Bridge spectra of iterated torus knots

We define the bridge spectrum $\mathbf{b}(K)$ of a knot $K$ in $S^3$ to be $\mathbf{b}(K) = (b_0(K), b_1(K), \dots)$, where $b_g(K)$ is the bridge number of $K$ with respect to a genus $g$ Heegaard surface for $S^3$. A well-known construction shows that when $b_{g-1}(K)$ is positive, $b_g(K) \lt b_{g-1}(K) - 1$; hence, we say that $\mathbf{b}(K)$ has a gap at index $g$ if this inequality is strict. An open question of Yo’av Rieck asks whether there are knots in $S^3$ 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 $\mathbf{b}(K)$ 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.

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Published 29 October 2014