International Congress of Mathematicians

390 Yu. V. ChekanovFigure 2: Lagrangian projections of two Legendrian 52 knotslb. 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(a,) = 0for i > 5, 9(ai) = 1 + a 7 + a 5 + a 7 a 6 a 5 + a 9 a s a 5 , d(a 2 ) = 1 + a 8 + a 5 a 6 a 9 ,9(03) = 1 + a%a,T, 9(04) = 1 + a%ag, 9(a») = 0 for i > 5.An explicit computation shows that I(L) = {t^ + ^ + t 2 }, I(L') = fôfl + t 1 }and hence Theorem 1.1 follows from Corollary 2.3.Figure 3: Lagrangian projection of a Legendrian 62 knot2. [14] Let (A, 9) = (T(ai,..., an), 9) be the DGA of the Legendrian knot K givenin Figure 3. We have m(K) = 0, ß(K) = -7, deg(aj) = 1 for i £ {1,2,7,9,10},deg(a,) = 0 for i £ {3,4}, deg(a,) = —1 for i £ {5,6,8,11}; 9(ai) = 1 + 0100503,d(a 2 ) = 1 + a 3 + a 3 a 6 aio + a 3 ana 7 , d(a i ) = a 5 + an + ana 7 a 5 , d(a 6 ) = ana 8 ,d(a 7 ) = a$aio, d(ag) = 1 + aioan, d(a,i) = 0 for i £ {3,5,8,10,11}. Denoteby K the 'Legendrian mirror' of if — the image of K under the map (q,p,u) >-¥(—q,p, —u). The Legendrian knots K,K have the same classical invariants. However,they are not Legendrian isotopie, and it is possible to distinguish them bymeansof their DGAs. There exist homology classes £+,£- in the graded homologyring H(A,d) such that deg(£ + ) = 1, deg(£_) = —1, and £+£- = 1 (choose£ + = [aio], C- = [ a n])- It follows from the definitions that the DGA for K is obtainedfrom (A, 9) by applying the anti-automorphism reversing the order of gener-