Using tropical geometry, Mikhalkin has proved that every smooth complex hypersurface in $mathbb{CP}^{n+1}$ decomposes into pairs of pants: a pair of pants is a real compact $2n$-manifold with cornered boundary obtained by removing an open regular nei
ghborhood of $n+2$ generic hyperplanes from $mathbb{CP}^n$. As is well-known, every compact surface of genus $ggeqslant 2$ decomposes into pairs of pants, and it is now natural to investigate this construction in dimension 4. Which smooth closed 4-manifolds decompose into pairs of pants? We address this problem here and construct many examples: we prove in particular that every finitely presented group is the fundamental group of a 4-manifold that decomposes into pairs of pants.
We prove the equivalence of the invariants EH(L) and LOSS-(L) for oriented Legendrian knots L in the 3-sphere equipped with the standard contact structure, partially extending a previous result by Stipsicz and Vertesi. In the course of the proof we r
elate the sutured Floer homology groups associated with a knot complement and knot Floer homology, and define intermediate Legendrian invariants.
We give new tightness criteria for positive surgeries along knots in the 3-sphere, generalising results of Lisca and Stipsicz, and Sahamie. The main tools will be Honda, Kazez and Matics, Ozsvath and Szabos Floer-theoretic contact invariants. We comp
ute the Ozsvath and Szabos invariant of positive contact surgeries along Legendrian knots in the 3-sphere in terms of the classical invariants of the knot. We also combine a Legendrian cabling construction with contact surgeries to get results about rational contact surgeries.
