The salient crossings of a crown diagram

The author would like to thank José Romàn Aranda Cuevas, Yasha Eliashberg, Bob Gompf, Rob Kirby, Robert Lewis, Robert Lipshitz, Cary Malkiewich, Jeff Nye, Tim Perutz, and Lisa Piccirillo for helpful conversations during the course of this work. The author would especially like to thank Umur Çiftçi for his crucial contribution of the software EQMaker.

A Morse 2-function is a smooth map $\alpha\colon\thinspace M\to S$, where $S$ is a smooth 2-manifold, such that each critical point of $f$ is locally modeled on a fold critical point,

or a cusp critical point,

The critical locus $\operatorname{crit}f$ forms a smooth 1-submanifold of $M$, stratified into a 1-dimensional stratum of fold points and a finite stratum of cusp points. Crown diagrams come from what this paper calls crown maps, after a suggestion of R. Kirby (In previous work such as [B, BH, W1, W2], a crown diagram was called a surface diagram). A crown map is a Morse 2-function $f$ for which $S=S^{2}$, the signs above are both chosen to be negative (that is, the critical points are indefinite folds and cusps), and $\operatorname{crit}f$ is a single circle on which $f$ is injective (Figure 1(c)). A crown map induces a singular fiber bundle called a crown fibration on $M$ as follows. The critical values of $f$ partition $S^{2}$ into a pair of disks, the higher-genus side whose regular fibers are diffeomorphic to a closed oriented genus-$g$ surface $\Sigma$, and the lower-genus side, which has closed oriented genus $g-1$ fibers. The two disks meet along the critical image of $f$, which is a circle with corners, where each corner is the image of a cusp point.

The local model for an indefinite fold point is $(h,\operatorname{id}_{\mathbb{R}})\colon\thinspace\mathbb{R}^{3}\times\mathbb{R}\to\mathbb{R}\times\mathbb{R}$, where $h$ is the local model for a 3-dimensional Morse critical point of index 1 or 2 (in Figure 1(a), it is index 1 with $h$ understood to increase from left to right along any horizontal arc). Such a Morse function induces a fibration of $R^{3}$ by the one- and two-sheeted hyperboloids $x^{2}+y^{2}-z^{2}=const$, and each one-sheeted hyperboloid $\{x^{2}+y^{2}-z^{2}=const>0\}$ intersects the $xy$ plane in a distinguished circle. In the image of a Morse 2-function, the region of regular values corresponding to $const>0$ is called the higher genus side of the fold arc and the distinguished circle is a simple closed curve in the regular fiber called a vanishing cycle. All of the vanishing cycles are depicted as simple closed curves in a fixed reference fiber $f^{-1}(p)$, measured by choosing a reference path from $p$ to the image of a fold arc, and a connection for the fibration, then marking the points in the regular fiber that flow to the fold point. In general, the isotopy class of the vanishing cycle depends on these choices; however, the crown condition implies that the vanishing cycles in this paper are all non-separating and lie in a smooth, closed oriented surface $\Sigma$ of genus at least 3, and different choices of higher-genus reference fiber, connection and reference path yield isotopic vanishing cycles [BH, Section 4].

With this understood, a crown diagram of the smooth, closed oriented 4-manifold $M$ is the oriented diffeomorphism class of a pair $(\Sigma,\Gamma)$, where $\Sigma$ is a closed, oriented surface of genus greater than 2 and $\Gamma$ is a collection of simple closed curves in $\Sigma$ which occurs as the counter-clockwise ordered sequence of vanishing cycles of a crown fibration whose oriented higher-genus fiber is $\Sigma$. A diffeomorphism of crown diagrams is required to preserve the cyclic ordering of $\Gamma$. According to [W1, Corollary 2], a crown diagram specifies $M$ up to orientation-preserving diffeomorphism.

The local model for an indefinite cusp is the same as the local model for the introduction via homotopy of a canceling pair of index-1 and index-2 Morse critical points in a 3-manifold (the homotopy parameter would increase from left to right in Figure 1(b)). For this reason, the vanishing cycles of the two fold arcs which meet at a cusp point can be depicted to have exactly one transverse crossing in $\Sigma$.

At various points it will be useful to designate a vanishing cycle as the “first” element of $\Gamma$, indexing with integers as $(\gamma_{1},\ldots,\gamma_{k})$, where the ordering of any three consecutive elements respects the cyclic order of $\Gamma$. Such a choice will be called a labeling of $\Gamma$.

It may be possible (and would be interesting to find) that two non-homotopic crown maps $M\to S^{2}$ have the same crown diagram. In that case, both homotopy classes of maps would be homotopy classes of the diagram.

