Knot theory and 3-manifold topology are closely intertwined, and few invariants stand so firmly in the intersection of these two subjects as the tunnel number of a knot, denoted t(K). We describe two very general constructions that result in knot and link pairs which are subbaditive with respect to tunnel number under connect sum. Our constructions encompass all previously known examples and introduce many new ones. As an application we describe a class of knots K in the 3-sphere such that, for every manifold M obtained from an integral Dehn filling of E(K), g(E(K))>g(M).

This dissertation is on a study of 2-string free tangle decompositions of knots with tunnel number two. As an application, we construct infinitely many counter-examples to a conjecture in the literature stating that the tunnel number of the connected sum of prime knots doesn't degenerate by more than one: t(K_1#K_2)≥ t(K_1)+t(K_2)-1, for K_1 and K_2 prime knots. We also study 2-string free tangle decompositions of links with tunnel number two and obtain an equivalent statement to the one on knots. Further observations on tunnel number and essential tangle decompositions are also made. / text