Surgery description of colored knots

The pair (K,r) consisting of a knot K and a surjective map r from the knot group onto a dihedral group is said to be a p-colored knot. D. Moskovich conjectured that for any odd prime p there are exactly p equivalence classes of p-colored knots up to surgery along unknots in the kernel of the coloring. We show that there are at most 2p equivalence classes. This is a vast improvement upon the previous results by Moskovich for p=3, and 5, with no upper bound given in general. T. Cochran, A. Gerges, and K. Orr, in "Dehn surgery equivalence relations of 3-manifolds", define invariants of the surgery equivalence class of a closed 3-manifold M in the context of bordisms. By taking M to be 0-framed surgery of the 3-sphere along K we may define Moskovich's colored untying invariant in the same way as the Cochran-Gerges-Orr invariants. This bordism definition of the colored untying invariant will be then used to establish the upper bound.Comment: 41 pages, 23 figures (Version 3) Minor revisions and typos fixed. Proofs of Propositions 4.1 and 4.8 revise

A characterization of alternating links in thickened surfaces

We use an extension of Gordon-Litherland pairing to thickened surfaces to give a topological characterization of alternating links in thickened surfaces. If $\Sigma$ is a closed oriented surface and $F$ is a compact unoriented surface in $\Sigma \times I$, then the Gordon-Litherland pairing defines a symmetric bilinear pairing on the first homology of $F$. A compact surface in $\Sigma \times I$ is called definite if its Gordon-Litherland pairing is a definite form. We prove that a non-split link $L$ in a thickened surface is alternating if and only if it bounds two definite surfaces of opposite sign.Comment: 19 pages, 9 figures. Revision includes new proofs of Lemmas 6 and 18, a slight improvement to Theorem 8, the addition of Corollary 20, and many other minor changes. This version is to appear in Proc. Roy. Soc. Edinburgh Sect.

Semantic web learning technology design: addressing pedagogical challenges and precarious futures

Semantic web technologies have the potential to extend and transform teaching and learning, particularly in those educational settings in which learners are encouraged to engage with ‘authentic’ data from multiple sources. In the course of the ‘Ensemble’ project, teachers and learners in different disciplinary contexts in UK Higher Education worked with educational researchers and technologists to explore the potential of such technologies through participatory design and rapid prototyping. These activities exposed some of the barriers to the development and adoption of emergent learning technologies, but also highlighted the wide range of factors, not all of them technological or pedagogical, that might contribute to enthusiasm for and adoption of such technologies. This suggests that the scope and purpose of research and design activities may need to be broadened and the paper concludes with a discussion of how the tradition of operaismo or ‘workers’ enquiry’ may help to frame such activities. This is particularly relevant in a period when the both educational institutions and the working environments for which learners are being prepared are becoming increasingly fractured, and some measure of ‘precarity’ is increasingly the norm

Signature and concordance of positive knots

We derive a linear estimate of the signature of positive knots, in terms of their genus. As an application, we show that every knot concordance class contains at most finitely many positive knots.Comment: 10 pages, 7 figure