There is a **impossible knot** to untie? It seems incredible to believe but: no! **All knots can be untied** and it is mathematics that tells us. Yes because mathematically, by studying the shape of the nodes, we can say that these structures **they are all equivalent** between them. This means that, even if two knots seem very different from each other in terms of the number of overlaps of the rope, in truth they can be considered equal from the point of view of knot theory because we are able to pass from one to the other simply **deforming** the rope, **without** the need for **break it.**

To understand this, let’s look together at the case of the node shown in **video.** The initial knot cannot be untied due to the presence of the thorn, which prevents the intersection from being “untied”. To be able to free the thorn, we can transform the knot into another **apparently different composition**but **mathematically equivalent**: we insert a new overlap thanks to which we manage to “cancel” the one already present, thus returning the cable to its “initial state”, i.e. the straight one.

In short, any node within a **open rope,** however complex it may seem – think of headphones when they remain in your pocket – it can always be **“jointed”**. And this is the reason why, mathematically, these types of “open” nodes are all equivalent to each other and indeed, they cannot even be defined as nodes!

**Mathematically** in fact, and more precisely in **topology** – the branch of mathematics that studies the properties of figures and geometric shapes when we deform them – i **knots** are defined as **closed curves** in space, and differ from each other depending on the **entanglements** that they present.

A **circumference** it’s a **trivial knot,** that is, without entanglements, while the famous **node** **clover** that he has **3 weaves** and they are not mathematically equivalent because to go from one to the other we would have to cut the thread. All the nodes shown in the figure are not equivalent to each other, because it is not possible to move from one to the other without performing a tear.

Fine, but what are these considerations for? There **knot theory**a branch of topology that studies these structures, has various applications. One of all – of great importance – is fundamental for **study natural structures such as DNA and proteins**.