A Topological Framework for the Computation of the HOMFLY Polynomial and Its Application to Proteins

TL;DR

This paper introduces a topological framework for computing the HOMFLY polynomial, enabling the detection of knotted proteins and other polymers with handedness sensitivity, improving upon existing methods.

Contribution

It presents a novel approach coupling a reduction scheme with HOMFLY polynomial computation, allowing for the detection of right-handed knots in proteins and broader applications.

Findings

Validated on known knots and links.

Discovered two new right-handed trefoil knots in proteins.

Produced an updated database of knotted proteins.

Abstract

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.