2011/06/24 by Ekaterina Pervova, Pervova, Ekaterina, Carlo Petronio +3
Computer Science · Mathematics · #57M15 (secondary) #57M20 #57M25 #57M27 (primary) #Advanced Combinatorial Mathematics #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.1106.4952
openalex publication_date 2011/06/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.