The Architecture Of Polyhedral Links And Their HOMFLY Polynomials

Many biomolecules with polyhedral shape have been discovered and synthesized in the nature and experiment, polyhedral links are presented to study their underlying mathematical mechanism. Polyhedral links are the structure model of these biomolecules, and reflect their topological properties in a degree. Knot invariant is an important means to study polyhedral links. Based on knot theory and graph theory, the paper simplifies the computation for HOMFLY polynomial of polyhedral links. The main results are as follows:1. Two base operations, 'X-tangle covering'and'tangle-covering', are proposed to construct polyhedral links. For an arbitrary polyhedral graph, four classes of polyhedral links can be obtained by applying the operation of'X-tangle covering' to the related reduced sets. On the other hand, a family of polyhedral links can be obtained by applying the operation of 'tangle-covering' to an arbitrary polyhedral graph, including the constructed four classes of polyhedral links as special case.2. First, we give the the relationships between the Tutte polynomial of a polyhedral graph and the HOMFLY polynomials of four polyhedral links. Based on that, the relationships between the W-polynomial of a polyhedral graph and the HOMFLY polynomials of four kinds of polyhedral links are established. These relationships not only simplify the computation but also provide a method of constructing a general formula for the HOMFLY polynomial of polyhedral links.3. The relationship between the Zw-polynomial of a polyhedral graph and the HOMFLY polynomial of a family of polyhedral links are established. By special parameter of Zw-polynomial, we will obtain the relationships between the Zw-polynomial of a polyhedral graph and the HOMFLY polynomials of four kinds of polyhedral links. In addition, we give the relationships between the Tutte polynomial of a polyhedral graph and the HOMFLY polynomials of some special kinds of polyhedral links. These relationships not only expand the general formula of HOMFLY polynomial into a family of polyhedral links but also simply the computation for HOMFLY polynomial of some special polyhedral links.