations in these monoids, if not needed.
2. The Φ -splitting
n
It is shown in [6] how associate with every braid β of B+ a unique sequence of braids in B+ ,
n n−1
calledtheΦ -splitting ofβ,thatcompletelydeterminesβ. Asmentionedintheintroduction,our
n
algorithm is basedon thisoperation. Inthissection werecall thedefinition andtheconstruction
of the Φ -splitting of a braid.
n
We recall that the positive braid monoid B+ is a Garside monoid with Garside element ∆
n n
defined by
∆ = (σ ... σ )·(σ ... σ )·... ·(σ σ )·σ .
n 1 n−1 1 n−2 1 2 1
See [7, 13] for a definition of a Garside monoid.
We denote by Φ the flip automorphism of B+, i.e., the application defined on B+ by Φ (β) =
n n n n
∆ β∆−1. The initial observation of the construction of the alternating normal form is that
n n
each braid of B+ admits a unique maximal right divisor lying in Φk(B+ ) for all k.
n n n−1
Lemma 2.1. For n > 3and k > 0, everybraid β of B+ admits a uniquemaximal right divisor β
n 1
lying in Φk(B+ ).
n n−1
Proof. The braid β is a maximal right divisor of β lying in Φk(B+ ) if and only if Φ−k(β )
1 n n−1 n 1
is the maximal right divisor of Φ−k(β) lying in B+ . As the submonoid B+ of B+ is closed
n n−1 n−1 n
under right divisors and left lcm, the braid Φ−k(β ) exists and is unique. (cid:3)
n 1
Definition 2.2. The braid β of Lemma 2.1 is called the Φk(B+ )-tail of β.
1 n n−1
By iterating the tail construction, we then associate with every braid of B+ a finite sequence
n
of braids of B+ .
n−1
A SIMPLE ALGORITHM FOR FINDING SHORT SIGMA-DEFINITE REPRESENTATIVES 5
Proposition 2.3. [6, Proposition 2.5] Assume n > 3. Then for each nontrivial braid β of B+,
n
there exists a unique sequence (β ,...,β ) in B+ satisfying
b 1 n−1
β 6= 1 and β = Φb−1(β )·...·Φ (β )·β (3)
b n b n 2 1
for each k >1, Φk−1(β ) is the Φk−1(B+ )-tail of Φb−1(β )·...·Φk−1(β ) (4)
n k n n−1 n b n k
Definition 2.4. The sequence (β ,...,β ) of Proposition 2.3 is called the Φ -splitting of β and
b 1 n
its length is called the Φ -breadth of β.
n
We give now an algorithm to compute the Φ -splitting of a braid given by a positive Σ -
n n
word w. More precisely the algorithm returns a sequence (w ,...,w ) of positive Σ -words
b 1 n−1
such that (w ,...,w ) is the Φ -splitting of w.
b 1 n
Algorithm 1. Compute the Φ -splitting of the braid represented by w
n
Input: A positive Σ -word w with n> 3
n
′
1. Put ß = (), w = w and k = 0.
′
2. While w 6= ε do
3. Put u= ε.
4. While there exists x ∈ Σ such that w′ ≻ Φk(x) do
n−1 n
5. Put w′ = w′/Φk(x) and u =xu
n
6. Insert u on the left of ß.
7. Put k = k+1
8. Return s.
Proposition 2.5. Running on w, Algorithm 1 ends in time O(|w|2) and returns a sequence
(w ,...,w ) of positive Σ -words such that (w ,...,w ) is the Φ -splitting of w.
b 1 n−1 b 1 n
′
Proof. We denote by β the braid represented by the value of w at Line 3. Lines 3, 4 and 5
compute the maximal right divisor of β lying in Φk(B+ ). At Line 6, the braid Φk(u) is equal
n n−1 n
to the Φk(B+ )-tail of β and w′ is equal to β/Φk(u).
n n−1 n
Therefore the algorithm applies successively the Φk(B+ )-tail construction for k = 1,2,....
n n−1
Then, by Proposition 2.3 it must stop and return the expected sequence of words.
As for time complexity, testing if w′ ≻ Φk(x) holds and computing w′/Φk(x) need to run the
n n
left revering process on w′(Φk(x))−1. Proposition 1.4 guarantees that these two operations can
n
bedoneintimeO(|w′|), andso,intimeO(|w|)since|w′|6 |w|holds. Thenaneasy bookkeeping
shows that the algorithm ends in time O(|w|2). (cid:3)
3. Garside quotient
In the previous section we have seen how to compute the Φ -splitting of a braid lying in B+.
n n
Of course there is no possible extension of the notion of Φ -splitting to the braid group B .
n n
However, we have the following.
