Not the Knot

The concept of a knot as a continuous thread in space is intuitively clear from common experience with string and shoelaces, but it is less clear how to decide when two knots should be viewed as equivalent or different. The novel idea from algebraic topology is to study the space where the knot is not, that is, the complement of a knot, which is a subset of Euclidean 3-space, $\mathbb{E} ^3,$ from which an embedded circle has been removed. The interest arises from the variety of ways in which a circle can be continuously embedded (ie as a homeomorphic image) into $\mathbb{E} ^3,$ even when we disallow infinite sequences of loops in the image (the so-called wild knots). In order to make a satisfactory attempt at classifying knots, we need to introduce some homotopy theory; this allows us to probe the complement of a knot with loop curves. The loops give rise to the knot group; then knots yielding different groups are different but the converse is not true so the classification is incomplete.