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Browsing by Subject "Yamada Polynomial"

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  • Lundström, Teemu (2022)
    Spatial graphs are graphs that are embedded in three-dimensional space. The study of such graphs is closely related to knot theory, but it is also motivated by practical applications, such as the linking of DNA and the study of chemical compounds. The Yamada polynomial is one of the most commonly used invariants of spatial graphs as it gives a lot of information about how the graphs sit in the space. However, computing the polynomial from a given graph can be computationally demanding. In this thesis, we study the Yamada polynomial of symmetrical spatial graphs. In addition to being symmetrical, the graphs we study have a layer-like structure which allows for certain transfer-matrix methods to be applied. There the idea is to express the polynomial of a graph with n layers in terms of graphs with n − 1 layers. This then allows one to obtain the polynomial of the original graph by computing powers of the so-called transfer-matrix. We introduce the Yamada polynomial and prove various properties related to it. We study two families of graphs and compute their Yamada polynomials. In addition to this, we introduce a new notational technique which allows one to ignore the crossings of certain spatial graphs and turn them into normal plane graphs with labelled edges. We prove various results related to this notation and show how it can be used to obtain the Yamada polynomial of these kinds of graphs. We also give a sketch of an algorithm with which one could, at least in principle, obtain the Yamada polynomials of larger families of graphs.