Two-edge cuts of a banana #
The elementary but important first part of the SameStrand geometry: a cut
which separates the two multivalent endpoints must cross every strand. Hence
a two-edge cut in a genus-at-least-two banana leaves the endpoints on the
same side. The remaining interval classification is deliberately kept
separate from this cardinality fact.
If a cut separates the two core vertices of a banana, it has at least one crossing unit step on each strand.
A two-edge cut in a banana with at least three strands cannot separate its two core vertices. This is the global half of the two-edge-cut classification used in the reduced-divisor argument.
If both core vertices lie outside a cut and the cut contains internal vertices on two different strands, then each strand must be entered and left. The resulting four crossing steps are distinct.
Consequently, a two-edge cut whose endpoints are both outside can contain interior vertices from at most one strand.
Once two distinct crossing steps on strand α exhaust a two-edge cut,
no other strand can cross that cut.
The local interval form of a two-edge cut. If two distinct steps of
strand α account for a cut of multiplicity two, every other strand is
constant on the cut: all its path vertices have the tail's membership.
Membership propagation along a strand #
If no unit step of a strand crosses a cut between two positions, the cut membership is constant at those positions. This is the induction principle needed to turn the two crossing steps of a multiplicity-two cut into an interval on the designated strand.
Under a multiplicity-two cut, two distinct crossing steps on one strand exhaust all crossings on that strand.
Between the two crossing steps of a multiplicity-two cut, membership on the designated strand is constant. The statement is oriented from the vertex immediately after the first crossing to any later position through the second crossing.