, which is dened as follows. Let be a closed essential (i.e., incom-pressible and not @ -parallel) surface in a knot exterior. We call accidental if it con-tains a non-trivial loop which is isotopic into the peripher al torus of the knot. Thereare some motivations to study the accidental surface from th e topological or the geo-metrical viewpoint. For example, it is known that accidenta l surfaces in a hyperbolicknot complement have a particular geometric behavior [11]. See [7] for more detail.First, we will consider the accidental slope for accidental surfaces. A slope on theperipheral torus of a knot is determined by the isotopy from a non-trivial loop on anaccidental surface into the torus. It is shown in [7, Theorem 1] that the slope is inde-pendent of the choice of the non-trivial loop on the surface. Hence we call this slopethe accidental slope for the accidental surface. In contrast, the accidental slo pe is notdetermined uniquely for a knot. In fact, an example of a knot a dmitting two accidentalsurfaces with accidental slopes 0 and 1 was given in [7, Figure 1]. We know that anyaccidental slope is integral or meridional [1, Lemma 2.5.3] , and the example showsthat there is a knot with integral and meridional accidental slopes. Hence, it is naturalto ask how many integral accidental slopes exist for a knot. In this paper, we give abound of the minimal intersection number of accidental slop es.Theorem 3.2. Let