Proposition 3.1. Each braid β admits a unique decomposition ∆−tβ′ where t is a nonnegative
n
integer and β′ is a braid belonging to B+, which is not left divisible by ∆ , unless t = 0.
n n
Proof. The monoid B+ is a Garside monoid with Garside elements ∆ , see [13]. As B is the
n n n
group of fractions of B+, there exist a smallest nonnegetive integer t such that ∆t β lies in B+.
n n n
6 JEANFROMENTINANDLUISPARIS
If t is positive, the minimality hypothesis on t implies ∆ 6≺ ∆t β. Then we define β′ to be the
n n
braid ∆t β.
n
Assume now that ∆−t′β′′ is another decomposition of β satisfying the hypothesis of the
n
proposition. As ∆t′β belongs to B+, we have t′ > t. Assume t′ > t. Then we have t′ > 0. As
n n
the braid β′′ is equal to ∆t′−tβ′, the relation ∆ ≺ β′′ holds, that is in contradiction with t′ > 0
n n
and the hypothesis of the proposition. Hence t′ is equal to t and then β′ is equal to β′′. (cid:3)
Algorithm 2. Compute the decomposition ∆−tv given in Proposition 3.1 of the braid repre-
n
sented by the Σ -word w.
n
Input: An Σ -word w
n
1. Write w as w x−1w ... w x−1w (where w is a positive word and x is a letter).
0 1 1 t−1 t t i j
3. For i = 1... t compute u such that ∆ = u x .
i n i i
4. Put v = Φt(w )Φt−1(u w )... Φ (u w )u w .
n 0 n 1 1 n t−1 t−1 t t
5. While ∆ ≺ v and t > 0 hold do
n
6. Put v = ∆ \v and t = t−1.
n
7. Return ∆−tv.
n
Proposition 3.2. Running on w, Algorithm 2 ends in time O(|w|2) and has the correct output.
Moreover we have |∆−tu| 6 (n2−n−1)·kwk , where kβk is the minimal length of a Σ-word
n σ σ
representing β.
Proof. For i = 1,...,t, we denote by x−1 the negative letters occurring in w. Then we replace
i
each x−1 by ∆−1u to obtain
i n i
w ≡ w ∆−1u w ... w ∆−1u w . (5)
0 n 1 1 t−1 n t t
The definition of Φ implies u∆−1 ≡ ∆−1Φ (u) for every positive Σ -word u. From relation (5),
n n n n n
we obtain
w ≡ ∆−tΦt(w )Φt−1(u w )... Φ (u w )u w . (6)
n n 0 n 1 1 n t−1 t−1 t t
So the word v introduced in Line 4 is equivalent to ∆t w. After Lines 5 and 6, the braid v is not
n
left divisible by ∆ unless t = 0. At the end, we have w ≡ ∆−tv, hence the algorithm returns
n n
the correct output.
As for the length, replacing x−1 by ∆−1u multiplies it by at most 2|∆ | − 1, i.e., by at
i n i n
most n2−n−1. Indeed, the relations in the presentation (1) preserve the length, hence we
have |u | = |∆ |−1. By Proposition 3.1, theinteger t andthebraidv dependonly of thebraidw
i n
and not on the word w. Hence, applying the algorithm to a geodesic word representing w gives
|∆−tv| 6 (n2−n−1)·kwk .
n σ
As for time complexity, the word of Line 4 is obtained in time O(|w|). The while command
of Line 5 needs at most |v| steps. Testing if ∆ ≺ v holds and computing ∆ \v need to run
n n
the right reversing process on ∆−1v. Proposition 1.4 guarantees that these two operations can
n
be done in time O(|v|). Then, from |v| 6 (n2−n−1)·|w|, we deduce that the algorithm ends in
time O(|w|)2. (cid:3)
Now, for a braid β, the decomposition which we shall introduce in the next proposition is
called the Garside–Thurston normal form of β. We use this normal form for computing the
minimal k such that β lies in B . Indeed, as we will see, the Garside-Thurston normal form
k
depends only on β and not on the group B in which it is viewed.
n
A SIMPLE ALGORITHM FOR FINDING SHORT SIGMA-DEFINITE REPRESENTATIVES 7
Proposition 3.3. [7, Corollary 7.5] Each braid β of B admits a unique decomposition β′−1β′′
n
where β′,β′′ belong to B+ and such that β′ ∧ β′′ is trivial. Moreover if β is represented
n L
by w then the braid β′ is represented by D (N (w)D (w)−1) and the braid β′′ is represented
L R R
by N (N (w)D (w)−1).
