Incompressible surfaces in tunnel number one knot complements (Q1807578)

An important problem in knot theory is that of determining all incompressible surfaces in a given knot complement. The author studies this problem for the case of tunnel number one knot complements. A knot \(K\) in the standard 3-sphere \(\mathbb{S}^3\) has tunnel number one if there exists an arc \(\tau\) embedded in \(\mathbb{S}^3\) such that \(K\cap\tau =\partial \tau\) and the complement in \(\mathbb{S}^3\) of an open regular neighborhood of \(K\cup\tau\) is a genus 2 handlebody. The following question was settled by \textit{C. McA. Gordon} and \textit{A. W. Reid} in [J. Knot Theory Ramifications 4, No.~3, 389-409 (1995; Zbl 0841.57012)]: Can the complement of a tunnel number one knot in \(\mathbb{S}^3\) contain a closed incompressible surface of genus \(\geq 2\)? The author gives an affirmative answer to the question above. More precisely, he constructs for any integer \(g\geq 2\) infinitely many tunnel number one knots whose complements contain a closed incompressible surface of genus \(g\).