Description
Knot theory has a long history yet remains a vibrant field today. One of its central questions is how to distinguish knots up to isotopy.
In this talk, we focus on combinatorial approaches. We study knots through knot diagrams, which are projections of knots onto the plane. Remarkably, any two diagrams of the same knot can be related by just three local changes, known as the Reidemeister moves.
By constructing quantities that remain unchanged under these moves, we obtain knot invariants !
We introduce several combinatorial knot invariants, such as tricolorability and the Kauffman bracket (leading to the famous Jones polynomial), which are powerful for telling knots apart.