<https://youtu.be/co78AEqsv3s?t=1901>


Within this video at 31:41, Professor Ronald Brown handles knots algebraically, I am confused to how he deals with the crossing indices. He substitutes and simplifies a way of interlacing a separate rope in the trefoil and then joining the ends. Algebraically it equals "1", which means that the trefoil is able to be untangled from the other rope, resulting in two separate knots; a trefoil and an unknot. What is this type of knot theory algebra called so that I can look it up and study it further?


For example, is the difference between x and x^-1 the direction, or is it the difference between overpass and underpass?



What is this type of knot algebra called within knot theory?
He is computing the [fundamental group](https://en.wikipedia.org/wiki/Fundamental_group) of the complement of the knot. This is also known as the [knot group](https://en.wikipedia.org/wiki/Knot_group) and is a key invariant in knot theory.


More specifically, he is computing the [Wirtinger presentation](https://en.wikipedia.org/wiki/Wirtinger_presentation) of this group.


