##
**Special positions for essential tori in link complements.**
*(English)*
Zbl 0833.57004

Topology 33, No. 3, 525-556 (1994); erratum ibid. 37, No. 1, 225 (1998).

The paper is devoted to the study of essential (incompressible not boundary parallel) tori in the complement of a link in the 3-sphere. The basic setting is the representation of links as closed braids; such representations have been studied by the authors in a previous paper [Studying links via closed braids, cf. Part VI, Pac. J. Math. 156, No. 2, 265-286 (1992; Zbl 0739.57002)]. The techniques developed there are used and extended in the present paper and involve the study of certain induced foliations of embedded essential tori coming from the canonical foliation by disks of the complement of the axis of a closed braid. This results in three canonical types of embeddings of essential tori into the complement of a closed braid (which are best explained by pictures for which we refer to the introduction of the paper). The first main result then states that, starting with an arbitrary closed braid representation of a link, after modifying the closed braid by a finite sequence of “exchange moves”, each torus of the Jaco-Shalen-Johannson decomposition of the link complement is of one of these three types. As a consequence, the closed 3- and 4-braids admitting an essential torus can be exactly described. Next, for a given essential torus bounding a solid torus \(V\) in the 3-sphere, the relative local position in \(V\) of the intersection with \(V\) of the braid (link) and its axis are analysed. An application is a new proof and a generalization of a theorem of Schubert about the braid index of satellite links. Finally, the type of the essential torus associated to a doubled knot is described.

Reviewer: B.Zimmermann (Trieste)

### MSC:

57M25 | Knots and links in the \(3\)-sphere (MSC2010) |

57N10 | Topology of general \(3\)-manifolds (MSC2010) |