Notes on the complexity of 3-valent graphs in 3-manifolds (Q2890231)

In [J. Knot Theory Ramifications 18, No. 10, 1439--1458 (2009; Zbl 1184.57011)], \textit{E. Pervova} and \textit{C. Petronio} introduced a theory of complexity for pairs \((M,G)\), \(M\) being a 3-manifold and \(G \subset M\) a 3-valent graph, which extends the classical one for compact 3-manifolds due to \textit{S. V. Matveev} [Acta Appl. Math. 19, No. 2, 101--130 (1990; Zbl 0724.57012)].NEWLINENEWLINEIn [\textit{C. Petronio}, Topology Appl. 153, No. 11, 1658--1681 (2006; Zbl 1114.57019)], the complexity function was proved to be additive under connected sum away from the graph; however, the additivity does not always hold under connected sum along unknotted arcs of the graphs (as the quoted paper by Pervova and Petronio shows).NEWLINENEWLINEThe present paper gives an explicit characterization of the circumstances under which complexity is additive under connected sum along graphs; as a basic fact, the authors prove that if an edge \(e\) of a graph \(G \subset M\) intersects transversely at a point a sphere that meets \(G\) at a point only, then \(e\) can be canceled without affecting the complexity of \((M,G)\).NEWLINENEWLINEMoreover, the particular case of pairs \((M,K)\), \(K \subset M\) being a knot, is taken into account: any function that is fully additive under connected sum along knots turns out to be a function of the ambient manifold only.