Knot spinning

This exposition is intended to provide some introduction to higher-dimensional knots embeddings of Sn−2 in S through spinning constructions. Once our shoe laces, those archetypical hand tools of knot theory, have been turned into spheres, how can we construct and visualize concrete examples of such knots? There are many important ways to construct higher-dimensional knots. If we are interested in algebraic knots, we can look at the links of singularities of complex algebraic varieties in C (see, e.g., [27, 7]). We can also construct knots by surgery theory (see [25] for a recent survey). There are powerful and complex tools for studying the knots that arise in these manners, but such construction methods frequently do not allow one to “see” the knot. Often these knots can be described only in terms of their algebraic invariants. We want to be able to visualize our knots, at least as far as it is possible to do so with our three-dimensional brains. This brings us to a series of constructions known as knot spinnings. Many extensions have been made to Artin’s original spinning technique, which dates back to 1925, but the various spinning constructions all have the appeal of being completely geometric in nature and thus highly visual. On top of providing a myriad of examples of importance in knot theory, these constructions provide an excellent introduction to thinking about higherdimensional knots and higher-dimensional topology in general.