Heegaard genera of high distance are additive under annulus sum

Abstract

Let M i be a compact orientable 3-manifold, and A i a non-separating incompressible annulus on ∂ M i , i = 1, 2. Let h : A 1 → A 2 be a homeomorphism, and M = M 1 ∪ h M 2 the annulus sum of M 1 and M 2 along A 1 and A 2 . In the present paper, we show that if M i has a Heegaard splitting V i ∪ S i W i with distance d(S i ) 2g(M i ) + 3 for i = 1, 2, then g(M) = g(M 1 ) + g(M 2 ). Moreover, if g(F i ) 2, i = 1, 2, then the minimal Heegaard splitting of M is unique.

Some Sufficient Conditions for Heegaard Genera to Be Additive Under Annulus Sum

“…And with the weaker assumptions, we obtain the stronger conclusion that the minimal Heegaard splitting of M is in some sense unique. Hence we remark that the situations here are quite different from those in [2], [8] and [17], and the arguments there are not applicable to the main cases here.…”

“…Lemma 2.9 ( [8], [17]). Let N be a compact orientable 3-manifold which is not a compression body, and F = ∂N.…”

“…Some sufficient conditions for tunnel number of knots not to go down under connected sum are given in [17]. When both A 1 and A 2 are non-separating on the corresponding boundary component, there are some sufficient conditions for the equality g(M) = g(M 1 ) + g(M 2 ) to hold, see [2], [8].…”

“…Note that papers [8] and [17] only consider the case that both A 1 and A 2 are non-separating. In [2], the bound of the Heegaard distance for the additivity to hold is "d(S i ) ≥ 2(g(M 1 ) + g(M 2 )), i = 1, 2", in other words, the bound of d(S 1 ) (d(S 2 ), resp.)…”

Equivariant Heegaard genus of reducible 3-manifolds

“…The first examples of pairs of knots producing super-additivity of tunnel number were given in [19,20]. Setting w = ∞ produces other examples, using the same method as in [14,41].…”