A classical problem in knot theory is the computation of the slice
genus. After the introduction of link homology theories, new tools to
compute the slice genus have been found. One of the most famous among
these tools is the Rasmussen invariant s. This is an integer-valued
link invariant that provides a lower bound to the value of the slice
genus. The computation of s is often difficult. This led to the
necessity of simply computable combinatorial bounds (i.e. bounds which
can be computed directly from a diagram).

In this talk we will discuss a new combinatorial bound, which arises
from a Bennequin-type inequality. First, we will prove the
independence of this bound on the known bounds. Then, we will discuss
an application to the computation of the genus of almost-positive
knots. Finally we will discuss how this bound is related to some
invariants for transverse links, and to a Bennequin-type inequality.