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.