  We give a method for constructing a Legendrian representative of a knot in
$S^3$ which realizes its maximal Thurston-Bennequin number under a certain
condition. The method utilizes Stein handle decompositions of $D^4$, and the
resulting Legendrian representative is often very complicated (relative to the
complexity of the topological knot type). As an application, we construct
infinitely many knots in $S^3$ each of which yields a reducible 3-manifold by a
Legendrian surgery in the standard tight contact structure. This disproves a
conjecture of Lidman and Sivek.
