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

For a connected oriented topological surface of finite type with negative Euler characteristic, every orientation-preserving mapping class has a periodic representative, preserves an essential multicurve, or has a pseudo-Anosov map representative. In the usual disjoint classification, the reducible case means a nonperiodic class preserving such a multicurve. The assertion concerns representatives up to isotopy, rather than requiring an arbitrary homeomorphism in a periodic class to have finite order.

For a torus with one puncture, the action on first homology identifies its mapping class group with $S{L}_2(\mathbb{Z})$. For the corresponding matrix $M$, the periodic classes have $|\mathrm{tr}M|<2$ or $M=\pm I$. Nonperiodic reducible classes have $|\mathrm{tr}M|=2$ and $M\ne \pm I$; pseudo-Anosov map classes have $|\mathrm{tr}M|>2$. For example, $\left(\begin{array}{cc}1& 1\\ 0& 1\end{array}\right)$ is a Dehn twist, while $I$ is periodic despite having the same trace.^[ref]

A puncture is distinguished from a boundary circle fixed pointwise: boundary twists give additional mapping classes invisible to the displayed $S{L}_2(\mathbb{Z})$ action. For the closed torus, the hyperbolic case has an Anosov homeomorphism representative.

A periodic surface homeomorphism has a positive power equal to the identity. A mapping class is periodic when it has such a representative. Allowing motion of boundary circles is a different convention from requiring every isotopy to fix the boundary pointwise.

A pseudo-Anosov homeomorphism preserves two transverse measured foliations and stretches one while contracting the other by the same factor greater than one. Unlike the Anosov homeomorphism case, the foliations may have singularities.

An Anosov surface homeomorphism has nonsingular invariant expanding and contracting foliations. In the closed orientable surface setting it occurs on the torus, with a hyperbolic integral linear model.

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.