Alternating knots satisfy strong property P

Abstract

Abstract. Suppose a manifold is produced by finite Dehn surgery on a non-torus alternating knot for which Seifert's algorithm produces a checkerboard surface. By demonstrating that it contains an essential lamination, we prove that such a manifold has R 3 as universal cover and, consequently, is irreducible and has infinite fundamental group. Together with previous work of Roberts, who proved this result in the case of alternating knots for which Seifert's algorithm does not produce a checkerboard surface, and Moser, who classified the manifolds produced by surgery on torus knots, this paper completes the proof that alternating knots satisfy Strong Property P.Mathematics Subject Classification (1991). Primary 57M25; Secondary 57R30.

Exceptional surgeries on alternating knots

“…In an unpublished preprint [7], Delman gave a construction of essential lamination in a Montesinos knot exterior. See [8] for a part of his construction. Actually if the Montesinos knot we consider has the form M (1/p, r 1 , r 2 ) with p and all the denominators of r i 's are all odd, then the construction given in [8] can be applied in our case.…”

“…See [8] for a part of his construction. Actually if the Montesinos knot we consider has the form M (1/p, r 1 , r 2 ) with p and all the denominators of r i 's are all odd, then the construction given in [8] can be applied in our case. Also see [6] for prototype of the construction in [7].…”

Lower Bounds on Boundary Slope Diameters for Montesinos Knots

“…In fact, these are known to be essential by [1], [5]. Let R 1 and R 2 denote the boundary slopes of F 1 and F 2 .…”

All exceptional surgeries on alternating knots are integral surgeries

“…See Delman-Roberts [8] for a part of his construction. Actually if the Montesinos knot we consider has the form M.1=p; r 1 ; r 2 / with p and all the denominators of the r i are odd, then the construction given in [8] can be applied in our case. Also see Delman [6] for prototype of the construction in [7].…”