Remarks on Suzuki’s knot epimorphism number

Abstract

A partial order on prime knots can be defined by declaring [Formula: see text], if there exists an epimorphism from the knot group of [Formula: see text] onto the knot group of [Formula: see text]. Suppose that [Formula: see text] is a 2-bridge knot that is strictly greater than [Formula: see text] distinct, nontrivial knots. In this paper, we determine a lower bound on the crossing number of [Formula: see text] in terms of [Formula: see text]. Using this bound, we answer a question of Suzuki regarding the 2-bridge epimorphism number [Formula: see text] which is the maximum number of nontrivial knots which are strictly smaller than some 2-bridge knot with crossing number [Formula: see text]. We establish our results using techniques associated with parsings of a continued fraction expansion of the defining fraction of a 2-bridge knot.