This is a followup to the answered question, “Can any sequence of consecutive integers be realized as winding numbers?.” Answer: Yes (domotorp).
Now I would like to define a local winding number as follows. For any region $ R$ delimited by arcs of a planar curve $ C$ , define the local winding number as the number of counterclockwise arcs bounding $ R$ minus the number of clockwise bounding $ R$ . If I’ve calculated correctly, here is that same example:
This “local winding number” differs from the winding number, as can be seen by comparing to that previous question. For example, unless I have counted incorrectly, no region $ 4$ separates $ 3$ from $ 5$ , so the “consecutive integers” requirement no longer holds. This concept may be relatively unexplored (likely, because it is useless!).
My question is:
Q. Which local winding number sequences are realizable by some curve $ C$ ?
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