Ribbon concordance of fibered knots and compressions of surface homeomorphisms

Abstract: We prove that simplicial volume and dilatation are monotone under ribbon concordance between fibered knots in $S<sup>3$, and that every fibered knot has only finitely many predecessors in the ribbon-concordance partial order, providing evidence for questions raised by Gordon. We also give an algorithm to enumerate, up to symmetries, all minimal compressions of a surface homeomorphism, extending a theorem of Casson--Long. This yields an algorithm to find all knots that are strongly homotopy-ribbon concordant to a given fibered knot in some homotopy $I\times S<sup>3$. Our study of minimal compressions also provides an alternative perspective on results of Miyazaki concerning nonsimple fibered ribbon knots.

• The paper proves that simplicial volume and monodromy dilatation do not increase under ribbon or strongly homotopy-ribbon concordance between fibered knots, while each fibered knot has finitely many predecessors.
• The authors reduce concordance questions to compressions of surface homeomorphisms and develop topological, bounded-cohomology, and fundamental-group methods to establish these monotonicity results.
• The paper extends Casson–Long by algorithmically classifying minimal compressions into six canonical forms, enabling effective enumeration of predecessors and new structural results for nonsimple fibered knots.

Overview

This paper, by Ian Agol and Qiuyu Ren, establishes monotonicity results for ribbon concordance between fibered knots in $S^3$ and develops the topological machinery needed to prove them. The main results answer affirmatively, in the fibered case, questions posed by Gordon in 1981: simplicial volume and dilatation are shown to be monotone under ribbon concordance, and every fibered knot is shown to have only finitely many ribbon predecessors. The proofs are purely topological, in contrast to independent Floer-theoretic proofs obtained by Baldwin–Sivek. A substantial secondary contribution is an algorithmic classification of minimal compressions of surface homeomorphisms that extends a theorem of Casson–Long by removing the pseudo-Anosov hypothesis, together with applications to nonsimple (satellite) fibered ribbon knots in the spirit of Miyazaki.

Background and Gordon's questions

A concordance from a knot $J$ to a knot $K$ is a smoothly embedded annulus in $I\times S^3$ cobounding them; it is a ribbon concordance if projection to $I$ restricts to a Morse function with no index-2 critical points. The relation $J\le K$, meaning $J$ is ribbon concordant to $K$, was introduced by Gordon, who conjectured it is a partial order — a conjecture proved by Agol using sutured Floer homology. Several invariants are known to be monotone under $\le$: the Alexander polynomial and $S$-equivalence class (Gilmer), Seifert genus and knot Floer homology (Zemke), and Khovanov homology (Levine). Gordon raised two further questions: whether simplicial volume satisfies $J$0 whenever $J$1, and whether any descending chain $J$2 must stabilize. Baldwin–Sivek later refined the second question to ask whether each knot has only finitely many ribbon predecessors.

The present paper answers all three affirmatively when $J$3 is fibered. Since fiberedness descends under ribbon concordance (Silver; Kochloukova), the restriction to fibered $J$4 automatically restricts to fibered $J$5. The precise statements are:

• Simplicial volume: if $J$6 is fibered and $J$7, then $J$8.
• Dilatation: with $J$9 defined as the maximal dilatation among pseudo-Anosov pieces of the monodromy $K$0 under its Nielsen–Thurston decomposition, $K$1 whenever $K$2.
• Finiteness: a fibered knot has only finitely many $K$3 with $K$4.

Baldwin–Sivek established analogues of the first two statements independently via Floer-theoretic methods; the authors note their topological arguments yield sharper bounds between fibered knots. The finiteness theorem was also obtained independently by Baldwin–Hanselman–Sivek as a corollary of the stronger statement that every knot has at most finitely many fibered ribbon predecessors.