The cyclic ordering of $\Gamma$ gives data reminiscent of crossings in knot diagrams as follows. For $i,j\in\mathbb{Z}/k\mathbb{Z}$, an intersection $x\in\gamma_{i}\cap\gamma_{j}$ is associated with the two arcs $[\theta_{i},\theta_{j}]\times\{x\}$ and $[\theta_{j},\theta_{i}]\times\{x\}$ whose union is $S^{1}\times\{x\}$. These are both oriented as subspaces of $S^{1}\times\Sigma$, connecting the circles $\{\theta_{i}\}\times\gamma_{i}$ and $\{\theta_{j}\}\times\gamma_{j}$. These arcs are called the chords of $x$.

Any chord $[\theta_{i},\theta_{j}]\times\{x\}$ singles out the sublink of $L$ contained in $[\theta_{i},\theta_{j}]\times\Sigma$ and the complimentary chord $[\theta_{j},\theta_{i}]\times\{x\}$ singles out the sublink of $L$ contained in $[\theta_{j},\theta_{i}]\times\Sigma$, because the interiors of the two intervals are disjoint positively oriented arcs in $S^{1}$. The crossing $x$ appears in both sublinks, and if $\gamma_{i}$ passes over $\gamma_{j}$ in one, then $\gamma_{i}$ passes under $\gamma_{j}$ in the other. In either case, given a chord $a$, one may choose a labeling of $\Gamma$ so that the under-crossing vanishing cycle at $x$ is $\gamma_{1}$. Then the over-crossing vanishing cycle would be $\gamma_{n}$ for some $n\in\{2,\ldots,k\}$ and that corresponding sublink is

For each labeling, the two strands of each crossing in the crown diagram necessarily come from an intersection of two distinct circles, so this paper adopts the visual convention that, having chosen a crossing and a labeling, the under-crossing strand is the one with the lesser subscript. Using this convention, the corners of the disks adjacent to each crossing in a sublink $L_{a}$ can be given signs as in Figure 2.

Since $\gamma_{i}$ intersects $\gamma_{i+1}$ at a unique point for each $i\in\mathbb{Z}/k\mathbb{Z}$, there are $k$ distinguished chords whose associated sublinks each have two components $\gamma_{i},\gamma_{i+1}$ and one crossing, which is a salient crossing. These chords are the salient chords of $\mathcal{A}(\Sigma,\Gamma)$.

Suppose $(\Sigma,\Gamma)$ is allowable, choose a labeling of $\Gamma$ and let $U$ be the collection of disks forming the disjoint union $\Sigma\setminus\cup_{i}\gamma_{i}=\sqcup D^{2}$. For any crossing there is a unique chord which does not intersect $[\theta_{k},\theta_{1}]\times\Sigma$; in other words, it is the chord which is contained in the link diagram whose over-crossing strands are designated by the labeling. Thus, a labeling gives signs to all the corners in the diagram. Given a labeling and a disk $u\in U$, denote by $Q_{+}(u)$ the set of chords corresponding to positive corners of $u$ and denote by $Q_{-}(u)$ the set of chords corresponding to negative corners of $u$. Note that allowability implies $Q_{+}(u)$ and $Q_{-}(u)$ are disjoint. Assign a variable $\operatorname{gr}(a)$ to each chord $a$ and impose the following grading equations for every labeling of $\Gamma$ and for every region $u$:

The system of equations given by Equation 3 will be referred to as the grading system. It is defined as a collection of equations associated to disks, but Equation 3 also makes sense for any region $u\subseteq\Sigma$ whose boundary is a union of arcs of vanishing cycles that meet at crossings. The right hand side of Equation 3 assigns an integer to each $u\in U$, and it is not hard to check that this assignment is additive in the sense that the grading equation for a union of disks is the sum of the grading equations for the disks.

It is now possible to define the main invariant of the paper.

The grading system is defined as a system of equations over the integers, but it may only have solutions in a finite cyclic group, and this solution is not necessarily unique.

According to [FS, Theorem 1.9], if $K$ is a fibered knot, then $E(1)$ is homeomorphic to the knot surgery manifold $E(1)_{K}$ and the Seiberg-Witten invariant of $E(1)_{K}$ is the Alexander polynomial $\Delta_{K}(t)$. In the same paper, Fintushel and Stern described a Lefschetz fibration structure on $E(1)_{K}$. The current paper uses a monodromy factorization of this Lefschetz fibration due to Yun [Y], and it refers to KnotInfo for the Alexander polynomials and monodromy for the knots $7_{6}$ and $10_{133}$. These are genus-2 fibered knots with the same Alexander polynomial. Genus-1 knots would have simpler diagrams, but they are not used because the two fibered genus-1 knots are the trefoil $3_{1}$ and the figure-8 knot $4_{1}$, which are distinguished by their Alexander polynomials.

