This master's thesis is based on a theoretical research approach. We focus on the parametrization of knots using the polygonal model. The core objective is to investigate invariants of polygonal knots, with particular emphasis on the bridge index, superbridge index and the polygonal index.
We begin by defining key concepts from topology and knot theory, which are then used to prove results regarding bounds of knot invariants and relationships between them. We show that every nontrivial knot has a polygonal index not smaller than 6. Furthermore, we provide estimates of the polygonal index of torus knots and review several known values of this index. Finally, we explore estimates of the polygonal index for specific families of knots, such as twist knots, pretzel knots and connected sum of knots.
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