On spatial theta-curves with the same $(Z_2\oplus Z_2)$-fold and $2$-fold branched covering

Abstract

In this note, we study two types of spatial theta-curves having two $(1,1)$-knotswhose each has two $(1,1)$-knotsand a trivial knot or two trivial knots and a $2$-bridge knot as constituent knots. Weshow that there is a $3$-manifold $M$ such that $M$ is the $(Z_2\oplus Z_2)$-fold and $2$-fold covering of $S^3$ branched over each type of spatial theta-curve. Furthermore, weinvestigate certain relations between the spatial theta-curves and between the closed $3$-manifolds which are coverings of $S^3$ branched over them.