Boundary slopes of knots

Abstract

Mathematics Subject Classification (1991). Primary 57M40, 57M25; Secondary 57M50, 57N10. Abstract.The results in this paper show that simple connectivity of a 3-manifold is reflected in the behavior of essential surfaces in exteriors of knots in the manifold. A corollary of the main theorem is that any non-trivial knot, with irreducible complement, in a homotopy 3-sphere must have two boundary slopes that differ by at least 2. This statement is false for knots in a homology 3-sphere. The main theorem itself applies more generally to knots in closed orientable 3-manifolds with cyclic fundamental group.

Cyclic surgery, degrees of maps of character curves, and volume rigidity for hyperbolic manifolds

“…Since this is not quite explicit there, let me elaborate. In the notation of [CS3], let v 1 and v 2 be the vertices of B which are the ends of the edge containing µ. Then µ = tv 1 + (1 − t)v 2 for some t ∈ [0, 1], and they show that the diameter of the set of boundary slopes is bounded below by 1/(2t(t − 1)) ≥ 2.…”

The diameter of the set of boundary slopes of a knot

“…We show that either (i) d K 2 or (ii) K is a generalized iterated torus knot. The proof combines results from Culler and Shalen [3] with a result about the effect of cabling on boundary slopes. 57M15, 57M25; 57M50…”

“…Theorem B is proved by combining one of the results of [3] with another new result, Theorem A, which asserts that a q -strand cabling of a knot increases the invariant d K by a factor of at least q 2 .…”

“…(For further discussion of this connection, see Shalen [11].) It was shown in Culler and Shalen [3] that if 1 . †/ is trivial and the knot K is nontrivial, then d K 2.…”

Lower Bounds on Boundary Slope Diameters for Montesinos Knots

“…Consider a compact possibly non-orientable surface properly embedded in a knot exterior in the 3-sphere S 3 . It is called essential if it is incompressible and boundary-incompressible.…”