Prime knot complements with meridional essential surfaces of arbitrarily high genus

Abstract

Abstract. We show the existence of infinitely many prime knots each of which having in their complements meridional essential surfaces with two boundary components and arbitrarily high genus.

Knot exteriors with all possible meridional essential surfaces

“…Let J be a prime knot as in the main theorem of [17], that is with meridional essential surfaces of any positive genus and two boundary components. The knot J is obtained by identifying the boundaries of two particular solid tori, say H 1 and H 2 , attaching meridian to longitude, and by identifying the boundaries of the respective essential arc each contains.…”

“…Besides being prime, the knots h(K) can be decomposed into two essential arcs by surfaces of genus higher than zero keeping the properties of Theorem 1 in [17] used in their construction. Proposition 6.…”

“…In this section we use the satellite operation described in Definition 3 and the knots from the main theorem of [17] to prove Theorem 1. First, we start with the following proposition where we show that for any knot with a meridional essential surface in its exterior there is a knot with meridional essential surfaces of the same genus and with an unlimited number of boundary components.…”

“…Other examples were later obtained, for instance, by Oertel [19], and, more recently, by Li [13] or by Eudave-Muñoz and Neumann-Coto [5]. Similarly, the author proved in [17] the existence of prime knot exteriors such that each contains meridional essential surfaces with two boundary components and arbitrarily high genus. On the other hand, one might wonder if the unbounded Euler characteristics of essential surfaces in a knot exterior can be from the number of boundary components instead of the genus.…”

“…For the construction we use satellite knots together with handlebody-knots of genus two. In section 3 we show a process to obtain knot exteriors with meridional essential surfaces of arbitrarily many boundary components as in Proposition 7, and use the knots from the main theorem of [17] to prove Theorem 1. The main methods are classical in 3-manifold topology; we use innermost curve arguments and branched surface theory.…”