  Mach reflection generally produces a contact discontinuity whose circulation
has previously been analyzed using "thermodynamic" arguments based on the
Hugoniot relations across the shocks. We focus on "kinematic" techniques that
avoid assumptions about the equation of state, using only jump relations for
conservation of mass and momentum, but not energy. We give a new short proof
for non-existence of pure (no contact) triple shocks, recovering a result of
Serre. For MR with a zero-circulation but nonzero-density-jump contact we show
that the incident shock must be normal. Nonexistence without contacts
generalizes to two or more incident shocks if we assume all shocks are
compressive. The sign of circulation across the contact has previously been
controlled with entropy arguments, showing the post-Mach-stem velocity is
generally smaller. We give a kinematic proof assuming compressive shocks and
another condition, for example backward incident shocks, or a weak form of the
Lax condition. We also show that for 2+2 interactions (two "upper" shocks with
clockwise flow meeting two "lower" shocks with counterclockwise flow in a
single point) the circulation sign can generally not be controlled. For
$\gamma$-law pressure we show 2+2 interactions without contact must be either
symmetric or antisymmetric, with symmetry favored at low Mach number and low
shock strength. For full potential flow instead of the Euler equations we
surprisingly find, contrary to folklore and prior results for other models,
that pure triple shocks without contacts are possible, even for $\gamma$-law
pressure with $1<\gamma<3$.
