Piecewise-linear pseudodiagrams

There are 2^n possible resolutions of a smooth pseudodiagram with n precrossings. If we consider piecewise-linear (PL) pseudodiagrams and resolutions that themselves are PL, certain resolutions of the pseudodiagram may not exist in three-space. We investigate this situation and its impact on the weighted resolution set of PL pseudodiagrams as well as introduce a concept specific to PL pseudodiagrams, the forcing number. Our main result classifies the PL shadows whose weighted resolution sets differ from the weighted resolution set that would exist in the smooth case.Comment: 7 pages, 8 figures; V2 fixes a minor typo in both statement of Cor. 2.8 and in its proo

On link diagrams that are minimal with respect to Reidemeister moves I and II

In this paper, a link diagram is said to be minimal if no Reidemeister move I or II can be applied to it to reduce the number of crossings. We show that for an arbitrary diagram D of a link without a trivial split component, a minimal diagram obtained by applying Reidemeister moves I and II to D is unique. The proof also shows that the number of crossings of such a minimal diagram is unique for any diagram of any link. As the unknot admits infinitely many non-trivial minimal diagrams, we see that every link has infinitely many minimal diagrams, by considering the connected sums with such diagrams. We show that for a link without a trivial split component, an arbitrary Reidemeister move III either does not change the associated minimal diagram or can be reduced to a special type of a move up to Reidemeister moves I and II

Avoided level crossing spectroscopy with dressed matter waves

We devise a method for probing resonances of macroscopic matter waves in shaken optical lattices by monitoring their response to slow parameter changes, and show that such resonances can be disabled by particular choices of the driving amplitude. The theoretical analysis of this scheme reveals far-reaching analogies between dressed atoms and time-periodically forced matter waves.Comment: 4 pages, 3 figure