UROP Proceedings 2020-21

School of Science Department of Mathematics 54 Cluster Algebra Supervisor: IP Ivan Chi Ho / MATH Student: SAMIN Thanic Nur / MATH Course: UROP3100, Spring In this report, we discuss Cluster Algebra, a function called the dilogarithm function and how they can be used to calculate complex volumes of hyperbolic knot complements. Cluster Algebra is an algebraic structure equipped with cluster variables, frozen variables and an exchange matrix. We can mutate the variables using rules that are dictated by the exchange matrix. These mutations are related to the dilogarithm functions, specially several identities associated with dilogarithm functions. We can also define the R operator from the mutations, which we can use to compute the complex volume of hyperbolic knot complements using the dilogarithm function. Finally, we showcase an example of calculating the volume of a knot complement. Cluster Algebra Supervisor: IP Ivan Chi Ho / MATH Student: HO Sui Kei / COSC Course: UROP1100, Summer We review the definition of cluster algebra of geometric type and cluster algebra with coefficients. Quivers and mutation formulas would be introduced to define the cluster algebra of geometric type. The abstract algebraic structure would be used to define the cluster algebra with coefficients. Laurent phenomenon, which is present in all cluster algebra, would be discussed. We then explore the phenomenon by an example. Finally we would describe cluster algebra from triangulation of unpunctured surfaces. We review the explicit formula of any cluster variable in terms of a given cluster with the aid of snake graph and perfect matchings. A concrete example would be given to demonstrate the findings. Cluster Algebra Supervisor: IP Ivan Chi Ho / MATH Student: HUANG Luyi / MATH Course: UROP1100, Summer In this paper, we firstly give a brief motivation and definition of cluster algebra with principle coefficient in section 1. Then in section 2, we are going to examine Laurent Phenomenon. In section 3, perfect matchings and related cluster expansion formulas will be studied. In section 4, an example concerning section 3 will be given.

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