UROP Proceedings 2020-21

School of Science Department of Mathematics 53 Cluster Algebra Supervisor: IP Ivan Chi Ho / MATH Student: SAMIN Thanic Nur / MATH-PMA Course: UROP2100, Fall In this report we discuss the connection between Cluster Algebra and Jones Polynomials from knot theory. Cluster Algebra is related to Continued Fractions, in the sense that we can study continued fractions as quotients of cardinalities of perfect matching of certain graphs, that appear naturally in Cluster Algebra. The graphs in question are called Snake Graphs. A class of knots: the two bridge link, can be mapped to continued fractions. The main result is that, in terms of normaliztion upto the leading term, the Jones polynomial of the knot is equal to the F-polynomial of the cluster variable under a specific substitution. Cluster Algebra Supervisor: IP Ivan Chi Ho / MATH Student: CHOY Sin Hang Sonia / MATH-IRE Course: UROP3100, Spring Most of this report aims to fill in the gaps of [1], which explains the relationship between cluster algebra mutations and complex volumes of knots. The dilogarithm function provides a complexification of the usual volume function, generalizing to the Bloch-Wigner function. Cluster algebra is used to give a combinatorical interpretation of the R-operator; each cluster mutation corresponds to a tetrahedron. Four tetrahedrons combine to become a hyperbolic octahedron, which are, in turn, associated to knot crossings, and are ultimately used to compute the complex volume of a knot. We expand on Section 4 of [1], which gives the numerical confirmation for Conjecture 2.4 in the two simplest cases, that of the figure-eight knot and trefoil knot. [1] K. Hikami, R. Inoue, Braids, complex volume and cluster algebras. Algebr. Geom. Topol. 15 (2015), 21752194. Cluster Algebra Supervisor: IP Ivan Chi Ho / MATH Student: MAN Ryuichi / MATH-IRE Course: UROP3100, Spring In this project, we first follow [KN11] to survey the basics in the study of the periodicity of cluster algebras and the associated quantum dilogarithm identities. Inspired by the related papers, we are motivated to investigate the conditions in which a quantum dilogarithm identity is a consequence of the trivial and pentagon relations, with the central focus on the cases in which the initial quiver is a square product AkAn. We shall introduce the relevant tools from the study of maximal green sequences, summarize the results from several recent journal articles and elaborate on their relations to the main conjecture (Conjecture 3.1) that we study. [KN11] R. M. Kashaev, T. Nakanishi, Classical and Quantum Dilogarithm Identities, SIGMA, 7, 2011. arXiv: 1104.4630.