This paper gives a partial description of the homotopy type of K, the space of long knots in R3. The primary result is the construction of a homotopy equivalence K â‰ƒ C2(P âŠ” {âˆ—}) where C2(P âŠ” {âˆ—}) isâ€¦ (More)

In this paper we construct a faithful representation of the mapping class group of the genus two surface into a group of matrices over the complex numbers. Our starting point is the Lawrence-Krammerâ€¦ (More)

We study Emb(Sj,Sn) the space of Câˆž -smooth embeddings of spheres in spheres, Kn,j the space of â€˜longâ€™ embeddings of Rj in Rn , and spaces of embeddings of spheres in euclidean space Emb(Sj, Rn), andâ€¦ (More)

This paper gives a detailed description of the JSJ-decompositions of knot complements in S3 . Formulated in the language of trees, the result is the construction of a bijective correspondence betweenâ€¦ (More)

The musical realm is a promising area in which to expect to find nontrivial topological structures. This paper describes several kinds of metrics on musical data, and explores the implications ofâ€¦ (More)

This paper is a computation of the homotopy type of K , the space of long knots in R the same space of knots studied by Vassiliev via singularity theory. Each component of K corresponds to an isotopyâ€¦ (More)

A non-singular sesquilinear form is constructed that is preserved by the Lawrence-Krammer representation. It is shown that if the polynomial variables q and t of the Lawrence-Krammer representationâ€¦ (More)

This is a collection of notes on embedding problems for 3-manifolds. The main question explored is â€˜which 3-manifolds embed smoothly in S4 ?â€™ The terrain of exploration is theâ€¦ (More)

Consider the space of 'long knots' in R n K n,1. This is the space of knots as studied by V. Vassiliev. Based on previous work [5, 12], it follows that the rational homology of K 3,1 is freeâ€¦ (More)

Abstract The operad of framed little discs is shown to be a cyclic operad. This answers a conjecture of Salvatore in the affirmative, posed at the workshop â€˜Knots and Operads in Roma,â€™ at UniversitÃ â€¦ (More)