L R R
Sincefor k 6 nthelattice operation∧ inB+ coincideswiththatofB+,adirectconsequence
L k n
of Proposition 3.3 is the following.
Corollary 3.4. Let k 6 n, β in B and β = β′−1β′′ be the Garside-Thurston normal form of β.
n
We have β ∈ B if and only if β′,β′′ lie in B+.
k k
Definition 3.5. We define the index of a Σ -word w to be the maximal i such that w contains
n
a letter σ . The index of a braid β is the minimal index of a word which represents β.
i−1
Obviously, the index of a braid β is the minimal integer n such that β lies in B .
n
′′
Algorithm 3. Compute the index k of w and a Σ -word w equivalent to w.
k
Input: An Σ -word w
n
′
1. Right reverse w into w .
′ ′′
2. Left reverse w into w .
′′
3. Let k be the index of w .
′′
4. Return (k,w ).
The correctness of this algorithm is a direct consequence of Proposition 3.3 together with
Corollary 3.4. Moreover, by Proposition 1.4 it ends in time O(|w|2).
4. The main algorithm
Putting all pieces together, we can now describeour algorithm which returns a quasi-geodesic
word equivalent to a given word. However, we first recall the definition of σ-definite words and
give the result of [6] which will be used to prove the correctness of the algorithm. As ever we
∞
assume Σ ⊂ Σ (as well as B ⊂ B ) for all n >2, and we set Σ = Σ .
n n+1 n n+1 n=2 n
Definition 4.1.
S
(i) A Σ-word is said to be σ-positive (resp. σ-negative) if it contains at least one letter of
i i
the form σ, no letter σ−1 (resp. at least one letter σ−1, no letter σ) and no letter σ with j > i.
i i i i j
(ii) A Σ-word is said to be σ-definite if it is either trivial, or σ-positive or σ-negative for a
i i
certain i.
(iii)Abraidissaidtobeσ-positive(resp. σ-negative) ifitcanberepresentedbyaσ-positive
i i i
word (resp. a σ-negative word).
i
Recall form [4] that the celebrated Dehornoy ordering on B is defined by β < γ if β−1γ is
n
σ-positive for some i 6 n. The key property that will be used on the Φ -splitting operation is
i n
its coincidence with the Dehornoy ordering <.
Proposition 4.2. [6] Let β and γ be two braids of B+. Let (β ,...,β ) and (γ ,...,γ ) be the
n b 1 c 1
Φ -splittings of β and γ respectively. Then β < γ holds if and only if we have either b < c or
n
b = c and, for some t 6 b, we have βt′ = γt′ for t < t′ 6 b together with βt < γt .
In [6], Dehornoy proves that the minimal positive braid of a given Φ -breadth b+2 (see 2.4)
n
with b > 0 is ∆ = ∆b ∆−b , and the Φ -splitting of the latter is the following sequence of
n−1,b n n−1 n
b
8 JEANFROMENTINANDLUISPARIS
length b+2
(σ ,σ ...σ σ2,...,σ ...σ σ2,σ ...σ σ ,1). (7)
1 n−1 2 1 n−1 2 1 n−1 2 1
As the braid ∆ lies in B+ , we deduce that the Φ -splitting of ∆b is the following sequence
n−1 n−1 n n
of length b+2
(σ ,σ ...σ σ2,...,σ ...σ σ2,σ ...σ σ ,∆b−1). (8)
1 n−1 2 1 n−1 2 1 n−1 2 1 n−1
The idea of our algorithm is that we can easily decide if a quotient ∆−tβ with β lying in B+
n n
is σ -negative or not.
n−1
Lemma 4.3. Assume that β is a braid of B+ such that ∆ 6 β 6 ∆b holds, then the
n n−1,b n
quotient ∆−bβ lies in B .
n n−1
b
Proof. The relation ∆ 6 β 6 ∆b and Proposition 4.2 imply that the Φ -splitting of β is
n−1,b n n
the following sequence of length b+2
b (σ ,σ ...σ σ2,...,σ ...σ σ2,σ ...σ σ ,β ), (9)
1 n−1 2 1 n−1 2 1 n−1 2 1 1
with 1 6 β 6 ∆b . Hence β is equal to ∆ β where β belongs to B+ . Then, as the
1 n−1 n−1,b 1 1 n−1
quotient ∆−bβ is equal to ∆−b ∆ β, we obtain ∆−bβ = ∆−b β . As β and ∆ lie
n n−1 n−1,b n n−1 1 1 n−1
in B the braid ∆−bβ lies in B . b (cid:3)
n−1 n n−1
b
Proposition 4.4. Assume n > 3 and β is a braid of B+. Let t be a positive integer and b
n
the Φ -breadth of β. If t > b−1 holds then the quotient ∆−tβ is σ -negative. Otherwise it is
n n n−1
not σ -negative.
