Crosscap number and epimorphisms of two-bridge knot groups

Abstract

We consider the relationship between the crosscap number [Formula: see text] of knots and a partial order on the set of all prime knots, which is defined as follows. For two knots [Formula: see text] and [Formula: see text], we say [Formula: see text] if there exists an epimorphism [Formula: see text]. We prove that if [Formula: see text] and [Formula: see text] are 2-bridge knots and [Formula: see text], then [Formula: see text]. We also classify all pairs [Formula: see text] for which the inequality is sharp. A similar result relating the genera of two knots has been proven by Suzuki and Tran. Namely, if [Formula: see text] and [Formula: see text] are 2-bridge knots and [Formula: see text], then [Formula: see text], where [Formula: see text] denotes the genus of the knot [Formula: see text].