A Simple and Efficient Algorithm to Identify the Chirality of Polymer Knots Based on the Alexander Polynomial

Abstract

Recent experimental observations of knotting in DNA and proteins have stimulated the simulation studies of polymer knots. Simulation studies usually identify knots in polymer conformations through the calculation of the Alexander polynomial. However, the Alexander polynomial cannot directly discriminate knot chirality, while knot chirality plays important roles in many physical, chemical, and biological properties. In this work, we discover a new relationship for knot chirality and accordingly, develop a new algorithm to extend the applicability of the Alexander polynomial to the identification of knot chirality. Our algorithm adds an extra step in the ordinary calculation of the Alexander polynomial. This extra step only slightly increases the computational cost. The correctness of our algorithm has been proved mathematically by us. The implication of this algorithm in physical research has been demonstrated by our studies of the tube model for polymer knots. Without this algorithm, we would be unable to obtain the tubes for polymer knots.