n−1
Proof. Let (β ,...,β ) be the Φ -splitting of β. Then the braid ∆−tβ is equal to
b 1 n n
∆−t·Φb−1(β )·... ·Φ (β )·β . (10)
n n b n 2 1
Pushing b−1 powers of ∆ to the right in (10) and dispatching them between the factors β ,
n k
we find
∆−tβ ≡ ∆−t·Φb−1(β )·... ·Φ (β )·β
n n n b n 2 1
≡ ∆−t+b−1·∆−b−1Φb−1(β )·... ·Φ (β )·β
n n n b n 2 1
≡ ∆−t+b−1·β ·∆−1·∆−b−2·... ·Φ (β )·β
n b n n n 2 1
≡ ... ≡ ∆−t+b−1 β ∆−1 β ∆−1... β ∆−1 β .
n b n b−1 n 2 n 1
If the relation t > b−1 holds then the braid
∆−t+b−1 β ∆−1 β ∆−1... β ∆−1 β , (11)
n b n b−1 n 2 n 1
is σ -negative. Indeed, by definition, the braid ∆−1 is σ -negative, while for each k the
n−1 n n−1
braid β lies in B+ . So, as −t+b−1 is nonpositive, the expression (11) contains t letters σ−1
k n−1 n−1
and no letter σ .
n−1
Now assume t < b−2. The Φ -breadth of ∆t is t+2 and we have t+2 < b. Then Proposi-
n n
tion 4.2 implies ∆t < β, i.e., ∆−tβ is σ-positive for a certain i, hence it is not σ -negative.
n n i n−1
Finally assume t = b−2. If the relation ∆t < β holds we concluse as in the previous case.
n
Then assume β 6 ∆t. As the Φ -breadth of β is b, which is equal to t+2, Proposition 4.2
n n
implies ∆ 6 β. Then by Lemma 4.3 the quotient ∆−tβ lies in B , hence it is not
n−1,t n n−1
σ -negative. (cid:3)
n−1
b
A SIMPLE ALGORITHM FOR FINDING SHORT SIGMA-DEFINITE REPRESENTATIVES 9
The following algorithm takes in entry a braid word w representing a braid β and it returns a
σ-definite word representing β. The main idea is to bring all possible cases to the case where β
is σ -negative, i.e., when β satisfy the conditions of Proposition 4.4.
n−1
Algorithm 4. Compute a σ-definite representative.
Input: A Σ -word w
n
1. Put e = 1.
2. Let (k,v) be the output of Algorithm 3 applied to w.
3. Use Algorithm 2 to compute ∆−tu ≡ ve.
k
4. If t = 0 or k = 2 then return ∆−tu.
k
5. Use Algorithm 1 to compute the Φ -splitting (u ,...,u ) of u.
k b 1
6. If t > b−1 then return (∆−t+b−1 u ∆−1 u ∆−1... u ∆−1 u )e.
k b k b−1 k 2 k 1
7. Else put e= −1 and goto Line 3.
Proposition 4.5. Algorithm 4 ends and returns in time O(|w|)2 a σ-definite word w′ equivalent
to w with |w′|6 (n2−n−1)·kwk , where kβk is the minimal length of a Σ-word representing β.
σ σ
Proof. We use Algorithm 3 to compute the index k of w and a Σ -word v equivalent to w. In
k
particular, the braid w is either σ -positive or σ -negative.
k−1 k−1
Next, we use Algorithm 2 to compute a quotient ∆−tu that is equivalent to v and so to w.
k
By Proposition 3.1 the exponent t and the positive braid u depends only of the braid w and not
on the word v. Moreover, we have
|∆−tu| 6(k2−k−1)kwk 6 (n2−n−1)kwk .
k σ σ
If t is equal to 0 then the quotient ∆−tu is equal to u, that is a positive word, hence a σ-definite
k
word. If n is equal to 2 with t 6= 0 then u is empty and ∆−tu is equal to ∆−t, that is a negative
k k
word, hence a σ-definite word.
Next, we use Algorithm 1 to compute the Φ -splitting (u ,...,u ) of u. Then the word ∆−tu
k b 1 k
′
is equivalent to u defined by
u′ = ∆−t+b−1 u ∆−1 u ∆−1... u ∆−1 u . (12)
k b k b−1 k 2 k 1
Iftherelationt > b−1holdsthenthewordu′ isσ -negative—see proofofProposition4.4—
k−1
hence it is σ-definite. So in this case the algorithm returns a σ-definite word equivalent to w.
