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I recently uploaded a paper to the arXiv entitled Knots with small rational genus, joint with Cameron Gordon. The genesis of this paper was a couple of nice (and related) talks at Caltech by Matthew Hedden and Jake Rasmussen in 2007. They both talked about potential applications of the theory of knot Floer homology to the Berge conjecture. A Berge knot is a (tame) knot $K$ in the 3-sphere which lies on a genus two Heegaard surface, and with the property that on each side of the Heegaard surface there is a meridian disk that the knot intersects exactly once. Equivalently, the inclusion of the knot into each (closed) handlebody sends the generator of $\pi_1(K)$ to a generator of $\pi_1(\text{handlebody})$. Note that since the 3-sphere admits a unique (up to isotopy) Heegaard splitting of any genus, one may think of such a knot as lying on a specific genus 2 surface in $S^3$. Such knots were classified by Berge; they admit (Dehn) surgeries which result in (nontrivial) Lens spaces. The Berge conjecture is the converse; i.e.:

Berge Conjecture: Let $K$ be a knot in $S^3$ which admits a nontrivial Lens space surgery; i.e. there is a Lens space $L$ and a knot $K'$ in $L$ for which $S^3 - K$ is homeomorphic to $L - K'$. Then $K$ is a Berge knot.

An equivalent formulation (of course) is to try to classify knots in Lens spaces which admit an $S^3$ surgery, i.e. to identify the knots $K'$ as in the formulation of the conjecture above. The equivalent formulation says that these knots should be 1-bridge. The strategy of Hedden-Rasmussen (building on work of Ken Baker and Eli Grigsby) to approach the Berge conjecture depends on characterizing such knots by properties which can be detected by topological invariants that behave well under surgery. An example of such a topological invariant is the Casson invariant $\lambda(\cdot)$, a $\mathbb{Z}$-valued invariant of integer homology spheres which satisfies the surgery formula $\lambda(M_{n+1}) - \lambda(M_n) = \text{Arf}(K)$ where $M_i$ denotes the result of $1/i$ surgery on some integral homology sphere $M$ along a fixed knot $K$, and $\text{Arf}(K)$ is the Arf invariant. For more sophisticated invariants like knot Floer homology, the surgery formula is replaced by an exact triangle. One important piece of topological information that is detected by knot Floer homology is the genus of a knot. The approach to the Berge conjecture thus rests on Ken Baker’s impressive paper showing that small genus knots (in a sense to be made precise) in Lens spaces have small bridge number.

Hedden remarked in his talk that his work, and that of his collaborators “gave the first examples of an infinite family of knots that were characterized by their knot Floer homology”. Though technically true, I think this overstates the role of knot Floer homology in this case, since the knots (1-bridge knots in Lens spaces) are entirely characterized (up to isotopy) by their genus (and therefore by any topological invariant which detects genus). My immediate instinct was to think that knots with small genus in any 3-manifold should always be quite special, and that a complete classification might even be feasible. My paper with Cameron confirms this suspicion, and gives such a classification. Let me admit at this point that I am not especially interested in the Berge conjecture per se, although I find it interesting that new ideas in 3-manifold topology are starting to have something meaningful to say about it. In any case, I shall not have anything else to say about it (meaningful or otherwise) in this post.

First I should say that I have been using the word “genus” in a somewhat sloppy manner. For an oriented knot $K$ in $S^3$, a Seifert surface is a compact oriented embedded surface $\Sigma \subset S^3$ whose boundary is $K$. The genus of such a surface is a non-negative integer, and the least such genus over all Seifert surfaces is (said to be) the genus of $K$, denoted $g(K)$. Such a surface represents the generator in the relative homology group $H_2(S^3, K)$ which equals $H_1(K) = \mathbb{Z}$ since $S^3$ has vanishing homology in dimensions 1 and 2. This relative homology group is dual to $H^1(S^3 - K)$, which is parameterized by homotopy classes of maps from $S^3 - K$ to a circle (which is a $K(\mathbb{Z},1)$). The preimage of a regular value under a smooth map dual to the homology class is a smooth proper surface in $S^3 - K$ whose closure is a Seifert surface. It is immediate that $g(K)=0$ if and only if $K$ is an unknot; in other words, the unknot is “characterized” by its genus. There are infinitely many knots of any positive genus in $S^3$; on the other hand, there are only two fibered genus 1 knots — the trefoil and the figure 8 knot (three if you distinguish the left-handed from the right-handed trefoil), and it is worth remarking (from the point of view of the motivation of characterizing knots by topological invariants) that a theorem of Yi Ni says that fiberedness of knots can be detected by knot Floer homology.

