Local Maxima and the Shape of a Graph

In first-year calculus, we learn how local maxima and minima provide insights into the shape of a graph. In discrete mathematics, a graph also refers to a structure made up of vertices and edges. When these vertices and edges are realized in three dimensions, the graph can form intriguingly knotted shapes. In this talk, I will explore how the arrangement of local maxima influences the geometry of a knotted graph. For example, if the sequence of certain local maxima and minima cannot be altered, it suggests that the polymer cannot fit within a tight tube.