Fibred torti-rational knots

TL;DR

This paper investigates fibred properties and genus of fibred torti-rational knots, providing algebraic and geometric methods to determine these features and constructing explicit minimal genus Seifert surfaces.

Contribution

It introduces new algebraic and geometric techniques to determine the genus and fibredness of torti-rational knots using continued fractions, and explicitly constructs minimal Seifert surfaces.

Findings

Most cases' genus and fibredness are determined by Alexander polynomials.

Explicit minimal genus Seifert surfaces are constructed.

Results apply to satellite knots of tunnel number one.

Abstract

A torti-rational knot, denoted by K(2a,b|r), is a knot obtained from the 2-bridge link B(2a,b) by applying Dehn twists an arbitrary number of times, r, along one component of B(2a,b). We determine the genus of K(2a,b|r) and solve a question of when K(2a,b|r) is fibred. In most cases, the Alexander polynomials determine the genus and fibredness of these knots. We develop both algebraic and geometric techniques to describe the genus and fibredness by means of continued fraction expansions of b/2a. Then, we explicitly construct minimal genus Seifert surfaces. As an application, we solve the same question for the satellite knots of tunnel number one.