Now, assume t < b−1. In this case we redo the same process with the word w−1. Note that,
as the index of w and w−1 are the same, we can directly go to Line 3 of the algorithm. In this
case, by Proposition 4.4, the braid w is not σ -negative, i.e., it is σ -positive or it lies in
k−1 k−1
B with m < k. As k is the index of w, the braid w is σ -positive. So the braid represented
m k−1
by v−1 is σ -negative. Hence the new value of t and b satisfy the relation t > b−1 and the
k−1
algorithm ends.
′
For length complexity, the length of the Σ -word u given in (12) is equal to the length of the
k
Σ -word ∆−tu. By Proposition 3.1 we have
k k
∆−tu6 (k2−k−1)·kvek = (k2−k−1)·kwek 6 (n2−n−1)·kwek .
k σ σ σ
Then, kw−1k = kwk implies |u′| 6 (n2−n−1)kwk . (cid:3)
σ σ σ
10 JEANFROMENTINANDLUISPARIS
5. Dual braid monoid
The dual braid monoid is another submonoid of B . It is generated by a subset of B
n n
that properly contains {σ ,...,σ }, and consists of the so-called Birman-Ko-Lee generators
1 n−1
introduced in [1].
Definition 5.1.
(i) For 1 6 p < q, we put a = σ ...σ σ σ−1 ...σ−1.
p,q p q−2 q−1 q−2 p
(ii) For n > 2, the set A is defined to be {a | 1 6 p < q 6 n}.
n p,q
(iii) The dual braid monoid B+∗, is the submonoid of B generated by A .
n n n
For p < q, we denote by Jp,qK the interval {p,...,q} of N, and we say that Jp,qK is nested
in Jr,sK if we have r < p < q < s. A presentation of B+∗ in terms of a is as follows.
n p,p
Proposition 5.2. [1] In terms of the a , the monoid B+∗ is presented by
p,q n
a a = a a for Jp,qK and Jr,sK disjoint or nested,
p,q r,s r,s p,q
a a = a a = a a for 16 p < q < r 6 n.
p,q q,r q,r p,r p,r p,q
As the positive braid monoid, we can endow the Birman–Ko–Lee monoid B+∗ with a Garside
n
structure. The corresponding Garside element is
δ = a a ... a .
n 1,2 2,3 n−1,n
We denote by φ the Garside automorphism of B+∗, i.e., the application defined on B+∗
n n n
by φ (β) = δ βδ−1.
n n n
An analog of the alternating normal form of the positive braid monoid B+ exists for the dual
n
braid monoid B+∗: the rotating normal form (see [11, 10] for more details about this normal
n
form). The rotating normal form is also based on an operation of splitting: the φ -splitting.
n
Moreover, for each result on the alternating normal form used in this paper there exists a
counterpart in the B+∗context, see [11] or [12] for more details.
n
±
AA -wordw is saidto beσ-definiteif all theletters a withhighestq appearonly positively
n p,q
or only negatively. This definition coincides with that given for Σ -words if we translate each
n
letter a of an A -word to the Σ -word given in Definition 5.1(i). Hence all the previous
p,q n n
algortihms can be translated to the dual language, replacing Φ by φ , ∆ by δ and Σ by A .
n n n n n n
One of the advantage of the dual braid monoid is that its Garside element δ has length n−1,
n
n(n−1)
while ∆ has length . Therefore in the dual context, Algorithm 2 runing on w returns
n 2
a word δ−tu whose length is at most (2n−3)kwk ,where, for β ∈ B , kβk denotes the word
n A n A
length of β with respect to A (as A contains Σ , we have necessary kβk 6 kβk for all
n n n A σ
braid β of B ). Hence Algorithm 4 runningon w returns a word of length at most (2n−3)kwk .
n A
References
[1] J Birman, K. H Ko, and S.J. Lee, A new approach to the word and conjugacy problems in the braid groups,
Adv.Math. 139 (1998), no. 2, 322–353.
[2] X.Bressaud, A normal form for braids, J. Knot Theory Ramifications 17 (2008), no. 6, 697–732.
[3] S.Burckel, The wellordering on positive braids, J. Pure Appl.Algebra 120 (1997), no. 1, 1–17.
[4] P. Dehornoy, Braid groups and left distributive operations, Trans. Amer. Math. Soc. 345 (1994), no. 1,
115–150.
[5] , A fast method for comparing braids, Adv.Math. 125 (1997), no. 2, 200–235.