The Least Spanning Area of a Knot and the Optimal Bounding Chain Problem

TL;DR

This paper demonstrates that finding the least area surface bounded by a knot in certain 3-manifolds can be done in polynomial time using linear programming, contrasting with NP-completeness in the general case.

Contribution

The paper introduces a polynomial-time method for computing the least area surface in specific 3-manifolds using linear programming, and analyzes the complexity of related problems.

Findings

Polynomial-time algorithm for least area surface in certain 3-manifolds.

NP-completeness of the Optimal Homologous Chain Problem with integer coefficients.

Conditions under which the Optimal Bounding Chain Problem is solvable in polynomial time.

Abstract

Two fundamental objects in knot theory are the minimal genus surface and the least area surface bounded by a knot in a 3-dimensional manifold. When the knot is embedded in a general 3-manifold, the problems of finding these surfaces were shown to be NP-complete and NP-hard respectively. However, there is evidence that the special case when the ambient manifold is R^3, or more generally when the second homology is trivial, should be considerably more tractable. Indeed, we show here that a natural discrete version of the least area surface can be found in polynomial time. The precise setting is that the knot is a 1-dimensional subcomplex of a triangulation of the ambient 3-manifold. The main tool we use is a linear programming formulation of the Optimal Bounding Chain Problem (OBCP), where one is required to find the smallest norm chain with a given boundary. While the decision variant of OBCP is NP-complete in general, we give conditions under which it can be solved in polynomial time. We then show that the least area surface can be constructed from the optimal bounding chain using a standard desingularization argument from 3-dimensional topology. We also prove that the related Optimal Homologous Chain Problem is NP-complete for homology with integer coefficients, complementing the corresponding result of Chen and Freedman for mod 2 homology.