International Congress of Mathematicians

Invariants of Legendrian Knots 391ators in all monomials. Thus, if K and K are Legendrian isotopie then the gradedhomology ring H (A, 9) is anti-isomorphic to itself, and there exist £+,£!_ G H (A, 9)such that deg(£|_) = 1, deg(£L) = —1, £-£+ = 1. But one can check that suchclasses do not exist (see [14, 15] for details) and hence K and K are not Legendrianisotopie. Note that 'first order invariants' such as Poincaré polynomials are uselessin distinguishing Legendrian mirror knots.3. Admissible decompositions of fronts3.1. DefinitionsIn this section, we present the invariants of Legendrian knots constructed in [2].These invariants are defined in terms of the front projection.Given a a-generic oriented Legendrian knot L, denote by C(L) the set ofits points corresponding to cusps of a(L). The Maslov index p:L\ C(L) —¥ Y =Z/ro(L)Z is a locally constant function, uniquely defined up to an additive constantby the following rule: the value of p jumps at points of C(L) by ±1 as shown inFigure 4. We call a crossing of S = a(L) Maslov if p takes the same value on bothits branches.ß = i+l~^^ / \ ^~-li = i+l[1 = i-^ \ / ^-// = iFigure 4: Jumps of the Maslov index near cuspsAssume that S = a(L) is a union of closed curves Xi,..., X n that have finitelymanyself-intersections and meet each other at finitely many points. Then we callthe unordered collection {Xi,... ,X n } a decomposition of S. A decomposition{Xi,...,X n } is called admissible if it satisfies certain conditions, which we aregoing to define. The first two are as follows:(1) Each curve Xi bounds a topologically embedded disk: X t = dB t .(2) For each i £ {l,...,n}, q £ R, the set Bi(q) = {« G R | (q,u) £ Bi} is eithera segment, or consists of a single point « such that (q,u) is a cusp of S, or isempty.Conditions (1) and (2) imply that each curve X t has exactly two cusps (and hencethe number of curves is half the number of cusps). Each X t is divided by cuspsinto two pieces, on which the coordinate q is a monotone function. Near a crossingx £ XiCi Xj, the decomposition of S may look in one of the three ways representedin Figure 5. Conditions (1) and (2), in particular, rule out the decomposition shownin Figure 5a. We call the crossing point x switching if X t and Xj are not smoothnear x (Figure 5b), and non-switching otherwise (Figure 5c).(3) If (qo, u) £ Xi fl Xj (i ^ j) is switching then for each q ^ q 0 sufficiently closeto qo the set Bi(q) fl Bj(q) either coincides with Bi(q) or Bj(q), or is empty.