A Legendrian knot is a curve in $\mathbb{R}^3$ on which $dz - ydx$ vanishes identically. Its projection to the $x,z$ plane is called a front diagram; as we can recover $y = dz/dx$ this determines the curve. Note that since $y$ is finite, we can never have vertical tangents; instead there are cusps in the diagram.

A $\mathbb{Z}$-Maslov potential is an assignment of integers to the arcs connecting cusps such that the number below a cusp is one less than the number above a cusp. The condition to admit such a thing is that the 'rotation number' -- which can be computed by counting the difference between the number of times you go from the top to the bottom of a cusp minus the number of times you go from the bottom to the top as you traverse the knot diagram -- be zero. Evidently a Maslov potential must take on at least two values.

Which rotation number zero Legendrian knots admit a diagram which admits a Maslov potential taking values in {$0,1$}?

For instance, the closure of any positive braid is such a knot.