Chapter 1 Braids and Permutations
In everything from hairstyles to Ethernet cables, you can find braids. Braids originate as sets of parallel strands, fixed on two ends, which are then "twisted" to introduce a series of crossings. Braids can be "closed" by joining the ends of the strands to create a knot or link.
In our course, we will use braids as an entry point to how algebraic structure arises from the process, and product, of creating crossing strands. First we'll explore how a simple type of braids arise from the study of permutations - the simplest of which are anagrams of words. Then we'll see how, though this simple construction gives rise to the structure of a finite group, a closer look at the crossings of a braid shows the need for infinite groups if we want to study them in their fullness. Finally, we explore some of the algebraic properties of these braid groups and their connections to groups of permutations, and to the construction of knots and links.