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We will start by reviewing some of the special appearances of knots in various areas of physics. From statistical mechanics and condensed matter physics to quantum field theory and integrability.
Then we will talk about a new and very promising direction to study knots and the physical systems associated with them, namely, the knots-quivers correspondence. The great advantage of the correspondence is that it enables us to construct complicated knot invariants from simple building blocks, known as quiver diagrams (directed graphs).
For our grand finale, we will dive into state-of-the-art research when we realise that not only we can associate a single quiver to build knot invariants, but for some of the simplest knots we found up to 142,368 quivers that produce the same knot invariant. That's where my collaborators and I derived new colourful symmetries to explain such a phenomenon. At the same time we make a surprising new connection with other combinatorial structures known as permutahedron.