Knots with unknotting number 1 and essential Conway spheres

Abstract

First we clarify the extension of the main result of [11] to knots in solid tori that is described in the Appendix of [11]. In Section 3, we define a family of hyperbolic knots J " .`; m/ in a solid torus, " 2 f1; 2g and`; m integers, each of which admits a half-integer surgery yielding a toroidal manifold. (The knots J " .`; m/ in the solid torus are the analogs of the knots k.`; m; n; p/ in the 3-sphere.) We then use [11] to show that these are the only such:Theorem 4.2 Let J be a knot in a solid torus whose exterior is irreducible and atoroidal. Let be the meridian of J and suppose that J. / contains an essential torus for some with . ; / 2. Then . ; / D 2 and J D J " .`; m/ for some ";`; m.This theorem along with the main result of [11] then allows us to describe in Theorem 5.2 the relationship between the torus decomposition of the exterior of a knot K in S 3 and the torus decompositon of any non-integral surgery on K . In particular, Theorem 5.2 says that the canonical tori of the exterior of K and of the Dehn surgery will be the same unless K is a cable knot (in which case an essential torus of the knot exterior can become compressible), or K is a k.`; m; n; p/, or K is a satellite with pattern J " .`; m/ (in the latter cases a new essential torus is created).We apply these theorems about non-integral Dehn surgeries to address questions about unknotting number. The knots k.`;m;n;p/ (J " .`; m/) are strongly invertible. Their quotients under the involutions give rise to EM-knots, K.`;m;n;p/ (EM-tangles, A " .`; m/ resp.) which have essential Conway spheres and yet can be unknotted (trivialized, resp.) by a single crossing change. Theorem 6.2 descibes when a knot with an essential Conway sphere or 2-torus can have unknotting number 1. This is naturally stated in the context of the characteristic decomposition of a knot along

The rational Witt class and the unknotting number of a knot

“…We see that each of the three cases we just discussed, leads to formula (5), thus proving the next proposition. Proposition 3.2.…”

“…The last three decades have furnished an impressive array of tools for studying unknotting numbers, tools stemming from varied sources such as gauge theory [3,21,20], polynomial knot invariants [22], linking forms [16] and 3-manifold theory [5]. In this article we propose to add yet another tool to this list by using the rational Witt class ϕ(K) to extract information about u(K).…”

Pretzel knots with unknotting number one

“…A complete treatment of algebraic knots can be found in [7,25], but in brief, the distinct types are 2-bridge, large algebraic, and Montesinos length three, with the characterization being split according to the topology of their double covers. To wit, we have the following division.…”

Heegaard splittings of knot exteriors

“…However computing the crossing number C(K) of an arbitrary knot is in general very difficult. For example, computing the unknotting number for all knots with 10 or fewer crossings has only recently been accomplished by Gordon and Luecke [14] and has required work of Ozsvath-Szabo [62].…”