Primitive, proper power, and Seifert curves in the boundary of a genus two handlebody

Abstract

A simple closed curve $\alpha$ in the boundary of a genus two handlebody $H$ is primitive if adding a 2-handle to $H$ along $\alpha$ yields a solid torus. If adding a 2-handle to $H$ along $\alpha$ yields a Seifert-fibered space and not a solid torus, the curve is called Seifert. If $\alpha$ is disjoint from an essential separating disk in $H$, does not bound a disk in $H$, and is not primitive in $H$, then it is said to be proper power. As one of the background papers of the classification project of hyperbolic primitive/Seifert knots in $S^3$ whose complete list is given in [BK20], this paper classifies in terms of R-R diagrams primitive, proper power, and Seifert curves. In other words, we provide up to equivalence all possible R-R diagrams of such curves. Furthermore, we further classify all possible R-R diagrams of proper power curves with respect to an arbitrary complete set of cutting disks of a genus two handlebody.

29 pages, 30 figures

References in corpus (1)

• Some knots with surgeries yielding lens spaces

Cited by in corpus (3)

• On nonhyperbolicity of P/P and P/SF knots in $S^3$
• Tunnel-number-one knot exteriors in $S^3$ disjoint from proper power curves
• Hyperbolic tunnel-number-one knots with Seifert-fibered Dehn surgeries