A study of projections of 2-bouquet graphs

A new field of mathematical research has emerged in knot theory, which considers knot diagrams with missing information at some of the crossings. That is, the observer does not know which strand lies over or underthe other at a crossing, and this new type of crossing is known as a precrossing. Pseudodiagrams are knot-like diagrams that contain precrossings and crossings, while projections are knot-like diagrams that only contain precrossings. The trivializing number (or knotting number, respectively) of a pseudodiagram is the number of precrossings that need to be changed to acrossing to obtain a diagram that represents the unknot (or a nontrivial knot,respectively) regardless of how the remaining precrossings are resolved.Spatial graph theory is a sub eld of knot theory that focuses onembeddings of graphs in three-dimensional space. In this thesis, we extend the concepts of trivializing and knotting numbers to spatial graph theory,where we focus on 2-bouquet graphs. Specically, we calculate the trivializing and knotting number for projections and pseudodiagrams of 2-bouquet spatial graphs based on the number of precrossings and the placement of the precrossings in the pseudodiagram of a spatial graph.