# Immersed Seifert surfaces of minimal genus

Let $K\subset S^3$ be a knot. We denote by $X=S^3\setminus \nu K$ the knot exterior, i.e. the complement of an open tubular neighborhood of $K$. An immersed Seifert surface for a knot $K$ is an immersion $f\colon S\to X$ such that $f(\partial S)$ is a longitude of $K$. We refer to the genus of $S$ as the genus of the immersed Seifert surface.

Gabai showed that the minimal genus of an immersed Seifert surface is the same as the minimal genus of an embedded Seifert surface. I was wondering whether a stronger statement holds: every immersed Seifert surface of minimal genus is an embedded Seifert surface in disguise', i.e. homotopic (through immersions) to an embedded Seifert surface.

For higher genus I can think of immersed Seifert surfaces that certainly do not from Seifert surfaces, e.g. there exists an immersed Seifert surface $f\colon S\to X_U$ of genus one for the unknot $U$ such that there are curves on $f(S)$ which are non-trivial in $H_1(X_U)$.

• Maybe I'm missing something but why can't you just take a minimal genus embedded SS and then just re-immerse it by inserting some self-intersections? You could have a self-intersection curve parallel to the knot, which bounds an annulus in the surface, whose only double points are on the boundary. This immersed annulus gives a torus in the knot complement, and the torus bounds a solid torus in the knot complement. Initially I do not think there's a path in the space of immersions to an embedding. – Ryan Budney Feb 28 '14 at 1:14

I think this should be false for fibered knots. Consider a fibered knot, which is the mapping torus of a mapping class $\phi: S\to S$. Suppose there is a non-separating simple curve $c\subset S$ such that $\phi(c)\cap c=\emptyset$. Then one can form a "crossjoin" surface, by removing annulus neighborhoods $\mathcal{N}(c)$ and $\mathcal{N}(\phi(c))$, and inserting two crossing annuli which connect one boundary of $\mathcal{N}(c)$ to the opposite boundary of $\mathcal{N}(\phi(c))$. This was used by Cooper-Long-Reid to construct immersed surfaces in fibered manifolds. This immersed crossjoin surface will have a curve on it which is homologically non-trivial, and therefore the surface cannot be homotoped to an embedding.
Addendum: Actually, I realized that there is a simple way to form counterexamples in torus knots. The monodromy of a torus knot is finite-order, say $k$. Take a non-separating simple closed curve, and remove an annulus neighborhood. Then insert an annulus that winds $k$ times around the mapping torus direction, and connect up with the other boundary component of the annular neighborhood. The cross-cut is not needed, since this produces one component which is an immersed torus which is homologically trivial. So this works even for the trefoil knot.