International Congress of Mathematicians

Invariants of Legendrian Knots 389the behaviour of the DGA associated with a Legendrian knot when its Lagrangiandiagram goes through elementary bifurcations (Legendrian Reidemeister moves).It turns out that one cannot replace the coefficient ring Z/2Z by Z: in somesense, our homology theory is not oriented. However, the construction describedabove can be modified to associate with a Legendrian knot L a DGA graded by Z andhaving Z[s, s^1](where deg(s) = m(Lj) as a coefficient ring [10]. After reducing thegrading to Z/ro(L)Z, and applying the homomorphism Z[s, s^1]—¥ Z/2Z sendingboth s and 1 G Z to 1 G Z/2Z, this Z^s^-DGA becomes the Z/2Z-DGA of theknot L.2.2. Poincaré polynomialsHomology rings of DGAs can be hard to work with. We are going to define aneasily computable invariant I, which is a finite subset of the group monoid No[F],where No = {0,1,...}, F = Z/ro(L)Z. Assume that do = 0. Then 9 2 = 0. Since9(Ai) c Ai, we can consider the homology H(Ai,di) = ker(9i| J 4 1 )/im(9i| J 4 1 ),which is a vector space graded by the cyclic group F. Define the Poincaré polynomialP {A , d) £No[Y]byP (A,d)(t) = J2dim(H X (Ai,di))t X ,Aerwhere H\(Ai,di) is the degree À homogeneous component of H(Ai,di). Definethe group Auto (A) to consist of graded automorphisms of A such that for each i £{1,... ,n} we have #(a,) = a, + c», where c, G A 0 = Z/2Z. (of course, c, = 0 whendeg(ctj) 7^ 0). Consider the set Uo(A,d) consisting of automorphisms g £ Auto(A)such that (9 s )o = 0 (where d 9 = g^1o 9 o g). DefineI(A,d) = {P {A9g) \g£Uo(A,d)}.Since Auto (A) has at most 2" elements, this invariant is not hard to compute. Wecan associate with every (7r-generic) Legendrian knot L the set I(L) = I(AL,8L)-Note that P( — l) = ß(L) for P £ I(L). One can show that J is an invariant ofstable tame DGA isomorphism. Hence Theorem 2.2 implies the followingCorollary 2.3. If L is Legendrian isotopie to L' then I(L) =I(L').The set I(L) can be empty (cf. Section 4) but no examples are known whereI(L) contains more than one element. Also, for all known examples of pairs L, L' ofLegendrian knots with coinciding classical invariants we have F(l) = P'(l), whereP £ 1(E), P' £ I(L'). Other, more complicated invariants of stable tame isomorfismwere developed and applied to distinguishing Legendrian knots in [15].2.3. Examplesla. Let (A,d) = (T(ai,... ,a®),d) be the DGA of the Legendrian knot L givenin Figure 2. We have m(L) = 0, ß(L) = 1, deg(a,) = 1 for i < 4, deg(as) = 2,deg(ag) = —2, deg(a,) = 0 for i > 7, d(ai) = 1 + a-j + a-ja^a^, d(a 2 ) = 1 + ag +a 5 a 6 a 9 , 8(0,3) = 1 + a s a 7 , d(afi) = 1 + a s a 9 , d(a t ) = 0 for i > 5.