On the image of the Lawrence-Krammer representation

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 representation are chosen to be appropriate algebraically independant unit complex numbers, then the form is negative-definite Hermitian. Since unitary matrices diagonalize, the conjugacy class of a matrix in the unitary group is determined by its eigenvalues. It is shown that the eigenvalues of a Lawrence-Krammer matrix satisfy some symmetry relations. Using the fact that non-invertible knots exist, the symmetry relations imply that there are matrices in the image of the Lawrence-Krammer representation that are conjugate in the unitary group, yet the braids that they correspond to are not conjugate. The two primary tools involved in constructing the form are Bigelow's interpretation of the Lawrence-Krammer representation, together with Morse theory on manifolds with corners.Comment: 17 pages, 11 figure

Topology of spaces of knots in dimension 3

This paper is a computation of the homotopy type of K, the space of long knots in R^3, the same space of knots studied by Vassiliev via singularity theory. Each component of K corresponds to an isotopy class of long knot, and we `enumerate' the components via the companionship trees associated to the knot. The knots with the simplest companionship trees are: the unknot, torus knots, and hyperbolic knots. The homotopy-type of these components of K were computed by Hatcher. In the case the companionship tree has height, we give a fibre-bundle description of those components of K, recursively, in terms of the homotopy types of `simpler' components of K, in the sense that they correspond to knots with shorter companionship trees. The primary case studied in this paper is the case of a knot which has a hyperbolic manifold contained in the JSJ-decomposition of its complement.Comment: 22 pages, 16 figure

An operad for splicing

A new topological operad is introduced, called the splicing operad. This operad acts on a broad class of spaces of self-embeddings N --> N where N is a manifold. The action of this operad on EC(j,M) (self embeddings R^j x M --> R^j x M with support in I^j x M) is an extension of the action of the operad of (j+1)-cubes on this space. Moreover the action of the splicing operad encodes Larry Siebenmann's splicing construction for knots in S^3 in the j=1, M=D^2 case. The space of long knots in R^3 (denoted K_{3,1}) was shown to be a free 2-cubes object with free generating subspace P, the subspace of long knots that are prime with respect to the connect-sum operation. One of the main results of this paper is that K_{3,1} is free with respect to the splicing operad action, but the free generating space is the much `smaller' space of torus and hyperbolic knots TH \subset K_{3,1}. Moreover, the splicing operad for K_{3,1} has a `simple' homotopy-type as an operad.Comment: 34 pages, 16 diagrams. V3->V4: Explicit definition of sigma^* \wr G-operads given, and proof included splicing operad is such. Re-wrote proof of Theorem 5.13 to make the equivariance of the maps more explicit. Cut down on extraneous notation. Fixed references, some small typo

A family of embedding spaces

Let Emb(S^j,S^n) denote the space of C^infty-smooth embeddings of the j-sphere in the n-sphere. This paper considers homotopy-theoretic properties of the family of spaces Emb(S^j,S^n) for n >= j > 0. There is a homotopy-equivalence of Emb(S^j,S^n) with SO_{n+1} times_{SO_{n-j}} K_{n,j} where K_{n,j} is the space of embeddings of R^j in R^n which are standard outside of a ball. The main results of this paper are that K_{n,j} is (2n-3j-4)-connected, the computation of pi_{2n-3j-3} (K_{n,j}) together with a geometric interpretation of the generators. A graphing construction Omega K_{n-1,j-1} --> K_{n,j} is shown to induce an epimorphism on homotopy groups up to dimension 2n-2j-5. This gives a new proof of Haefliger's theorem that pi_0 (Emb(S^j,S^n)) is a group for n-j>2. The proof given is analogous to the proof that the braid group has inverses. Relationship between the graphing construction and actions of operads of cubes on embedding spaces are developed. The paper ends with a brief survey of what is known about the spaces K_{n,j}, focusing on issues related to iterated loop-space structures.Comment: This is the version published by Geometry & Topology Monographs on 22 February 200

On the homology of the space of knots

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 of the authors, it follows that the rational homology of K_{3,1} is free Gerstenhaber-Poisson algebra. A partial description of a basis is given here. In addition, the mod-p homology of this space is a `free, restricted Gerstenhaber-Poisson algebra'. Recursive application of this theorem allows us to deduce that there is p-torsion of all orders in the integral homology of K_{3,1}. This leads to some natural questions about the homotopy type of the space of long knots in R^n for n>3, as well as consequences for the space of smooth embeddings of S^1 in S^3.Comment: 36 pages, 6 figures. v3: small revisions before publicatio

An obstruction to a knot being deform-spun via Alexander polynomials

We show that if a co-dimension two knot is deform-spun from a lower-dimensional co-dimension 2 knot, there are constraints on the Alexander polynomials. In particular this shows, for all n, that not all co-dimension 2 knots in S^n are deform-spun from knots in S^{n-1}.Comment: 6 Pages, 1 Figure. to appear in Proceedings of the AM

Embeddings of 3-manifolds in S^4 from the point of view of the 11-tetrahedron census

This is a collection of notes on embedding problems for 3-manifolds. The main question explored is `which 3-manifolds embed smoothly in the 4-sphere?' The terrain of exploration is the Burton/Martelli/Matveev/Petronio census of triangulated prime closed 3-manifolds built from 11 or less tetrahedra. There are 13766 manifolds in the census, of which 13400 are orientable. Of the 13400 orientable manifolds, only 149 of them have hyperbolic torsion linking forms and are thus candidates for embedability in the 4-sphere. The majority of this paper is devoted to the embedding problem for these 149 manifolds. At present 41 are known to embed. Among the remaining manifolds, embeddings into homotopy 4-spheres are constructed for 4. 67 manifolds are known to not embed in the 4-sphere. This leaves 37 unresolved cases, of which only 3 are geometric manifolds i.e. having a trivial JSJ-decomposition.Comment: 58 pages, 80+ figures. V6: Included references to libraries valid in Regina 5.0+. Incorporated changes suggested by Ahmed Issa, following from his techniques developed with McCoy. Included a few recent references. To appear in Experimental Mathematic