# Knots and links

The history of knot theory goes back to the nineteenth century, when Lord Kelvin proposed a description of atoms as knotted vortices in the ether. Although this physical theory was later falsified and ultimately abandoned, the mathematical framework it inspired evolved into a key component of low-dimensional topology. Knot-theoretic concepts can be found whenever linking, entanglement, and topological constraints affect physical behavior, including statistical mechanics, plasma physics, and molecular biology.  The topology of magnetic field lines is crucial for understanding solar activity, and DNA knotting and supercoiling have emerged in research on biophysics. Indeed, according to research by De Witt Sumners, Nicholas Cozzarelli, and others, knot theory can be used to explain how enzymes alter DNA topology during transcription and replication {{< cite "sumnersAnalysisMechanismDNA1995" >}} and {{< cite "sumnersDNAKnotsTheory2011" >}}.




There is much literature to be found on knot theory. We mainly follow {{< cite "lickorishIntroductionKnotTheory1997" >}}. The proofs of all the theorems in the coming sections can be found there.

## What is a link/knot?
Intuitively, one can think of a knot as a piece of string, which is in some way tangled or knotted up with connected endings. This is only part of the story; we might just as well move another piece of string through any loops of the original, knot and tangle it up, and connect the ends. This would be an intuitive view of what we call a link. To avoid exotic pathologies we define a knot in a rigorous manner, where we focus on its topological aspects.

{{< definition label="link"  >}}
A *link* of $m$ components is a subset of $S^3$, or of $\R^3$, consisting of $m$ disjoint, piecewise linear, simple closed curves. When the link has only one component, it is called a *knot*.
{{< /definition >}}

In the definition above, piecewise linear refers to the fact that we only consider the link or knot to be composed of a finite number of straight line segments, placed end to end. However, in practice, the knots or links are rounded off when drawn in a diagram to maintain simplicity. The reason for considering only finite piecewise linear curves is to prevent a link from having an infinite number of kinks. If we were not so stringent, it would be possible to construct a link where the kinks converge to a point. Links of this type are called *wild*.

## Equivalence
One cannot simply project a three-dimensional knot onto two-dimensional paper, as this would make the crossings ambiguous. Knots are therefore illustrated as in Figure 1. This knot is called the *trefoil*.



{{< figure-svg src="diagram1.svg" alt="System diagram" caption="**Figure 1:** The trefoil." width="20%" >}}


As in many areas of mathematics, we are interested in classifying the objects that we define. In knot theory, in makes sense to classify links according to deformations of the background space. From a topological viewpoint, we therefore consider homeomorphisms that preserve the links themselves. To be more precise, we define the following.

{{< definition label="equivalence"  >}}
Two links $L_1$ and $L_2$ in $S^3$ are called *equivalent* if there is an orientation preserving piecewise linear homeomorphism $h \colon S^3 \to S^3$ such that $h(L_1) = L_2$.
{{< /definition >}}


This definition allows one to express any knot in different arrangements, while maintaining equivalence. From an intuitive viewpoint this is quite clear: one would want a knot to remain the "same" after stretching or moving it. In this sense, the map $h$ defined above is a rearrangement of the same knot. In piecewise linear topology, one may just as well call $h$ *isotopic* to the identity. This implies the existence of a map $h_t \colon S^3 \to S^3$ for $t \in [0,1]$ so that $h_0 = 1$ and $h_1 = h$ and the map $H \colon  S^3 \times [0,1] \to S^3 \times [0,1]$ defined as $(x,t) \mapsto (h_tx,t)$ is a piecewise linear homeomorphism. This provides a more intuitive sense of our ability to move, stretch and shrink the knot while maintaining equivalence. Since we have to transform the ambient space $S^3$ with the link, we cannot shrink any complication in the knot away. Consequently, we would not be able to create the trefoil from a loop (or the ``unkot''), since this would require rupture.

## Reidemeister moves
The type of links that we shall consider in this article can be represented in a diagram without too much difficulty (meaning that they are not overly complex). As we shall see shortly, whether two diagrams of links are in fact equivalent can be deduced by performing a finite amount of Reidemeister moves in conjunction with an orientation-preserving homeomorphism of the plane. In total there are three types of Reidemeister moves, which transform one configuration of crossings to another configuration. The moves are illustrated in Figure 2.

{{< figure-svg src="reidemeister.svg" alt="System diagram" caption="**Figure 2:** Reidemeister moves." width="50%" >}}

{{< theorem label="reidemeister_theorem"  >}}
  Two link diagrams represent the same ambient isotopy class of a link in $S^{3}$ if and only if they are related by a finite number of Reidemeister moves and a planar isotopy.
{{< /theorem >}}

This theorem allows us to check whether two simple knots are in fact equivalent. However, in knot theory, the knots can get very complicated indeed, which makes it difficult or impossible to find such a sequence of Reidemeister moves. Additionally, the Reidemeister theorem does not provide a way to check whether two knots are not equivalent.


## References

{{< references >}}