Let k be a knot (i.e. an embedding of S^1 into S^3). Once S^3 is seen as the boundary of D^4, one can ask which kind of (properly) embedded surfaces in D^4 have k as boundary. Finding the minimal genus of such a surfaces (called slice genus) is a central topic in low dimensional topology.
In this talk I wish to describe some inequalities, arising from contact topology and quantum homologies. These inequalities, called Bennequin-type inequalities, can be used to estimate the slice genus of a knot in term of other invariants.