Minimal Homotopies in Three Dimensions: A Cable System Approach

TL;DR

This paper introduces a cable system approach to analyze null homotopies of immersed spheres in 3D, establishing a lower volume bound and providing an efficient algorithm for its computation.

Contribution

It defines the cable index, proves its relation to Brouwer degree, and constructs explicit homotopies that attain volume bounds, along with a linear-time computation method.

Findings

Cable index matches Brouwer degree on each region.

Derived a degree-weighted lower bound for swept volume.

Provided a linear-time algorithm to compute cable indices.

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.