The mathematics of knots

For millennia, sailors (and more recently climbers) have used knots to tie their ropes. Interlacements, already present in Roman decorative art, are found in Islamic and Celtic motifs. At the end of the 19th century, Tait and Thomson, through a hypothesis – which proved to be false – of chemical bonds that would interlace, brought knots into the realm of mathematics. Do you like tying strings or untangling the hanks of your balls of wool? Knot theory is made for you! Mathematical challenges are not lacking there. Without knowing it, Gauss, in studying certain curves, had laid the groundwork for this theory, which makes it possible to generate knots systematically, to classify them, to understand their connections with graph theory or to distinguish them 'at first glance' with the help of a complete invariant... which still remains to be constructed.

The sphere is one of the most elementary objects in geometry. To study it, since Archimedes, treasures of ingenuity have been deployed. The most famous of them is the one we inhabit, but it took centuries to become convinced that the Earth is 'round'! And the multiple attempts at cartographic representations have given rise to mathematical developments that have sometimes held quite a few surprises. Each projection of the globe onto the plane generates its share of deformations, transformations of areas, modifications of properties. Thus, the stereographic projection preserves angles, but not distances. How to choose 'the' right way to visualize this object? Depending on the characteristics to be preserved, various types of transformations have been devised. And if we took the time to revisit the properties of the sphere?