  Given a front projection of a Legendrian knot $K$ in $\mathbb{R}^{3}$ which
has been cut into several pieces along vertical lines, we assign a differential
graded algebra to each piece and prove a van Kampen theorem describing the
Chekanov-Eliashberg invariant of $K$ as a pushout of these algebras. We then
use this theorem to construct maps between the invariants of Legendrian knots
related by certain tangle replacements, and to describe the linearized contact
homology of Legendrian Whitehead doubles. Other consequences include a
Mayer-Vietoris sequence for linearized contact homology and a van Kampen
theorem for the characteristic algebra of a Legendrian knot.
