We show that no difference between PL isotopy and TOP isotopy (as equivalence relations on PL links in S 3) can be detected by finite type invariants. Next, let L be a link with components K 1 ,. .. , Km, and let Q L denote either the Conway polynomial, or a certain exponential parametrization of the HOMFLY or the Kauffman two-variable polynomial, or any of certain 2 m−1 polynomials, in totalit...

Let $M_1$ and $M_2$ be closed connected orientable $3$-manifolds. We classify the sets of smooth piecewise linear isotopy classes embeddings $M_1\sqcup M_2\rightarrow S^6$.

One of Thurston’s important contributions to surface theory is his construction of measured lamination spaces. These spaces originally arose because their projectivizations form natural boundaries for Teichmüller spaces, but they are also of interest purely topologically. For example, they provide a nice global framework in which to view all the isotopy classes of simple closed curves in a surf...

Abstract We classify rational cuspidal curves of degrees 6 and 7 in the complex projective plane, up to symplectic isotopy. The proof uses topological tools, pseudoholomorphic techniques, birational transformations.

This work adresses the problem of the approximation of the normals of the offsets of general compact sets in euclidean spaces. It is proven that for general sampling conditions, it is possible to approximate the gradient vector field of the distance to general compact sets. These conditions involve the μ-reach of the compact set, a recently introduced notion of feature size. As a consequence, w...

In this paper we show how to construct an isomorphism between an alternative algebra A over a field of characteristic 6= 3 and its isotope A(1+c), where c is an element of Zhevlakov’s radical of A. This leads to the equivalence of any polynomial identity f = 0 in alternative algebras and the isotope identity f(s) = 0. Given an invertible element s of an alternative algebra A, we can form a new ...

We define combinatorial Floer homology of a transverse pair of noncontractibe nonisotopic embedded loops in an oriented 2-manifold without boundary, prove that it is invariant under isotopy, and prove that it is isomorphic to the original Lagrangian Floer homology.

Lagrangian spheres in the symplectic Del Pezzo surfaces arising as blow-ups of CP in 4 or fewer points are classified up to Lagrangian isotopy. Unlike the case of the 5-point blow-up, there is no Lagrangian knotting.

Let Bn denote the classical braid group on n strands and let the mixed braid group Bm,n be the subgroup of Bm+n comprising braids for which the first m strands form the identity braid. Let Bm,∞ = ∪nBm,n. We will describe explicit algebraic moves on Bm,∞ such that equivalence classes under these moves classify oriented links up to isotopy in a link complement or in a closed, connected, oriented ...