For knots in integral homology $3$-spheres, the situation is very similar: every knot admits a Seifert surface, and the least genus of such a surface is the genus of a knot. The unknot is (always) characterized by the fact that it has genus $0$, but there are infinitely many knots of every positive genus. For a knot $K$ in a general $3$-manifold $M$ it is not so easy to define genus. A necessary and sufficient condition for $K$ to bound an embedded surface in its complement is that $[K]=1$ in $H_1(M)$. However, if $[K]$ has finite order, one can find an open properly embedded surface $\Sigma$ in the complement of $K$ whose “boundary” wraps some number of times around $K$. Technically, let $\Sigma$ be a compact oriented surface, and $f:\Sigma \to M$ a map which restricts to an embedding from the interior of $\Sigma$ into $M-K$, and which restricts to an oriented covering map from $\partial \Sigma$ to $K$ (note that we allow $\Sigma$ to have multiple boundary components). If $p$ is the degree of the covering map $\partial \Sigma \to K$, we call $\Sigma$ a $p$-Seifert surface, and define the rational genus of $\Sigma$ to be $-\chi^-(\Sigma)/2p$, where $\chi$ denotes Euler characteristic, and $\chi^-(\Sigma) = \min(0,\chi(\Sigma))$ (for a connected surface $\Sigma$). The reason to use Euler characteristic instead of genus is that Euler characteristic is multiplicative under coverings (unlike genus), and behaves well with respect to “local” operations on surfaces like cut-and-paste. Moreover, (negative) Euler characteristic, unlike genus, is a good measure of complexity for surfaces with possibly many boundary components. The coefficient of $2$ in the denominator reflects the fact that genus is “almost” $-2$ times Euler characteristic. With this definition, we say that the rational genus of $K$, for any knot $K \subset M$ with $[K]$ of finite order in $H_1(M)$, is the infimum of $-\chi^-(\Sigma)/2p$ over all $p$-Seifert surfaces for $K$ and all $p$. The purpose of our paper is to give a complete classification of knots with sufficiently small rational genus, and to show that such knots are always “geometric” — i.e. they can be isotoped into a normal form which is sensitive to the geometric decomposition of the ambient $3$-manifold $M$. Thus the concept of rational genus makes contact between the homological world of the Thurston norm, knot Floer homology and such invariants, and the geometric world of hyperbolic structures, JSJ decompositions and so on.

It is worth pointing out at this point that knots with small rational genus are not special by virtue of being rare: if $K$ is any knot in $S^3$ (for instance) of genus $g(K)$, and $K'$ in $M$ is obtained by $p/q$ Dehn surgery on $K$, then the knot $K'$ has order $p$ in $H_1(M)$, and $\|K'\| \le (g-1/2)/2p$. Since for “most” coprime $p/q$ the integer $p$ is arbitrarily large, it follows that “most” knots obtained in this way have arbitrarily small rational genus.

There is a precise connection between rational genus and the Thurston norm. There is an exact sequence in homology, which contains the fragment $H_2(M,K) \to H_1(K) \to H_1(M)$. Since $H_1(K) = \mathbb{Z}$, the kernel of $H_1(K) \to H_1(M)$ is generated by some class $n[K]$, and one can define the affine subspace $\partial^{-1}(n[K]) \subset H_2(M,K)$. By excision, we identify $H_2(M,K)$ with $H_2(M-\text{int}(N(K)), \partial N(K))$ where $N(K)$ is a tubular neighborhood of $K$. Under this identification, the rational genus of $K$ is equal to $\inf \|[\Sigma]\|_T/2$ where $\|\cdot\|_T$ denotes the (relative) Thurston norm, and the infimum is taken over classes in $H_2(M-\text{int}(N(K)), \partial N(K))$ in the affine subspace corresponding to $\partial^{-1}(n[K])$. Since the Thurston norm is a convex piecewise rational function, this infimum is realized at some rational point. In other words, rational genus of any knot is rational, and is realized by some $p$-Seifert surface, where $n$ as above divides $p$ (note: if $M$ is a rational homology sphere, then necessarily $p=n$, but if the rank of $H_1(M)$ is positive, this is not necessarily true, and $p/n$ might be arbitrarily large). This relationship to the Thurston norm also gives a straightforward algorithm to compute rational genus, since one can compute Thurston norm e.g. by linear programming in normal surface space relative to any triangulation.

