This introduction aims to familiarize readers with basic dimensions of variation among pictorial and diagrammatic representations, as we understand them, in order to serve as a backdrop to the articles in this volume. Instead of trying to canvas the vast range of representational kinds, we focus on a few important axes of difference, and a small handful of illustrative examples. We begin in Section 1 with background: the distinction between pictures and diagrams, the concept of systems of representation, and that of the properties of usage associated with signs. In Section 2 we illustrate these ideas with a case study of diagrammatic representation: the evolution from Euler diagrams to Venn diagrams. Section 3 is correspondingly devoted to pictorial representation, illustrated by the comparison between parallel and linear perspective drawing. We conclude with open questions, and then briefly summarize the articles to follow. Types of IconicityAs early as 1868, Charles S. Peirce distinguished between at least two basic kinds of sign: symbols and icons. 1 As we shall understand these categories, SYMBOLIC 1 We use Peirce's taxonomy as the point of departure for our own, but we don't aspire to exegetical accuracy. In fact, Peirce famously distinguished up to three kinds of sign, including not only symbols and icons, but also indices. To simplify our presentation, we will only discuss the first two. Moreover, in 1868, Peirce used the term "likeness" instead of "icon", but later changed his terminology. V. Giardino ( )

The aim of this article is to explain why knot diagrams are an effective notation in topology. Their cognitive features and epistemic roles will be assessed. First, it will be argued that different interpretations of a figure give rise to different diagrams and as a consequence various levels of representation for knots will be identified. Second, it will be shown that knot diagrams are dynamic by pointing at the moves which are commonly applied to them. For this reason, experts must develop a specific form of enhanced manipulative imagination, in order to draw inferences from knot diagrams by performing epistemic actions. Moreover, it will be argued that knot diagrams not only can promote discovery, but also provide evidence. This case study is an experimentation ground to evaluate the role of space and action in making inferences by reasoning diagrammatically.