  We construct in Logspace non-zero circulations for H-minor free graphs where
H is any fixed graph with a single crossing. This extends the list of planar,
bounded genus and K_{3,3}-free, K_5-free graphs for which non-zero circulations
are already known. This implies static bounds of UL and SPL, respectively, for
the problems of reachability and maximum matching in this class of graphs.
  Further, we prove the following dynamic complexity bounds for classes of
graphs where polynomially bounded, skew-symmetric non-zero circulations are in
Logspace: (1) A DynFO[Parity] bound (under polylogarithmic changes per step)
for constructing perfect matching and maximum matching. (2) A DynFO[Parity]
bound (under polylogarithmic changes per step) for distance. (3) A DynFO bound
(under slightly sublogarithmic changes per step) for maximum cardinality
matching size.
  The main technical contributions include the construction of polynomially
bounded non-zero circulation weights in single crossing minor free graphs based
on existing results [AGGT16,DKMTVZ20]; using the approach from [FGT16] to
construct dynamic isolating weights and generalizing the rank algorithm from
[DKMSZ15,DKMSZ18] to handle slightly sublogarithmic many (O(log n/log log n))
changes.
