Small knots and large handle additions (Q5895028)

The authors construct knots \(K\) in a genus 3 handlebody \(H\) with the following properties: 1) \(K\) is small and simple, that is, the exterior of \(K\), \(H_{K}\), admits no closed essential surface, essential disk or essential annulus. 2) For every even integer \(g\), there are infinitely many non-separating slopes \(r\) on \(\partial H\) so that after adding a 2-handle to \(H_K\) the manifold obtained is small and contains an essential surface of genus \(g\). Condition~(1) alone is highly non-trivial as small knots (such as 2-bridge and torus knots) are very hard to construct, in general. By Thruston's work, condition (1) implies that after removing the torus boundary component from \(H_K\) it admits a complete hyperbolic metric with \(\partial H\) totally geodesic. The authors cite well known results about handles addition to manifold with boundary torus, or when \(g=0,1\), that contrast sharply with their construction.