The HOMFLY polynomials of odd polyhedral links

Abstract

This paper extends the methodology for the construction of odd polyhedral links. Building blocks are odd chain tangles, each of which consists of finitely many $$2n+1$$ -twist tangles for any nonnegative integer $$n$$ . For any polyhedral graph $$G$$ , replacing each edge with an odd chain tangle results in an infinite collection of odd polyhedral links. The relationship between the HOMFLY polynomials of these odd links and the $$Q^{d}$$ -polynomial of $$G$$ is established. It leads to the determination of the span of the HOMFLY polynomial, the bound on the braid index and the genus of each odd link. Our results show that these indices depend not only on the building blocks but also on the graph $$G.$$