On Class Diagrams, Crossings and Metrics

As a standardized software engineering diagram, the UML class diagram provides various information on the static structure of views on software while design, implementation and maintenance phase. This talk gives an overview on drawing UML class diagrams in hierarchical fashion. Therefore, common elements of class diagrams are introduced and aesthetic rules for drawing UML class diagrams are given. These rules are based on four disciplines involved in the reading process of diagrams. After a brief introduction to our drawing algorithm, an extensive extension of the well-known Sugiyama algorithm, two details are highlighted: A new crossing reduction algorithm is presented and compared to existing ones and issues on measuring the quality of a layout are discussed

Untwisting information from Heegaard Floer homology

The unknotting number of a knot is the minimum number of crossings one must change to turn that knot into the unknot. We work with a generalization of unknotting number due to Mathieu-Domergue, which we call the untwisting number. The p-untwisting number is the minimum number (over all diagrams of a knot) of full twists on at most 2p strands of a knot, with half of the strands oriented in each direction, necessary to transform that knot into the unknot. In previous work, we showed that the unknotting and untwisting numbers can be arbitrarily different. In this paper, we show that a common route for obstructing low unknotting number, the Montesinos trick, does not generalize to the untwisting number. However, we use a different approach to get conditions on the Heegaard Floer correction terms of the branched double cover of a knot with untwisting number one. This allows us to obstruct several 10 and 11-crossing knots from being unknotted by a single positive or negative twist. We also use the Ozsv\'ath-Szab\'o tau invariant and the Rasmussen s invariant to differentiate between the p- and q-untwisting numbers for certain p,q > 1.Comment: 21 pages, 11 figures; final version, accepted for publication in Algebraic & Geometric Topolog

Instantons and odd Khovanov homology

We construct a spectral sequence from the reduced odd Khovanov homology of a link converging to the framed instanton homology of the double cover branched over the link, with orientation reversed. Framed instanton homology counts certain instantons on the cylinder of a 3-manifold connect-summed with a 3-torus. En route, we provide a new proof of Floer's surgery exact triangle for instanton homology using metric stretching maps, and generalize the exact triangle to a link surgeries spectral sequence. Finally, we relate framed instanton homology to Floer's instanton homology for admissible bundles.Comment: 64 pages, 19 figure