A generalized bridge number for links in 3-manifolds
Schubert showed that [bridge number --1] is an invariant of knots in the three-sphere which is additive when the connected sum of two knots is formed. We generalize bridge number to links in arbitrary three-manifolds and prove some cases of a similar theorem for these new invariants. Let \(L\) be a link in a closed orientable three-manifold \(M\) and \(T\) a Heegaard-surface of \(M\). Define \(L\) to be in \(n\)-bridge-position with respect to \(T\) if \(L\) intersects the closure of each component of \(M\backslash T\) in a collection of \(n\) trivial arcs. The genus-\(g\)- bridge-number of \(L\), \(b_ g(L)\), is the smallest integer \(n\) for which \(L\) can be represented in an \(n\)-bridge-representation with respect to some Heegaard-surface of \(M\) of genus \(g\). Theorem. Let \(L=L_ 1\#L_ 2\) be a composite link in \(M\). Given an integer \(g\geq 0\), there are integers \(g_ 1\), \(g_ 2\geq 0\) with \(g_ 1+g_ 2=g\) so that (a) if \(M=S^ 3\) and either \(g=0\) or \(g=1\) and \(L\) is connected, then \(b_ g(L_ 1\#L_ 2)=b_{g_ 1}(L_ 1)+b_{g_ 2}(L_ 2)-1\), (b) if \(M\) is an irreducible non-Haken manifold of genus \(g\), then \(b_{g_ 1}(L_ 1)+b_{g_ 2}(L_ 2)-2\leq b_ g(L_ 1\#L_ 2)\leq b_{g_ 1}(L_ 1)+b_{g_ 2}(L_ 2)-1\). For both inequalities, examples have been found where equality holds.
scientific article; zbMATH DE number 86215
scientific article