2005/11/13 by Xian’an Jin, Jin, Xian'an, Fuji Zhang +1
Mathematics · #57M15 #Advanced Combinatorial Mathematics #Combinatorics (math.CO) #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.math/0511326
openalex publication_date 2005/11/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let G be a signed graph. Let G be the graph obtained from G by replacing each edge e by a chain or a sheaf. We first establish a relation between the Q-polynomial of G[6] and the W-polynomial of G [9]. Two special dual cases are derived from the relation, one of which has been studied in [8]. Based on the one to one correspondence between signed plane graphs and link diagrams, and the correspondence between the Q-polynomial of signed plane graph and the Kauffman bracket of link diagram, we can compute the Kauffman bracket of link diagram corresponding to G by means of the W-polynomial of G. By this way we use transfer matrix approach to compute the Kauffman bracket of rational links, and obtain their closed-form formulae. Finally we provide an example to point out that the relation we built can be used to deal with a wide type of link family.