Fibered knot

A fibered knot is a knot whose exterior fibers over the circle, with fiber a compact surface whose boundary is the knot longitude. Its exterior is the mapping torus of the fiber monodromy.

For a genus-one fibered knot, the fiber is a once-punctured torus and its monodromy acts on first homology by a matrix $M\in S{L}_2(\mathbb{Z})$. Up to a Laurent unit, the one-variable Alexander polynomial is

The Nielsen–Thurston classification says that an orientation-preserving surface homeomorphism is periodic, reducible, or pseudo-Anosov. For a once-punctured torus, the three cases correspond to $|\mathrm{tr}M|<2$, $|\mathrm{tr}M|=2$, and $|\mathrm{tr}M|>2$ for its action $M\in S{L}_2(\mathbb{Z})$ on first homology.

The interior of the mapping torus of a pseudo-Anosov homeomorphism of a compact surface with boundary admits a complete finite-volume hyperbolic metric. Periodic monodromy instead gives a Seifert fibered mapping torus, while reducible monodromy gives a manifold containing an essential torus.