QUASIPOSITIVITY AND RIGHT-INVARIANT

ORDER ON BRAID GROUPS

S.Yu. Orevkov

Let B_{n} be the braid group on n strings:

B_{n} = ⟨σ_{1}, …, σ_{n−1} | σ_{i}σ_{j} = σ_{j}σ_{i} if |i − j| > 1, σ_{i}σ_{j}σ_{i} = σ_{j}σ_{i}σ_{j} if |i − j| = 1⟩.

A braid β ∈ B_{n} is called quasipositive if β = (γ_{1}σ_{1}γ_{1}^{−1}) · . . . · (γ_{s}σ_{1}γ_{s}^{−1}) for some γ_{1}, …, γ_{s} ∈ B_{n}. This notion was introduced by Rudolph [6, 7]; he has shown that a braid is quasipositive if and only if it is the boundary braid of an algebraic function w(z) implicitely defined in a disk by w^{n} + a_{1}(z)w^{n−1} + ⋯ + a_{n}(z) = 0, a_{i} ∈ C[z]. The question of quasipositivity of a braid naturally arises in the topology of plane real algebraic curves [5]. An algorithm to decide if a given braid is quasipositive or not is known only for n = 3. Some necessary conditions for quasipositivity can be found in [7, 5]. Theorem A below provides one more necessary condition for the quasipositivity of a braid.

Dehornoy [1– 3] proved that there exists a total right invariant order on the elements of B_{n} (right invariant means α < β ⇒ αγ < βγ, ∀α, β, γ) uniquely defined by the condition

β_{0}σ_{i}β_{1} > 1 if β_{0}, β_{1} are words in σ_{1}^{±1}, …, σ_{n−1}^{±1}.

He gave also an efficient in practice algorithm to decide if a given braid is positive, trivial or negative (a braid β is called positive if β > 1).

Recently, the authors of [4] gave a nice geometric interpretation of the rightinvariant order (see §1 below) and other algorithms to compare braids — one of them is proven to be theoretically of quadratic time. Using this interpretation, we prove here the following

Theorem A. If a braid β is quasipositive then β > 1 (in the right-invariant order).

Denote by Q_{n} the set of quasipositive braids in B_{n} and let P_{n} = {β ∈ B_{n} | ∀α ∈ B_{n}, αβα^{−1} > 1}. Since the quasipositivity is invariant under the conjugation, Theorem A can be reformulated as Q_{n} ⊂ P_{n}. Note, that the braid β = σ_{1}^{2}σ_{2}^{−2} provides an example of a positive braid not belonging to P_{n} because αβα^{−1} < 1 for α = σ_{2}σ_{1}^{2}σ_{2} (see [4]).

The author thanks the Max-Planck-Institut

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§1. Geometric interpretation of the right-invariant order

All mappings below are piecewise smooth. Let D ⊂ C be the closed unit disk and X_{n} = {x_{0}, …, x_{n+1}} where −1 = x_{0} < x_{1} < ⋯ < x_{n} < x_{n+1} = 1 are real points in D. The braid group B_{n} can be naturally identified with the mapping class group of (D, X_{n}), that is the group of homeomorphisms (D, X_{n}) → (D, X_{n}) fixed on ∂D modulo the homeomorphisms isotopic to the identity mapping. The class [γ] ∈ B_{n} of a homeomorphism γ is uniquely defined by the curve Γ := γ([−1, 1]). Following [4], we call Γ a curve diagram of the braid [γ]. The curve diagram of σ_{i} is [−1, x_{i−1}] ∪ Γ_{+} ∪ [x_{i}, x_{i+1}] ∪ Γ_{−} ∪ [x_{i+2}, 1] where Γ_{+} is a path from x_{i−1} to x_{i+1} in the upper half-disk and Γ_{−} is a path from x_{i} to x_{i+2} in the lower half-disk. The convention on the order of multiplication is illustrated in Figure 1 (we reproduce this figure from [4]).

Figure 1: Curve diagrams of braids 1, σ_{1}, and σ_{1}σ_{2}^{−1}

Let Γ = γ([−1, 1]) be a curve diagram. Denote γ([x_{i}, x_{i+1}]) by Γ_{i}. Say that a curve diagram Δ is transverse to Γ if for every Γ_{i} either one has Γ_{i} ⊂ Δ or Γ_{i} is transverse to Δ.

Let Δ be transverse to Γ. A component of D \ (Γ ∪ Δ) is called bigon between Γ and Δ if it is homeomorphic to an open disk and is bounded by one open segment of Γ \ X_{n}, one open segment of Δ \ X_{n} and two points. We say that the pair (Γ, Δ) is reduced if there are no bigon between Γ and Δ.

Figure 2

Any pair of curve diagrams (Γ, Δ) can be reduced by a finite number of reduction moves shown in Figure 2 where Γ is solid and Δ is dashed. Each move reduces by 1 the number of bigons.

