Knots

Preliminary definitions

Let us begin by defining a knot.

A knot is a smooth embedding of a circle into ambient space $\mathbb {R} ^{3}$ . The most basic knot is the unknot, which is essentially just a circle $\bigcirc$ (we will define this properly shortly).

A link of $n\in \mathbb {N}$ components is a smooth embedding of $n$ disjoint circles into three-dimensional space. So obviously any knot can be seen as a link with one component. Analogously to the unknot, the most basic link is the unlink, which is essentially $n$ disjoint circles. We usually denote an $n$ -unlink as $\bigcirc ^{n}$ . So, a $1$ -unlink is an unknot.

Before this chapter, we mentioned that knot theory is the study of distinguishing knots and links. So we now need to define a notion of equivalence on links to be able to distinguish links.

Two links are equivalent if they are ambient isotopic. Here, ambient isotopy means that there exists a homotopy such that it is a smooth family of diffeomorphisms. Two links are said to have the same link type if they are equivalent; so link type is the equivalence class for links. Similarly, knot type is the equivalence class for knots. We included smoothness here, because homotopy is too weak for embeddings in ambient space since we can define any continuous map from one link to the unlink.

It is easy to see that link equivalence is in fact an equivalence relation.

Now we can define the $n$ -unlink as a link that is equivalent to $n$ disjointed circles.

We can now define a projection map from $\mathbb {R} ^{3}$ to $\mathbb {R} ^{2}$ such that no three edges pass through a point and no two edges are tangent to one another. We then indicate over and under crossings by putting breaks in the edge. We call this projected link a link diagram.

Theorem (Reidemeister's theorem). Every link has a link diagram. Also, two links are equivalent if and only if their link diagrams are related by a finite sequence of Reidemeister moves. Reidemeister moves consist of three very standard moves: RI - a twist, RII - moving one strand over the other, RIII - moving a strand over or under a crossing.

References

Queen, M., On the Jones Polynomial: A Monica Queen Thesis, arXiv:2108.13835, 2021.