Recent work of I Dynnikov and M Prasolov proposes a new method of comparing
Legendrian knots. In general, to apply the method requires a lot of technical work. In
particular, one needs to search all rectangular diagrams of surfaces realizing certain
dividing configurations. We show that in the case when the orientation-preserving
symmetry group of the knot is trivial, this exhaustive search is not needed,
which simplifies the procedure considerably. This allows one to distinguish
Legendrian knots in certain cases when the computation of the known algebraic
invariants is infeasible or not informative. In particular, we disprove that
when
is an annulus tangent to the standard contact structure
along , then the
two components of
are always equivalent Legendrian knots. A candidate counterexample was proposed
recently by Dynnikov and Prasolov, but the proof of the fact that the two components
of
are not Legendrian equivalent was not given. Now this work is accomplished. It
is also shown here that the problem of comparing two Legendrian knots
having the same topological type is algorithmically solvable provided that the
orientation-preserving symmetry group of these knots is trivial.