Strongly homotopy-ribbon concordances and compressions

A key reduction uses a consequence of Casson–Gordon: for fibered $K$5, the knot $K$6 admits a strongly homotopy-ribbon concordance to $K$7 in some homotopy $K$8 (written $K$9) if and only if the monodromy of $I\times S^3$0 compresses to that of $I\times S^3$1. Here a strongly homotopy-ribbon concordance is one whose complement admits a relative handle decomposition with only 1- and 2-handles; every ribbon concordance is of this type, and the arguments of Agol's partial-order proof show $I\times S^3$2 is itself a partial order. All three main theorems are actually proved in the stronger form with $I\times S^3$3 replaced by $I\times S^3$4. Whether $I\times S^3$5 is strictly finer than $I\times S^3$6 remains open.

A compression body $I\times S^3$7 is viewed as a cobordism from its interior boundary $I\times S^3$8 to its exterior boundary $I\times S^3$9; a homeomorphism $I$0 compresses in $I$1 when $I$2 extends over $I$3, yielding the restricted homeomorphism on $I$4. The necessity direction of the Casson–Gordon criterion follows by deleting a tubular neighborhood of an arc in the concordance to obtain a ribbon disk for $I$5 in a homotopy 4-ball, applying Casson–Gordon's compression theorem to the closed monodromy $I$6, and observing surjectivity of $I$7. Sufficiency is constructive: the mapping torus of the extension $I$8 has vertical boundary $I$9, and attaching $J\le K$0 killing meridional slopes produces a homotopy $J\le K$1 containing the desired concordance.

Monotonicity of simplicial volume

The volume proof proceeds through bounded cohomology. Using Gromov's duality between the $J\le K$2-norm on homology and the $J\le K$3-norm on bounded cohomology, the authors prove that for a homeomorphism $J\le K$4 of a connected compression body restricting to $J\le K$5 on the interior boundary and $J\le K$6 on the exterior boundary,

$J\le K$7

where $J\le K$8 denotes the mapping torus. After doubling along the vertical boundary, the argument exploits Gromov's theorem that the cyclic cover $J\le K$9 induces an isometric injection on bounded cohomology (since $J$0 is amenable), together with naturality of the averaging operator $J$1 appearing in Gromov's proof. Naturality follows because $J$2 is defined via a non-principal ultrafilter, whose limit operation commutes with induced maps once the ultrafilter is fixed. Since $J$3 retracts onto $J$4, pulling back a dual class from $J$5 through the retraction and averaging produces a class on $J$6 with no larger norm, which yields the inequality via the duality formula. Combined with the Casson–Gordon criterion, this gives monotonicity of simplicial volume for fibered knots under $J$7, hence under $J$8.

Monotonicity of dilatation

For dilatation, the bridge is the identity $J$9 relating the dilatation of a surface homeomorphism to the growth rate of the induced map on $K$0, where $K$1 is the exponential growth rate of word lengths under iterates of an endomorphism. Fathi–Laudenbach–Poénaru proved this for pseudo-Anosov maps; the authors extend the equality to arbitrary homeomorphisms by choosing a metric adapted to the Nielsen–Thurston decomposition — invariant metrics on periodic pieces, singular Euclidean metrics on pseudo-Anosov pieces, and controlled metrics on the annular neighborhoods of the canonical reduction system — and estimating loop lengths piecewise, with contributions from reducible annuli growing at most linearly.

The essential observation is then elementary: a compression body with connected nonempty boundary components yields intertwined maps $K$2. Word length does not increase under surjections, giving $K$3; and since $K$4 is a quasi-isometric embedding admitting a retract, coarse preservation of word length gives $K$5. Chaining these inequalities through the Casson–Gordon criterion yields $K$6.

