  This paper gives the first description of derived monodromy on the stringy
K\"ahler moduli space (SKMS) for a general irreducible flopping curve C in a
3-fold X with mild singularities. We do this by constructing two new infinite
helices: the first consists of sheaves supported on C, and the second comprises
vector bundles in a tubular neighbourhood. We prove that these helices
determine the simples and projectives in iterated tilts of the category of
perverse sheaves, and that all objects in the first helix induce a twist
autoequivalence for X. We show that these new derived symmetries, along with
established ones, induce the full monodromy on the SKMS.
  The helices have many further applications. We (1) prove representability of
noncommutative deformations of all successive thickenings of a length l
flopping curve, via tilting theory, (2) control the representing objects,
characterise when they are not commutative, and their central quotients, and
(3) give new and sharp theoretical lower bounds on Gopakumar-Vafa invariants
for a curve of length l. When X is smooth and resolves an affine base, we
furthermore (4) prove that the second helix classifies all tilting reflexive
sheaves on X, and thus that (5) all noncommutative crepant resolutions arise
from tilting bundles on X.
