Regular ${\mathbf{O}}\left( n \right)$-manifolds, suspension of knots, and knot periodicity

REGULAR O (n)-MANIFOLDS, SUSPENSION OF KNOTS, AND KNOT PERIODICITY

1. Statement of the main results. It will be convenient for us to define an n-knot to be a smooth, connected, oriented, n-dimensional (closed) submanifold IP of S (oriented). If X" is homeomorphic to S, then we call it a spherical knot. All manifolds in this note will be oriented and all constructions we consider will induce canonical orientations. This will be understood and not commented upon...

Cyclic Suspension of Knots and Periodicity of Signature for Singularities

By a knot we mean a pair (S, M~) with M~ a smooth closed oriented submanifold of S (m^3). If such a knot is given and i:S-> gm+z j s t j ^ standard embedding, then one can isotope / in an essentially unique way (Lemma 1 below) to an embedding j :s -+S whose intersection with iS is M c S transversally. The «-fold cyclic branched cover of (S, iS) branched along (jS, M~) exists uniquely and is a m...

Combinatorial Knot Contact Homology and Transverse Knots

We give a combinatorial treatment of transverse homology, a new invariant of transverse knots that is an extension of knot contact homology. The theory comes in several flavors, including one that is an invariant of topological knots and produces a three-variable knot polynomial related to the A-polynomial. We provide a number of computations of transverse homology that demonstrate its effectiv...

Link Manifolds and Periodicity 571

Periodicity of Goussarov-vassiliev Knot Invariants

The paper is a survey of known periodicity properties of finite type invariants of knots, and their applications.