### What is a Fibred Knot?

A knot $$K \in S^3$$ is fibred if there is a 1-parameter family $$F_\theta$$ of Seifert surfaces for $$K$$ called fibres or fibration surfaces where $$\theta \in [0,2\pi]$$ runs through the points of a unit circle $$S^1$$, such that if $$\theta$$ is not equal to $$\theta^*$$ then the intersection of their Fibres is exactly the knot, $$F_\theta \cap F_{\theta^*} = K$$ .

Therefore, we have a fibration map: $$f:S^3-K \rightarrow S^1$$ such that for each point on the cirle, $$\theta \in S^1$$, the inverse fibration map, $$f^{-1}(\theta)$$ , is a surface (3-manifold) with boundary $$K$$.

### Visualizing the Fibrations

On this page we visualize the fibration of a trefoil knot. Notice that the boundary of our fibration surface is always exactly the knot. This visualization shows members of the family of fibres$$F_\theta$$ by animating through values of $$\theta \in S^1$$.

In the subsequent pages we will understand and visualize how this fibration was constructed.

#### About Project

This is my final deliverable for the summer research component of a Master's in Mathematics at the University of Toronto supervised by Dr. Dror Bar-Natan.

#### About Me

My name is Jesse Bettencourt and I've recently acquired an M.Sc. in Mathematics from U of T. I'm interested in vizualizations from the intersection of math and computer science.