Erik López-García

Instituto de Matemáticas, Unidad Cuernavaca,
Universidad Nacional Autónoma de México,
Av. Universidad s/n, Col. Lomas de Chamilpa,
Cuernavaca, Mor. 62210, MEXICO.

Title: Knots with infinitely many closed essential surfaces

Abstract: Let $K$ be a knot in $S^{3}$. A meridional surface for $K$ is a properly embedded surface $F$ in the exterior of $K$ whose boundary consists of meridians of $K$. The surface $F$ is essential if it is incompressible and is not a boundary parallel annulus; and $F$ is meridionally compressible if there exist an annulus that connects a (non-boundary parallel) curve in the surface to a meridian of $K$. We show that if $K$ is a knot in $S^{3}$ that admits two disjoint, essential meridional surfaces $F_{1}$ and $F_{2}$, and $F_{1}$ is meridionally compressible, then $K$ admits infinitely many closed, essential surfaces, unless the meridian annulus for $F_{1}$ is ``centered'' with respect to $F_{2}$. This implies for example that any non-satellite knot that admits an essential 4-puntured sphere admits infinitely many closed essential surfaces.