Finiteness of predecessors

Theorem 3 (finiteness) is deduced twice. First it follows formally from the dilatation bound combined with genus and Alexander polynomial bounds. The second, more structural proof analyzes the JSJ decomposition of $K$7 directly. The knot complement is recovered from finite data: the rooted JSJ graph, the pieces, their boundary identifications, and meridian-longitude parametrizations. Each datum is shown to admit only finitely many possibilities:

• Graph size: JSJ tori correspond to orbits of components of the canonical reduction system $K$8 of $K$9, whose size is bounded since $\le$0 has no parallel components on a surface of bounded genus ($\le$1).
• Seifert fibered pieces: these are complements of keychain links, cable patterns, or torus knots. Cable and torus knot parameters $\le$2 are bounded using the satellite formulas for genus and Alexander polynomial together with divisibility $\le$3; the keychain parameter is bounded by graph size.
• Hyperbolic pieces: these correspond to mapping tori of pseudo-Anosov pieces of $\le$4, whose fiber complexity is bounded by $\le$5 and whose dilatations are bounded by $\le$6; finitely many such mapping tori exist.
• Boundary parametrizations: longitudes are controlled via Thurston-norm minimality of $\le$7 and non-degeneracy of the Thurston norm on hyperbolic pieces; meridians are controlled by the knot complement problem and bounds on boundary-reducible surgeries applied to Dehn fillings described by Fox's re-embedding theorem.

An important technical point is that $\le$8 has nonzero winding number in the solid torus bounded by any JSJ torus, since otherwise $\le$9 would inject into the commutator subgroup $S$0.

Algorithmic compressions of surface homeomorphisms

Casson–Long proved that a pseudo-Anosov homeomorphism admits at most finitely many minimal compressions, detectably and findable algorithmically. The pseudo-Anosov hypothesis is genuinely necessary: the identity on a surface with a complicated component admits infinitely many minimal compressions, one per isotopy class of essential curve, and even pseudo-Anosov homeomorphisms can admit infinitely many non-minimal compressions (e.g., a pseudo-Anosov on $S$1 built from Dehn twists about two filling curves compresses to $S$2 in infinitely many ways).

The paper's generalization removes the pseudo-Anosov assumption by quotienting by the symmetry group $S$3 of rel-boundary homeomorphisms commuting with $S$4 up to isotopy. The main theorem states that any orientation-preserving homeomorphism of a compact oriented surface admits at most finitely many minimal compressions up to symmetry, and there is an algorithm to enumerate them. A corollary drops minimality as well: every surface homeomorphism compresses to only finitely many homeomorphisms up to isotopy and conjugation, all computable. The remark that even for pseudo-Anosov inputs one must analyze compressions of periodic and reducible homeomorphies underscores that the generalization is not merely cosmetic.

The proof proceeds by choosing an essential compressing curve $S$5 in minimal position with respect to the canonical reduction system $S$6 and running a cut-and-paste argument against pulled-tight iterates $S$7. Minimality of $S$8 forces $S$9, and the case $J$00 is subsequently ruled out by exhibiting a disjoint essential compressing curve. This yields six mutually exclusive canonical forms for minimal compressions:

Form	Description
1.1	Compressing an orbit $J$01 of a component of $J$02, subject to a planar-piece condition ensuring minimality
1.2	Compressing an orbit of an essential curve in a periodic piece, characterized via the quotient orbifold $J$03
1.3	A minimal compression of a single pseudo-Anosov piece, glued to the product elsewhere
2.1.1	A product region identifying two distinct pseudo-Anosov pieces adjacent to a common reduction curve, via an orientation-reversing intertwining map $J$04
2.1.2	A twisted product region bounding a single pseudo-Anosov piece adjacent twice to a reduction curve, via an involution $J$05 squaring to $J$06
2.2	A twisted product region bounding a pseudo-Anosov piece adjacent on both sides to one reduction curve, via a free involutive conjugator $J$07

Two auxiliary algorithmic results are developed for forms 2.1.2 and 2.2: algorithms to find all square roots of a pseudo-Anosov mapping class, and all involutive conjugators, both via the periodic splitting sequence of the stable lamination in the sense of Agol's ideal triangulations work. A fractional Dehn twist coefficient condition (the sum of coefficients about the two boundaries arising from a reduction curve must vanish) is necessary for forms 2.1.x and 2.2, and a lemma shows this condition suffices to promote free commutativity with $J$08 to commutativity rel fixed boundaries after modifying by boundary twists. The enumeration algorithm combines these lemmas with Hemion's solution to the mapping class conjugacy problem and an algorithm for computing centralizers (symmetry groups) due to Rafi–Tao, adapted to the fixed-boundary setting.

