Additive invariants for knots, links and graphs in $3$–manifolds

Abstract

We define two new families of invariants for ($3$–manifold, graph) pairs which detect the unknot and are additive under connected sum of pairs and ($-\frac{1}{2}$) additive under trivalent vertex sum of pairs. The first of these families is closely related to both bridge number and tunnel number. The second of these families is a variation and generalization of Gabai’s width for knots in the $3$–sphere. We give applications to the tunnel number and higher-genus bridge number of connected sums of knots.