  We show that any knot which is smoothly the closure of a 3-braid cannot be
Lagrangian concordant to and from the maximum Thurston-Bennequin Legendrian
unknot except the unknot itself. Our obstruction comes from drawing the
Weinstein handlebody diagrams of particular symplectic fillings of cyclic
branched double covers of knots in $S^3$. We use the Legendrian contact
homology differential graded algebra of the links in these diagrams to compute
the symplectic homology of these fillings to derive a contradiction. As a
corollary, we find an infinite family of contact manifolds which are rational
homology spheres but do not embed in $\mathbb{R}^4$ as contact type
hypersurfaces.
