A cable system defines an index agreeing with Brouwer degree on complementary regions, providing a sharp lower bound on swept volume for null homotopies of immersed spheres in R^3 that is attained under sense-preserving monotonic conditions.
Measuring Similarity Between Curves on 2-manifolds via Homotopy Area
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Minimal Homotopies in Three Dimensions: A Cable System Approach
A cable system defines an index agreeing with Brouwer degree on complementary regions, providing a sharp lower bound on swept volume for null homotopies of immersed spheres in R^3 that is attained under sense-preserving monotonic conditions.