Omwinding

Omwinding is a generalized topological invariant associated with curves in a space that contains multiple reference centers or punctures. It extends the classical winding number by recording how a curve winds around each center, producing a vector of integers rather than a single integer.

Formal construction: Let X be the plane with a finite set P of distinguished points (centers). For a smooth curve gamma: [0,1] -> R^2 avoiding P, define for each p in P the angular function theta_p(t) = angle from p to gamma(t). The omwinding of gamma with respect to P is the vector W(gamma) = (w_p(gamma))_p, where w_p is the net change of theta_p(t) divided by 2π, taken as an integer. For closed curves, each w_p is an integer; for open curves, the corresponding values reflect the net angular change from start to end and need not be integers.

Properties: The omwinding is invariant under homotopies that avoid the centers P, and it is additive under concatenation of curves. It depends on the chosen labeling and positions of the centers, so different reference configurations yield different omwindings. Changes in the way the curve encircles a center can alter individual components by multiples of 2π.

Applications and examples: In topology, omwinding can help distinguish curves in punctured planes and study their interaction with multiple obstacles. In robotics and path planning, it provides a quantitative measure of how a trajectory winds around obstacles, informing strategies for clearance and safety margins. A simple loop encircling a single center once typically yields a unit entry for that center and zero for others in the corresponding component.

Terminology and reception: Omwinding is not a standard term in mainstream textbooks; it appears in some discussions as a generalization of the winding number or as a multi-center index on punctured spaces. See also winding number, index, and degree.