Abstract. In this paper, we study Manolescu's construction of the relative Bauer-Furuta invariants arising from the Seiberg-Witten equations on 4-manifolds with boundary. The main goal of this paper is to introduce a new gauge fixing condition in order to apply the finite dimensional approximation technique. We also hope to provide a framework to extend Manolescu's construction to general 4-manifolds.

Let Y be a closed and oriented 3-manifold. We define different versions of unfolded Seiberg-Witten Floer spectra for Y . These invariants generalize Manolescu's Seiberg-Witten Floer spectrum for rational homology 3-spheres. We also compute some examples when Y is a Seifert space.

We apply knot Floer homology to exhibit an infinite family of transversely nonsimple prime knots starting with 10 132 . We also discuss the combinatorial relationship between grid diagrams, braids and Legendrian and transverse knots in standard contact R 3 .