Notes on the complexity of 3-valent graphs in 3-manifolds

TL;DR

This paper investigates the complexity of 3-valent graphs in 3-manifolds, revealing conditions under which the complexity remains additive during connected sums and characterizing the behavior for knots.

Contribution

It proves that edges intersecting spheres can be canceled without changing complexity and characterizes when complexity is additive under connected sums along graphs.

Findings

Edges intersecting spheres can be canceled without affecting complexity

Complexity is additive under connected sum along graphs under specific conditions

Additive functions for knots depend only on the ambient manifold

Abstract

A theory of complexity for pairs (M,G) with M an arbitrary closed 3-manifold and G a 3-valent graph in M was introduced by the first two named authors, extending the original notion due to Matveev. The complexity c is known to be always additive under connected sum away from the graphs, but not always under connected sum along (unknotted) arcs of the graphs. In this article we prove the slightly surprising fact that if in M there is a sphere intersecting G transversely at one point, and this point belongs to an edge e of G, then e can be canceled from G without affecting the complexity. Using this fact we completely characterize the circumstances under which complexity is additive under connected sum along graphs. For the set of pairs (M,K) with K a knot in M, we also prove that any function that is fully additive under connected sum along knots is actually a function of the ambient manifold only.