The right-invariant order on the braid group can be described in termes of curve diagrams as follows (see [4]). Let braids γ, δ be defined by curve diagrams Γ, Δ which are transverse to each other. Suppose (Γ, Δ) is reduced. The set D\Γ consists of two components. The component containing \sqrt{−1} is called upper. Suppose that Γ and Δ coincide on the arc from −1 to x_{i} (maybe, i = 0) and do not coincide after x_{i}. Let a be the piece of Δ \ (Δ ∩ Γ) starting at x_{i}. Then δ > γ if and only if a lies in the upper part of D \ Γ. This means that moving along Δ from −1 to 1, one follows Γ till the point x_{i} and then turns to the left.

§2. Proof of Theorem A

If α > β then α · β > 1 · β = β > 1. Thus, the product of positive braids is positive, hence, it suffices to prove that γσ_{1}γ^{−1} > 1 for any γ ∈ B_{n}. By right invariance this is equivalent to γσ_{1} > γ.

Let γ be an arbitrary braid in B_{n}. We have to prove that δ > γ where δ = γσ_{1}. Let Γ be a curve diagram of γ. Let I = [−1, 1], it is a curve diagram of the identity braid. Suppose Γ is transverse to I and (Γ, I) is reduced. Let D_{ε} be the ε-neighbourhood of [x_{1}, x_{2}]. Since (Γ, I) is reduced, for a sufficiently small ε, the pair (D_{ε}, D_{ε} ∩ Γ) is isotopic to one of those depicted in Fugure 3 and the following condition is satisfied:

(*) If two points of Γ ∩ ∂D_{ε} are neighbouring in the sence of the cyclic order along the circle ∂D_{ε} then they can not be connected by an arc of Γ \ D_{ε} not containing points of X_{n}.

Figure 3

Figure 4

The curves D_{ε} ∩ Γ in Figure 3 can be deformed into the curves shown by solid lines in Figure 4. A curve diagram Δ_{ε} of the braid δ := γσ_{1} can be obtained from Γ by cutting D_{ε} ∩Γ and pasting the curves shown in Fugure 4 by dashed lines. The diagram Δ_{ε} is not yet transverse to Γ. Let Δ be obtained from Δ_{ε} applying the modification shown in Figure 5 at each component of Γ \ (D_{ε} ∪ X_{n}) one of whose ends lies on ∂D_{ε}. The diagram Δ is already transversal to Γ.

Figure 5

If the case shown in Figure 3.A occures then (Γ, Δ) is reduced and the desired inequality δ > γ follows immediately from the result of [4] described in §1. Thus, we suppose that one of the cases 3.B_{k} takes place.

Γ ∪ Δ cuts D_{ε} into several curvilinear poligons. Among them, 2k + 2 triangles and 2k + 2 pentagons (quadrilaterals for k = 0) are adjacent to the circle ∂D_{ε}.

Denote them respectively by T_{i}, P_{i}, i = 1, …, 2k + 2. Say that a triangle T_{i} is bad if it comes from a triangle in Figure 4 whose vertices lying on ∂D_{ε} are connected outside D_{ε} by an arc of Γ. By (*), this arc must contain a point of X_{n}. Attaching the region bounded by this arc to a bad triangle, we get a bigon between Γ and Δ, and all bigons between Γ and Δ are obtained in this way.

Figure 6

Let Δ′ be obtained from Δ by applying the modification shown in Figure 6 to each bad triangle. Since bad triangles can not be neighbouring, these modifications are independent. Let us show that (Γ, Δ′) is reduced. Indeed, for k = 0 this is obviouse (see Figure 6). For k > 0 only three new polygons appear: Q′, P′_{2}, and P_{1} ∪ T_{2} (see Figure 6). None of them can be contained in a bigon between Γ and Δ′.

We see in Figure 6 that Δ′ was obtained from Δ modifying only segments of Δ whose both ends are inside D_{ε}. However, the point x_{i} where Δ′ the first time deviates from Γ, lies on a segment starting at x_{0} and not entering into D_{ε}. Hence, Δ′ coincides with Δ near x_{i}. But near x_{i}, Δ is arranged with respect to Γ as in Figure 5. Thus, δ > γ according to §1. Theorem A is proven.

References

1. P. Dehornoy, Braid groups and left distributive operations, Trans. AMS 343 (1994), 115–150.

2. P. Dehornoy, From large cardinals to braids via distributive algebra, J. Knot Theory and its Ramifications 4 (1995), 33–79.

3. P. Dehornoy, A fast method of comparing braids, Adv. in Math. 125 (1997), 200–235.

4. R. Fenn, M.T. Greene, D. Rolfsen, C. Rourke, B. Wiest, Ordering the braid group, Preprint.

5. S.Yu. Orevkov, Link theory and oval arrangements of real algebraic curves, Topology (to appear).

6. L. Rudolph, Algebraic functions and closed braids, Topology 22 (1983), 191–202.

7. L. Rudolph, Some knot theory of complex plane curves, Noeuds, Tresses et Singularités, Mono. No. 31, L’Ens. Math., 1983, pp. 99–122.

Steklov Mathematical Institut, ul. Gubkina 8, Moscow, Russia

Max-Planck-Institut für Mathematik, Gottfried-Claren-Strasse 26, Bonn, Germany

   E-mail address: orevkov@mi.ras.ru