On the HOMFLY polynomials of even trigonal bipyramid links

Even trigonal bipyramid links ( ETB links ) are the mathematical models of DNA trigonal bipyramids such that each edge is composed of a twist tangle with 2 n -crossing number (Lin et al. in J Mol Graph Model 95: 107492, 2020). In this paper, a basis of ETB links has been introduced as a set of trivial links obtained from any ETB link by respectively smoothing and switching each crossing, which is used to calculate the HOMFLY polynomial of each ETB link. Hence through establishing the algorithm for computing the HOMFLY polynomials of ETB links, an ETBL program has been developed in the Python programming language. As a result, the HOMFLY polynomials of 104 families of ETB links are given in terms of the parameter n . Based on these polynomials, we show that all ETB links are all chiral. Also, the structural property of 30 pairs of link families is further explored due to each pair of links having the same HOMFLY polynomial. These results provide a necessary theoretical support for predicting and controlling the topological structures of DNA trigonal bipyramids.