The notion of equivalence of knots can be strengthened by saying that K and K' are isotopic if the above homeomorphism h is isotopic to the identity, or equivalently, orientation-preserving. The analog of Theorem 1 holds in this setting too: if two knots have complements which are homeomorphic by an orientation-preserving homeomorphism, then they are isotopic. Theorem 1 and its orientation-pres...

We study those Artin groups which, modulo their centers, are finite index subgroups of the mapping class group of a punctured sphere. In particular, we show that any injective homomorphism between these groups is parameterized by a homeomorphism of a punctured disk together with a homomorphism to the integers. The technique, following Ivanov, is to prove that every superinjective map of the com...

Given an orientation-preserving homeomorphism of the plane, a rotation number can be associated with each locally attracting fixed point. Assuming that the homeomorphism is dissipative and the rotation number vanishes we prove the existence of a second fixed point. The main tools in the proof are Carathéodory prime ends and fixed point index. The result is applicable to some concrete problems i...

We study full groups of minimal actions of countable groups by homeomorphisms on a Cantor space X, showing that these groups do not admit a compatible Polish group topology and, in the case of Z-actions, are coanalytic nonBorel inside Homeo(X). We point out that the full group of a minimal homeomorphism is topologically simple. We also study some properties of the closure of the full group of a...