Introducing Knot Fibration

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.

Project Repository

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.