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202358 https://hdl.handle.net/2286/R.2.N.202358 http://rightsstatements.org/vocab/InC/1.0/ All Rights Reserved 2025 2027-08-01T17:24:45 183 pages Doctoral Dissertation Academic theses en Barkataki, Kasturi Panagiotou, Eleni Danielli, Donatella Kauffman, Louis H Paupert, Julien Renaut, Rosemary Arizona State University Partial requirement for: Ph.D., Arizona State University, 2025 Field of study: Mathematics This dissertation focuses on the rigorous analysis of collections of open curves in 3-space by developing new theoretical and computational methods. Even though knot theory has been successful at classifying simple closed curves in 3-space, open curves have been much more difficult to characterize. Moreover, many filamentous material are composed of collections of open curves in 3-space whose entanglement complexity is important for their structure, properties and function.A major focus of this work is the development of invariants for linkoids which are multi-component generalizations of knotoids (open arc diagrams). More precisely, a new combinatorial definition of the Jones polynomial for linkoids is introduced, proving it to be a topological invariant and consistent with the classical Jones polynomial for links. Furthermore, by establishing a surjective virtual closure map between linkoids and virtual knots, new invariants of linkoids are developed. The introduction of a virtual spectrum for linkoids provides a new framework for analyzing their topological complexity, which does not depend on any particular closure and gives rise to stronger invariants of linkoids. The invariants of linkoids are used to create well-defined measures of entanglement of open curves in 3-space which are continuous functions of the coordinates of the curves and they converge to their classical counterparts as the endpoints merge. Periodic Boundary Conditions model physical systems with repeating, infinite entangled components. This dissertation introduces the Periodic Jones polynomial as an extension of the classical Jones polynomial, using a finite representative, which is shown to capture a repeating factor of the Jones polynomial for any finite cutoff of a system with one Periodic Boundary Condition. Computing the Jones polynomial is #P-hard, and only serial algorithms are known to exist. Quantifying structural complexity in filamentous matter using topological measures is often computationally prohibitive. This dissertation presents the first fully parallel algoirithm for its computation, achieving exponential speedup using multiple processors. The algorithm could be generalized to other knot invariants. A new Python package has also been developed, as a part of this research, to analyze the Jones polynomial for links, linkoids, open/closed curves in 3-space, and systems with periodic boundary conditions. Mathematics Computational Mathematics Geometry Knot Theory Topology Novel Measures of Entanglement of Collections of Open Curves in 3-Space and Their Applications