Applications to nonsimple fibered knots

The classification of minimal compressions gives conceptually simpler proofs of several of Miyazaki's results on nonsimple fibered ribbon knots, avoiding heavy use of Jaco–Shalen–Johannson characteristic submanifold theory. As an illustration, the authors classify all compressions of the monodromy of $J$09, the $J$10-cable of the figure-8 knot, showing the unique chain of minimal compressions terminates at a monodromy on $J$11 that admits no further compression; in particular the monodromy never compresses to $J$12, so $J$13 is not strongly homotopy-ribbon in any homotopy 4-ball. This knot has been the subject of recent work showing it is not slice, so it cannot provide a counterexample to the slice-ribbon conjecture.

The main structural result here is a factorization theorem: for prime fibered knots $J$14, one has $J$15 if and only if each $J$16 dominates a connected sum of factors drawn from the $J$17's, with the leftover $J$18's occurring in mirrored pairs $J$19. The necessity proof is an induction on total genus, analyzing which canonical form the minimal subcompressions take; the hardest cases involve torus-knot and cable-pattern monodromies, where fractional Dehn twist coefficient bookkeeping and orbifold analysis force the mirrored-pair conclusion. Two consequences follow conditionally on standard conjectures: assuming the slice-ribbon conjecture, each concordance class contains at most one $J$20-minimal fibered knot; assuming additionally the smooth 4-dimensional Poincaré conjecture, no torsion element of order greater than 2 in the knot concordance group can be represented by a fibered knot. Both conclusions were previously known from Miyazaki's work under the same hypotheses.

Limitations and open questions

Several boundaries of the results deserve emphasis. The monotonicity and finiteness theorems are proved only for fibered knots; whether simplicial volume is monotone under ribbon concordance in general remains Gordon's original open question. Whether $J$21 is strictly finer than $J$22 is unknown. The conditional corollaries depend on the slice-ribbon conjecture and the smooth 4-dimensional Poincaré conjecture, and the authors note the implication may run in the other direction: the predecessor-enumeration algorithm could, modulo those conjectures, decide smooth sliceness of fibered knots, and combining it with the Khovanov obstruction to strongly homotopy-ribbon concordance offers a potential route to detecting an exotic $J$23. It is also open whether every concordance class has a unique minimum with respect to $J$24 or $J$25, and whether the concordance group has torsion of order greater than 2 at all. Finally, the paper poses the question of whether concordant fibered knots $J$26 must dominate a common fibered knot $J$27 (equivalently, be related by a zigzag of ribbon concordances); a partial result shows that for hyperbolic knots of genus at most 3 that are $J$28-minimal, such a common dominator would force $J$29.

Conclusion

This paper settles three of Gordon's questions about ribbon concordance in the fibered case, proving monotonicity of simplicial volume and dilatation and finiteness of predecessors by purely topological means centered on the interplay between concordances and compressions of surface monodromies. Its algorithmic classification of minimal compressions into six canonical forms extends Casson–Long beyond the pseudo-Anosov setting and yields effective enumeration of all strongly homotopy-ribbon predecessors of a given fibered knot, while supplying streamlined proofs of Miyazaki's structure theorems for nonsimple fibered ribbon knots. The remaining questions — monotonicity without fiberedness, the relationship between $J$30 and $J$31, and the common-dominator question for concordant fibered knots — delineate the natural scope for further work.