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201667 https://hdl.handle.net/2286/R.2.N.201667 http://rightsstatements.org/vocab/InC/1.0/ http://creativecommons.org/licenses/by-nc-sa/4.0 2025-05 26 pages Balanay, Ron Panagiotou, Eleni Espanol, Malena Villarreal, Daniel Tolosa Barrett, The Honors College In this thesis, we explore quandle invariants as a means to classify knotoids, a gen- eralization of classical knots to diagrams with endpoints. The theory of knotoids is used to rigorously define measures of entanglement of open curves in 3-space. These invariants not only distinguish different knotoid types but also provide insight into their topological complexity and the complexity of open curves in 3-space. In particular, we focus on tricoloring invariants which can be computed by solving linear homogeneous systems of equations on diagrams. We define the tricoloring number of an open curve in 3-space and prove that it is a continuous function of the curve coordinates. We study computational approaches for calculating the number of tricolorings of polygonal curves embedded in R3. A python algorithm is developed for computing the tricoloring num- ber and it is applied to systems of random walks. The computational tools developed in this thesis may be used to validate theoretical results and support applications in physics, biology, or other fields where knot theory is relevant. Knotoids Knot Theory Quandle Invariants Tricoloring Open Curves 3-space Planar Diagrams Reidemeister moves Virtual Knots Computational topology Algebraic invariants Python algorithm Quandle Invariants of Knotoids and Open Curves in 3-space