Bridge spectra of iterated torus knots

Abstract

We define the bridge spectrum b(K) of a knot K in S3 to be b(K) = (b0(K), b1(K),...), where bg(K) is the bridge number of K with respect to a genus g Heegaard surface for S3. A wellknown construction shows that when bg-1(K) is positive, bg(K) ≤ bg-1(K) - 1; hence, we say that b(K) has a gap at index g if this inequality is strict. An open question of Yo'av Rieck asks whether there are knots in S3 whose bridge spectra have more than one gap. We determine the bridge spectra of many iterated torus knots, answering Rieck's question by showing that for every n, there are infinitely many iterated torus knots K, such that b(K) has precisely n gaps. In addition, we prove a structural lemma about the decomposition of a strongly irreducible bridge surface induced by cutting along a collection of essential surfaces.