First of all let me apologise for my lack of recent posts, finishing a PhD and moving countries is exhausting!

I recently gave a learning seminar on knots, and I would like to share some of my discoveries with you in this post.

### What is a knot?

We all create knots in our daily lives, by tying our shoelaces or tangling our headphones. But mathematically a knot is the following:

“A knot is an oriented, locally flat embedding of into “.

we can think of this as taking a string, tying some sort of knot and then gluing the two ends of the string together! We then add an arrow to ‘orient’ the knot: imagine the string flowing in one direction.

The ‘locally flat’ roughly means that zooming in to any section of our knot we see what looks like a straight line travelling through space, and the fact we are mapping the circle into the three-sphere can be ignored for now: we can just imagine that is the Euclidean space .

### Knot diagrams

We can view a knot in the form of a *knot diagram *$ latex D $. This is the projection of the knot onto a plane in , where we symbolise which part of our string is behind another part by using under and over crossings. Here are some examples of well known knots and their diagrams:

In 1932 Reidemeister proved that any two diagrams that represent equivalent knots are related by a sequence of *Reidemeister moves* along side smooth deformations that preserve the arcs and crossings. These moves are as follows:

### Knot adjectives!

When we define a new type of mathematical object, such as a group, topological space, or knot, it helps to be able to describe specific subsets of those objects and so we define characteristics that a group/space/knot may have. For instance a group could be cyclic or abelian, a space could be connected or compact and a knot could be…. below we will introduce some adjectives!

#### Prime

A prime knot is one that cannot be written as the *connect sum* of two or more knots, which aren’t the unknot. The connect sum operation takes two knots, cuts them, and then glues them to each other as below. We have to make sure that we do this in a way which agrees with the orientations.

The unknot and the trefoil are both prime knots, whereas the granny knot is the connect sum of a trefoil with itself and the reef knot is the connect sum of the trefoil with its reflection!

#### Ribbon

A knot is called *ribbon * if it bounds a self intersecting disc with only *ribbon singularities.* A ribbon singularity is shown below, where the knot is in red and the disk that it bounds is in blue.

The reef knot is an example of a ribbon knot:

**Alternating**

An alternating knot is one such that, when you travel along the knot diagram you encounter an ‘over then under then over then under’ pattern in the crossings. The trefoil is an example of an alternating knot: try placing your finger on the trefoil and flowing along the string!

That’s all the knotty adjectives I have time for now: I’ll be back soon to explore the relationship between knots and braids!