We compute the height of the third Stiefel--Whitney characteristic class of the canonical bundles over some infinite classes of Grassmann manifolds of five dimensional vector subspaces of real vector spaces.
In this short note we give an elementary combinatorial argument, showing that the conjecture of J. Fernández de Bobadilla, I. Luengo-Velasco, A. Melle-Hernández and A. Némethi [Proc. London Math. Soc. 92 (2006), 99-138, Conjecture 1] follows from Theorem 5.4 of Brodzik and Livingston [arXiv:1304.1062] in the case of rational cuspidal curves with two critical points.
Let K (resp. L) be a Montesinos knot (resp. link) with at least four branches. Then we show the unknotting number (resp. unlinking number) of K (resp. L) is greater than 1.
The Polish space Y constructed in [vM1] admits no nontrivial isotopy. Yet, there exists a Polish group that acts transitively on Y.
Heegaard splittings and Heegaard diagrams of a closed 3-manifold are translated into the language of Morse functions with Morse-Smale pseudo-gradients defined on . We make use in a very simple setting of techniques which Jean Cerf developed for solving a famous pseudo-isotopy problem. In passing, we show how to cancel the supernumerary local extrema in a generic path of functions when . The main tool that we introduce is an elementary swallow tail lemma which could be useful elsewhere.
In this paper we are interested in symmetries of alternating knots, more precisely in those related to achirality. We call the following statement Tait’s Conjecture on alternating achiral knots:Let be a prime alternating achiral knot. Then there exists a minimal projection of in and an involution such that:1) reverses the orientation of ;2) ;3) ;4) has two fixed points on and hence reverses the orientation of .The purpose of this paper is to prove this statement.For the historical...