Dimension four is the lowest dimension where smooth and topological manifolds can differ; any difference between these categories is known as exotica. In particular, a smooth 4-manifold is \emph{exotic} if there is another smooth 4-manifold which is homeomorphic but not diffeomorphic to it. There is a wealth of literature, mostly written between 1983 and 2008,…

By slicing a Heegaard diagram for a knot K in $S^3$, it is possible to retrieve the knot Floer homology of K as a tensor product of bimodules over an $\A_{\infty}$ algebra corresponding to the slice. The first step in this process is to assign an $\mathcal{A}_{\infty}$ algebra to this slice, which can also be…

Let M be the 3-manifold obtained by r-surgery on the right handed trefoil knot. Classification of contact structures on such manifolds have been mostly understood for r\geq 1 and r=0. Etnyre-Min-Tosun has an upcoming work on the classification of the tight contact structures for all r. The fillability of contact structures on M is mostly understood if r is not…