Braiding sequences occur naturally in the study of the canonical genus of knots. The first talk gives a description of the canonical Seifert surfaces of given genus, and a relation to graphs on surfaces and Bieulerian paths in graphs. As an application, some estimates on the number of positive and alternating knots of given genus are given. In the second talk, diagrams of canonical genus one and two are discussed in more detail, and a complete classification is given. If time permits, we will discuss an algorithm to make any diagram into a special one without altering the canonical genus. As a consequence, we have an inequality for the signature of a positive link related to its Murasugi sum decomposition.