Invariants of planar knotoids and linkoids

Knotoids are a natural generalization of mathematical knots in which one considers knot diagrams modeled on the interval rather than the circle. In this dissertation I introduce several new invariants of knotoids and of their multi-component counterparts called linkoids. I especially focus on constructing new invariants of knotoids drawn on the plane, called 'planar' knotoids, which are more difficult to classify than those drawn on the surface of a sphere. Additionally I define new generalizations of knotoids, including 'twisted' knotoids on arbitrary closed surfaces and 'biframed' knotoids. In particular the introduction of biframed knotoids allows for the construction of quantum knotoid invariants, which give powerful examples of planar knotoid invariants.