  Polymers can be modeled as open polygonal paths and their closure generates
knots. Knotted proteins detection is currently achieved via high-throughput
methods based on a common framework insensitive to the handedness of knots.
Here we propose a topological framework for the computation of the HOMFLY
polynomial, an handedness-sensitive invariant. Our approach couples a
multi-component reduction scheme with the polynomial computation. After
validation on tabulated knots and links the framework was applied to the entire
Protein Data Bank along with a set of selected topological checks that allowed
to discard artificially entangled structures. This led to an up-to-date table
of knotted proteins that also includes two newly detected right-handed trefoil
knots in recently deposited protein structures. The application range of our
framework is not limited to proteins and it can be extended to the topological
analysis of biological and synthetic polymers and more generally to arbitrary
polygonal paths.
