Structural and optical properties of MOCVD AllnN epilayers

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Knot Floer homology, genus bounds, and mutation

In an earlier paper, we introduced a collection of graded Abelian groups \HFKa(Y,K) associated to knots in a three-manifold. The aim of the present paper is to investigate these groups for several specific families of knots, including the Kinoshita-Terasaka knots and their ``Conway mutants''. These results show that \HFKa contains more information than the Alexander polynomial and the signature of these knots; and they also illustrate the fact that \HFKa detects mutation. We also calculate \HFKa for certain pretzel knots, and knots with small crossing number ($n\leq 9$). Our calculations prove that many of the knots considered here admit no Seifert fibered surgeries.Comment: minor revisions, updated reference

A refined Jones polynomial for symmetric unions

Motivated by the study of ribbon knots we explore symmetric unions, a beautiful construction introduced by Kinoshita and Terasaka in 1957. For symmetric diagrams we develop a two-variable refinement $W_D(s,t)$ of the Jones polynomial that is invariant under symmetric Reidemeister moves. Here the two variables $s$ and $t$ are associated to the two types of crossings, respectively on and off the symmetry axis. From sample calculations we deduce that a ribbon knot can have essentially distinct symmetric union presentations even if the partial knots are the same. If $D$ is a symmetric union diagram representing a ribbon knot $K$, then the polynomial $W_D(s,t)$ nicely reflects the geometric properties of $K$. In particular it elucidates the connection between the Jones polynomials of $K$ and its partial knots $K_\pm$: we obtain $W_D(t,t) = V_K(t)$ and $W_D(-1,t) = V_{K_-}(t) \cdot V_{K_+}(t)$, which has the form of a symmetric product $f(t) \cdot f(t^{-1})$ reminiscent of the Alexander polynomial of ribbon knots.Comment: 28 pages; v2: some improvements and corrections suggested by the refere

Colorings, determinants and Alexander polynomials for spatial graphs

A {\em balanced} spatial graph has an integer weight on each edge, so that the directed sum of the weights at each vertex is zero. We describe the Alexander module and polynomial for balanced spatial graphs (originally due to Kinoshita \cite{ki}), and examine their behavior under some common operations on the graph. We use the Alexander module to define the determinant and $p$-colorings of a balanced spatial graph, and provide examples. We show that the determinant of a spatial graph determines for which $p$ the graph is $p$-colorable, and that a $p$-coloring of a graph corresponds to a representation of the fundamental group of its complement into a metacyclic group $\Gamma(p,m,k)$. We finish by proving some properties of the Alexander polynomial.Comment: 14 pages, 7 figures; version 3 reorganizes the paper, shortens some of the proofs, and improves the results related to representations in metacyclic groups. This is the final version, accepted by Journal of Knot Theory and its Ramification