of a $k$-book is identified with the $z$-axis, for convention we say that the spine is oriented upwards. We can then describe the orientation of the arcs embedded into the $k$-book as being [*upwards*]{} or [*downwards*]{}, relative to the spine. A [*$k$-page oriented book embedding*]{} for an oriented graph $\overrightarrow{D}$ is an embedding into a $k$-book such that:

-   each vertex $v \in V(\overrightarrow{D})$ is embedded into $L$;

-   each arc $a \in A(\overrightarrow{D})$ is either completely embedded into $L$, or all points of $a$ (except the endpoints) are embedded into the interior of a single page.

-   the orientation of all arcs embedded into $L$ agree;

-   the orientation of all arcs embedded into a given page agree. (We call this restriction the [*direction rule*]{}.)

If all arcs embedded into a single page are upwards, we refer to the page as an *upwards page*, or simply say the *page is upwards*; similarly if all arcs embedded into a single page are downwards, we refer to the page as an *downwards page*, or simply say the *page is downwards*. We do not restrict the direction of all arcs in the oriented book embedding to be the same, just those in 