The precise statement of results depends on the geometric decomposition of the ambient manifold $M$. By the geometrization theorem (of Perelman), a closed, orientable $3$-manifold is either reducible (i.e. contains an embedded sphere that does not bound a ball), or is a Lens space, or is hyperbolic, or is a small Seifert fiber space, or is toroidal (i.e. contains an essential ($\pi_1$-injective) embedded torus). For the record, the complete “classification” is as follows:

Reducible Theorem: Let ${K}$ be a knot in a reducible manifold ${M}$. Then either

1. ${\|K\| \ge 1/12}$; or
2. there is a decomposition ${M = M' \# M''}$, ${K \subset M'}$ and either
1. ${M'}$ is irreducible, or
2. ${(M',K) = (\mathbb{RP}^3,\mathbb{RP}^1)\#(\mathbb{RP}^3,\mathbb{RP}^1)}$

Lens Theorem: Let ${K}$ be a knot in a lens space ${M}$. Then either

1. ${\|K\| \ge 1/24}$; or
2. ${K}$ lies on a Heegaard torus in ${M}$; or
3. ${M}$ is of the form ${L(4k,2k-1)}$ and ${K}$ lies on a Klein bottle in ${M}$ as a non-separating orientation-preserving curve.

Hyperbolic Theorem: Let ${K}$ be a knot in a closed hyperbolic ${3}$-manifold ${M}$. Then either

1. ${\|K\| \ge 1/402}$; or
2. ${K}$ is trivial; or
3. ${K}$ is isotopic to a cable of the core of a Margulis tube.

Small SFS Theorem: Let ${M}$ be an atoroidal Seifert fiber space over ${S^2}$ with three exceptional fibers and let ${K}$ be a knot in ${M}$. Then either

1. ${\|K\| \ge 1/402}$; or
2. ${K}$ is trivial; or
3. ${K}$ is a cable of an exceptional Seifert fiber of ${M}$; or
4. ${M}$ is a prism manifold and ${K}$ is a fiber in the Seifert fiber structure of ${M}$ over ${\mathbb{RP}^2}$ with at most one exceptional fiber.

Toroidal Theorem: Let ${M}$ be a closed, irreducible, toroidal 3-manifold, and let ${K}$ be a knot in ${M}$. Then either

1. ${\|K\| \ge 1/402}$; or
2. ${K}$ is trivial; or
3. ${K}$ is contained in a hyperbolic piece ${N}$ of the JSJ decomposition of ${M}$ and is isotopic either to a cable of a core of a Margulis tube or into a component of ${\partial N}$; or
4. ${K}$ is contained in a Seifert fiber piece ${N}$ of the JSJ decomposition of ${M}$ and either
1. ${K}$ is isotopic to an ordinary fiber or a cable of an exceptional fiber or into ${\partial N}$, or
2. ${N}$ contains a copy ${Q}$ of the twisted ${S^1}$ bundle over the Möbius band and ${K}$ is contained in ${Q}$ as a fiber in this bundle structure;
5. or

6. ${M}$ is a ${T^2}$-bundle over ${S^1}$ with Anosov monodromy and ${K}$ is contained in a fiber.

The constant $1/402$ is presumably not optimal, but reflects the coarseness of certain geometric estimates at a particular step in the argument. Broadly speaking, there are two cases to consider: when the knot complement $M-K$ is hyperbolic, and when it is not. The complement $M-K$ is hyperbolic unless it contains an essential subsurface of non-negative Euler characteristic.

The case that $M-K$ is hyperbolic is conceptually easiest to analyze. Let $\Sigma$ be a surface, embedded in $M$ and with boundary wrapping some number of times around $K$, realizing the rational genus of $K$. The complete hyperbolic structure on $M-K$ may be deformed, adding back $K$ as a cone geodesic. Just as a cone can be obtained from a wedge of paper by gluing the two edges together, the geometry of a cone geodesic is locally modeled on the quotient space obtained from a (3-dimensional hyperbolic) wedge by gluing the two flat faces together. The thinner the wedge, the smaller the cone angle along the geodesic. For all sufficiently small angles $\theta > 0$, Thurston proved that there exists a unique hyperbolic metric on $M$ which is singular along a cone geodesic, isotopic to $K$, with cone angle $\theta$. Call this metric space $M_\theta$. The cone angle can be increased, deforming the geometry in a family of spaces, until one of the following three things happens:

1. The cone angle is increased all the way to $2\pi$, resulting in the complete hyperbolic structure on $M$, in which $K$ is isotopic to an embedded geodesic; or
2. The volume of the family of manifolds $M_\theta$ goes to zero (and either converges after rescaling to a Euclidean cone manifold, or converges after rescaling to have fixed diameter and injectivity radius going to zero everywhere); or
3. The cone locus bumps into itself (this can only happen for $\theta > \pi$).

