School of Science Department of Mathematics 45 Cluster Algebra Supervisor: IP Ivan Chi Ho / MATH Student: CHAU Yu Foon Darin / MATH-IRE Course: UROP1100, Spring We investigate the construction of cluster algebras over the classical SL(V)-invariant ring of tensors . Fomin and Pylyavskyy pioneered this study of these rings in the context of cluster algebras using Kuperberg’s web basis. We provide a simplified, explicit construction for the procedures described in the original article. We classify the cluster algebra types for n = 9, and prove a structure theorem relating . We also suggest some applications to conjectures in the original reference by Fomin and Pylyavskyy. Cluster Algebra Supervisor: IP Ivan Chi Ho / MATH Student: CHIU Yan Ho / IRE Course: UROP1100, Summer In this report, we define a turn of a snake graph generated by an arc. The number of the perfect matchings of the snake graph gives an integer sequence of type which the even terms of it satisfy the mutation exchange relation . We also use the cluster expansion formula in terms of perfect matching to show that it has a linear recurrence relation by considering its snake graph. We could link up the exchange relation and the linear recurrence relation by the relationship between cluster variable from surface and its corresponding snake graph. Quantum Groups Supervisor: IP Ivan Chi Ho / MATH Student: CHUNG Soobeom / PHYS Course: UROP2100, Fall Since its discovery, the Jones polynomial of links was shown to be constructed using quantum groups by Turaev. It was also generalized to isotopy invariant of colored links and ribbon invariant in . Furthermore, Witten constructed a topological invariant of the closed 3-manifold using Chern-Simonseld theory and the mathematical counterpart was demonstrated by Reshetikhin and Turaev. For an oriented 3manifold obtained by surgery on a link, they used quantum groups and modular Hopf algebra to produce the invariant. The result is a linear sum of the generalized Jones Polynomial. In this report, derivation and an example of Jones polynomial will be shown. Furthermore, invariant for different 3-manifolds will be calculated.