Crossing changes

A naive but often useful way of thinking of a knot or link in R3 is to generically project it onto a plane, keeping account of which part of the knot goes under and which part goes over at any given crossing. Think of laying the knot on a table and taking note of how it crosses itself whenever one part of it lies on top of another. It’s natural to ask how the knot is changed by altering one of the crossings, reversing which arc goes over and which goes under at one of the crossing points. This survey article is meant to explore this question. Though it’s a naive question, it connects at a deep level to some of the most important ideas in modern low-dimensional topology.