As the cone angle along $K$ increases, so does the length of the cone geodesic. Simultaneously, the diameter of an embedded tube about this diameter decreases. While the diameter of the tube is big, the deformation can continue. Hodgson-Kerckhoff analyzed the kinds of degenerations that can occur, and obtained universal geometric control on how fast the tube diameter can shrink, or the length of the cone geodesic grow. They showed that the cone angle can be increased (giving rise to a family of singular hyperbolic structures $M_\theta$) either until $\theta = 2\pi$, or until the product $\theta \cdot \ell$, where $\ell$ is the length of the cone geodesic, is at least $1.019675$, at which point the diameter of an embedded tube about this cone geodesic is at least $0.531$. Since $\theta < 2\pi$ in the latter case, one obtains a lower bound on both the length of the cone geodesic and the diameter of an embedded tube, independent of $K$ or $M$.

Now, one would like to use this big tube to conclude that $\|K\|$ is large. This is accomplished as follows. Geometrically, one constructs a $1$-form $\alpha$ which agrees with the length form on the cone geodesic, which is supported in the tube, and which satisfies $\|d\alpha\|\le C$ pointwise for some (universal) constant $C$. Then one uses this $1$-form to control the topology of $\Sigma$. By Stokes theorem, for any surface $S$ homotopic to $\Sigma$ in $M-K$ one has an estimate

$1.019675/2\pi \le \ell = \int_K \alpha = \frac {1}{p} \int_S d\alpha \le \frac {C}{p} \text{area}(S)$

In particular, the area of $S$ divided by $p$ can’t be too small. However, it turns out that one can find a surface $S$ as above with $\text{area}(S) \le -2\pi\chi(S)$; such an estimate is enough to obtain a universal lower bound on $\|K\|$. Such a surface $S$ can be constructed either by the shrinkwrapping method of Calegari-Gabai, or the (related) PL-wrapping method of Soma. Roughly speaking, one uses the cone geodesic as an “obstacle”, and finds a surface $S$ of least area homotopic to $\Sigma$ (rel. boundary) subject to the constraint that it cannot cross the geodesic. Away from the cone geodesic, $S$ looks like an ordinary minimal surface. In particular, its intrinsic curvature is no more than the extrinsic curvature of hyperbolic space, which is $-1$ everywhere. Along the geodesic, $S$ looks like a bedsheet hanging on a clothesline; in particular, it does not accumulate any corners or atoms of positive curvature along this singularity, so the Gauss-Bonnet theorem gives the desired bound on $\text{area}(S)$.

This leaves the case that $M-K$ is not hyperbolic to analyze. As remarked above, this only occurs when $M-K$ contains an essential surface (which might be closed or proper) of non-negative Euler characteristic, i.e. a sphere, a disk, an annulus or a torus. In this case, one tries to make the intersection of $\Sigma$ with this essential surface as simple as possible; if one arranges this just right, every intersection contributes a definite amount to the topology of $\Sigma$, and one can conclude either that $\Sigma$ is complicated (in which case $\|K\|$ is large), or that the intersection is simple, and therefore draw some topological conclusion.

To actually do this in practice is quite complicated, but fortunately it relies on (largely combinatorial) methods developed at length by Gabai, Scharlemann, Gordon and others over the last 30 years to analyze (so-called) “exceptional surgeries”. Of course, the argument is still complicated, and this analysis takes up most of the length of the paper. It is also worth pointing out that every case provided for by the classification above actually occurs, with examples of arbitrarily small rational genus.

This paper raises several natural questions, the most obvious of which is whether the explicit (but quite small) constants can be improved in any way. The constant $1/402$ in the statement of the Toroidal Theorem is really only there to take care of a knot sitting inside a hyperbolic piece in the decomposition; a knot that interacts in a meaningful way with an essential torus necessarily has rational genus at least $1/24$ (for a precise statement, see the paper). As remarked above, knots of (ordinary) genus $1$ are very plentiful, even in $S^3$, and do not “see” any of the ambient geometry, so the wildest and most optimistic guess might be that there is a chance of classifying knots of rational genus at most $1/4$. There are some (very weak) reasons to think that this fraction is critical, at least in some cases, not least of which is the papers of Hedden and Ni mentioned above. But in the hyperbolic case, it is probably not easy to get a better estimate using purely geometric arguments.

Another approach might be to try to substitute another conclusion (again in the hyperbolic case) than that $K$ be isotopic to the cable of a core of a Margulis tube. For instance, one might ask for $K$ to admit an insulator family (of the kind Gabai used here), or one might merely ask that $K$ be unknotted in the universal cover, or satisfy some other condition. This goes to the heart of a very, very difficult and important question, namely how to identify geometric features of codimension 2 objects in (especially hyperbolic) geometric 3-manifolds from purely topological properties. If I am optimistic, then I can imagine that this paper makes a contribution, however small, to this ongoing project.