The Braid Index of Complicated DNA Polyhedral Links

The goal of this paper is to determine the braid index of two types of complicated DNA polyhedral links introduced by chemists and biologists in recent years. We shall study it in a more broad context and actually consider so-called Jaeger's links (more general Traldi's links) which contain, as special cases, both four types of simple polyhedral links whose braid indexes have been determined and the above two types of complicated DNA polyhedral links. Denote by b(L) and c(L) the braid index and crossing number of an oriented link L, respectively. Roughly speaking, in this paper, we prove that b(L) = c(L)/2 + 1 for any link L in a family including Jaeger's links and contained in Traldi's links, which is obtained by combining the MFW inequality and an Ohyama's result on upper bound of the braid index. Our result may be used to to characterize and analyze the structure and complexity of DNA polyhedra and entanglement in biopolymers.