2-bridge θ-curves in S3

In [Math. Z. 61 (1954) 245], Schubert introduced an invariant of knots in the 3-sphere, called the bridge number, and Goda extended this invariant to θ-curves in [Topology Appl. 79 (1997) 177]. It is known that knots with bridge number 2 are prime. However we show in this paper that θ-curves with bridge number 2 are not always prime. Actually we give a necessary and sufficient condition for θ-curves to be nonprime with bridge number 2.