Minimal Homotopies in Three Dimensions: A Cable System Approach

Abstract

We study null homotopies of immersed spheres in $\mathbb{R}^3$ and the volume they sweep during contraction. For a smooth immersion with finitely many transverse self-intersections, we introduce a cable system that connects each bounded region of the complement to the exterior. From this construction we define the cable index and prove that it agrees with the Brouwer degree on each complementary region. Using this identification, we derive a degree-weighted lower bound for the swept volume of any Lipschitz null homotopy. We show that the bound is attained whenever the homotopy is sense-preserving, meaning the surface moves in a consistent direction, and the index evolves monotonically along the homotopy. In addition, in the case where the immersion arises as the boundary of an immersed ball, we construct an explicit homotopy that realizes this lower bound via a deformation of the ball. Finally, we present a linear-time algorithm that computes all cable indices from a finite cable system, providing a concrete and computable method for evaluating the lower bound.