Gunawan, Emily2016-10-252016-10-252016-08https://hdl.handle.net/11299/182830University of Minnesota Ph.D. dissertation. August 2016. Major: Mathematics. Advisor: Gregg Musiker. 1 computer file (PDF); viii, 183 pages.We construct a periodic infinite frieze using a class of peripheral elements of a cluster algebra of type D or affine A􏰕. We discover new symmetries and formulas relating the entries of this frieze and bracelet elements. We also present a correspondence between Broline, Crowe and Isaacs’s classical matching tuples and various recent interpretations of elements of cluster algebras from surfaces. We extend a T-path expansion formula for arcs on an unpunctured surface to the case of arcs on a once-punctured polygon and use this formula to give a combinatorial proof that cluster monomials form the atomic basis of a cluster algebra of type D. We further generalize our work and present T-path formulas for tagged arcs with one or two notchings on a marked surface with punctures.encluster algebrascombinatoricsfrieze patternsThesis or Dissertation