Summary: Immersed, VirtuallyEmbedded, Boundary Slopes.
M. Baker and D. Cooper \Lambda
March 7, 1997
For the figure eight knot, we show that slopes with even numerator are slopes of immersed
surfaces covered by incompressible, boundaryincompressible embeddings in some finite cover.
Definition 1.1 A slope on a torus, T; is the isotopy class of an essential unoriented simple closed
curve, ff; on T: Suppose that X is a three manifold with a boundary component which is a torus T:
An immersed boundary slope on T is a slope, ff; on T such that there is a proper immersion of a
compact, oriented, surface into X which is ß 1 injective and which is an embedding in a neighborhood
of the boundary of X: We also require that the surface cannot be homotoped into the boundary of
X by a proper homotopy. The boundary of the surface consists of loops on T parallel to ff: If the
immersion is an embedding then we also call the slope an embedded boundary slope. If the
immersion is covered by an embedding in some finite cover, then we also call the slope a virtually
embedded boundary slope.
It is known that a knot has only finitely many embedded boundary slopes, . It is easy to see
that a torus knot has only two immersed boundary slopes, and these are also embedded boundary
slopes. Several examples of immersed boundary slopes which are not embedded boundary slopes
have been constructed. Some immersed boundary slopes have been found for the figure eight knot,