It has been known since at least the publication of [L] that one may convert a Lefschetz fibration over $S^{2}$ into a crown map by a homotopy of the fibration in which one converts the Lefschetz critical points into 3-cusped circles and performs cusp merges to unite the circles, resulting in a crown map whose crown diagram is called a coronation of the Lefschetz fibration (see [W4, Figure 7] for an example of how this looks in a crown diagram). Using the notion of surgered diagrams from [W2] to deduce properties of the resulting vanishing cycles, Behrens and Hayano gave a description of the full collection of coronations of a Lefschetz fibration in terms of mapping class groups of surfaces [BH]. In [W4] there is a fully diagrammatic interpretation of the same homotopy, yielding a single, much more explicit, member of this collection. Recording the vanishing cycles using the notation of train tracks, the weights of such a diagram reach the tens of thousands for a diagram with 22 vanishing cycles, such as the ones for knot surgery on $E(1)$. Instead, this paper will consider a simpler kind of diagram which is slide-equivalent to any coronation of Yun’s Lefschetz fibration. It comes with a collection of crossings that end up being the salient crossings of a coronation. Before discussing this, though, recall that the invariance of $\mathcal{S}$ applies to crown diagrams with a common homotopy class. The manifolds $E(1)_{7_{6}}$ and $E(1)_{10_{133}}$ are nominally different manifolds, so it is necessary to explain how their coronations would have a common homotopy class if their corresponding smoothings were isotopic.

Figure 5 depicts the relevant information to obtain Yun’s monodromy factorizations for $E(1)_{7_{6}}$ and $E(1)_{10_{133}}$. Starting with the five circles in Figure 5(a), cut the surface at the four circles shown in Figure 5(b). Let $t_{x}$ denote the right Dehn twist about the circle $x$. Then, according to KnotInfo, and using the circles shown in Figure 5(d), the monodromy of $7_{6}$ can be written as the composition of Dehn twists

applied from left to right, while the monodromy of $10_{133}$ can be written

According to [Y, Theorem 2.11], the Lefschetz monodromy factorization for $E(1)_{K}$ is

The collection of circles that results for $7_{6}$ is a messy picture, which can be simplified considerably by applying one Dehn twist and pushing the feet of two of the handles around, switching their places in a clockwise rotation as in Figure 6. The resulting collection of Lefschetz vanishing cycles can be seen in Figures 7(a)–7(j) by deleting the pair of long rectangles and connecting the resulting ends with vertical line segments. The simplifying diffeomorphism is not used for $E(1)_{10_{133}}$, and the resulting collection of circles appears analogously in Figure 8. Note that in those figures, the reversal of ordering of the circles comes from the fact that the circles in the above monodromy factorization are vanishing cycles of a Lefschetz fibration, while the circles in Figures 7 and 8 get their ordering from the merging algorithm depicted in [W4, Figure 6]. In short, the so-called pseudocoronation shown in Figure 7 or Figure 8 comes from choosing an arc in the fiber of the Lefschetz fibration that crosses each of the fibration’s Lefschetz vanishing cycles exactly once, replacing that arc with a tube over which all of those vanishing cycles run, adding two more vanishing cycles in a standard way as in Figures 7(k) and 8(k), then ordering the vanishing cycles as required by [W4, Figure 6]: In that figure, $r_{i}$ or $r_{i}^{s}$ corresponds to the $i^{th}$ Lefschetz vanishing cycle. Circle 20 in Figures in Figures 7(k) and 8(k) corresponds to $g_{1}$ in [W4, Figure 6], and circle 22 in Figures 7(k) and 8(k) corresponds to $b_{1}$ in [W4, Figure 6].

According to the algorithm in [W4] and the subsequent proofs that the algorithm is realizable as a sequence of slides between vanishing cycles in the pseudocoronation, a coronation of either of the Lefschetz fibrations of $E(1)_{7_{6}}$ or $E(1)_{10_{133}}$ can be obtained from applying slides among the vanishing cycles in Figures 7 and 8, while keeping the staircase-shaped arrangement in Figures 7(l) and 8(l) unchanged. Since $\mathcal{S}$ is invariant under such slides, one could instead use the grading system as given by the depicted pseudocoronations, taking the crossings between consecutive vanishing cycles in the staircase arrangement as the salient crossings: The first step in the algorithm for finding the vanishing cycle $\gamma_{i}$ of the coronation is to repeatedly slide $\gamma_{i}$ over $\gamma_{22}$ along $\gamma_{i+1}$ to eliminate all crossings between $\gamma_{i}$ and $\gamma_{i+1}$ other than the salient crossing in the staircase.

The number of regions in the pseudocoronations given by Figures 7 and 8 is large, and each region has multiple equations coming from the different choices of labeling. For this reason, there is software written by Umur Çiftçi, called EQMaker, which produces the grading system as a text file, and there is a list of Mathematica commands for extracting from this system a new system of equations which expresses the salient set as a sum of formal parameters, all available at the author’s website. The grading systems for these diagrams both have integer solutions, the rank of the system for $E(1)_{7_{6}}$ is 21, and the rank of the system for $E(1)_{10_{133}}$ is 20. For this reason, the salient sets are not equal, so the corresponding smoothings are not isotopic.