  A gentle algebra gives rise to a dissection of an oriented marked surface
with boundary into polygons and the bounded derived category of the gentle
algebra has a geometric interpretation in terms of this surface. In this paper
we study silting theory in the bounded derived category of a gentle algebra in
terms of its underlying surface. In particular, we show how silting mutation
corresponds to the changing of graded arcs and that in some cases silting
mutation results in the interpretation of the octahedral axioms in terms of the
flipping of diagonals in a quadrilateral as in the work of Dyckerhoff-Kapranov
in the context of triangulated surfaces. We also show that silting reduction
corresponds to the cutting of the underlying surface as is the case for
Calabi-Yau reduction of surface cluster categories as shown by Marsh-Palu and
Qiu-Zhou.
