Steven Sivek

There has been a great deal of interest in understanding which knots are characterized by which of their Dehn surgeries. We study a 4-dimensional version of this question: which knots are determined by which of their traces? We prove several results that are in stark contrast with what is known about characterizing surgeries, most notably that the 0-trace detects every L-space knot. Our proof combines tools in Heegaard Floer homology with results about surface homeomorphisms and their dynamics. We also consider nonzero traces, proving for instance that each positive torus knot is determined by its n-trace for any n≤0, whereas no non-positive integer is known to be a characterizing slope for any positive torus knot besides the right-handed trefoil.

The A-polynomial of a knot is defined in terms of SL(2,ℂ) representations of the knot group, and encodes information about essential surfaces in the knot complement. In 2005, Dunfield–Garoufalidis and Boyer–Zhang proved that it detects the unknot using Kronheimer–Mrowka's work on the Property P conjecture. Here we use more recent results from instanton Floer homology to prove that a version of the A-polynomial distinguishes torus knots from all other knots, and in particular detects the torus knot T_a,b if and only if one of |a| or |b| is 2 or both are prime powers. These results enable progress towards a folklore conjecture about boundary slopes of non-torus knots. Finally, we use similar ideas to prove that a knot in S³ admits infinitely many SL(2,ℂ)-abelian Dehn surgeries if and only if it is a torus knot, affirming a variant of a conjecture due to Sivek–Zentner.

We use instanton gauge theory to prove that if Y is a closed, orientable 3-manifold such that H₁(Y;ℤ) is nontrivial and either 2-torsion or 3-torsion, and if Y is neither #^r ℝℙ³ for some r≥1 nor ±L(3,1), then there is an irreducible representation π₁(Y) → SL(2,ℂ). We apply this to show that the Kauffman bracket skein module of a non-prime 3-manifold has nontrivial torsion whenever two of the prime summands are different from ℝℙ³, answering a conjecture of Przytycki (Kirby problem 1.92(F)) unless every summand but one is ℝℙ³. As part of the proof in the 2-torsion case, we also show that if M is a compact, orientable 3-manifold with torus boundary whose rational longitude has order 2 in H₁(M), then M admits a degree-1 map onto the twisted I-bundle over the Klein bottle.

Let Y be a closed, orientable 3-manifold with Heegaard genus 2. We prove that if H₁(Y;ℤ) has order 1, 3, or 5, then there is a representation π₁(Y)→SU(2) with non-abelian image. Similarly, if H₁(Y;ℤ) has order 2 then we find a non-abelian representation π₁(Y)→SO(3). We also prove that a knot K in S³ is a trefoil if and only if there is a unique conjugacy class of irreducible representations π₁(S³∖K)→SU(2) sending a fixed meridian to diag(i,-i).

Bill Thurston proved that taut foliations of hyperbolic 3-manifolds have Euler classes of norm at most one, and conjectured that any integral second cohomology class of norm equal to one is realised as the Euler class of some taut foliation. Recent work of the second author, joint with David Gabai, has produced counterexamples to this conjecture. Since tight contact structures exist whenever taut foliations do and their Euler classes also have norm at most one, it is natural to ask whether the Euler class one conjecture might still be true for tight contact structures. In this short note, we show that the previously constructed counterexamples for Euler classes of taut foliations in [Yaz20] are in fact realised as Euler classes of tight contact structures. This shows that the Euler class one conjecture has a chance of being true for tight contact structures.

We prove that 0 is a characterizing slope for infinitely many knots, namely the genus-1 knots whose knot Floer homology is 2-dimensional in the top Alexander grading, which we classified in recent work and which include all (-3,3,2n+1) pretzel knots. This was previously only known for 5₂ and its mirror, as a corollary of that classification, and for the unknot, trefoils, and the figure eight by work of Gabai from 1987.

We give new proofs that Khovanov homology detects the figure eight knot and the cinquefoils, and that HOMFLY homology detects 5₂ and each of the P(−3,3,2n+1) pretzel knots. For all but the figure eight these mostly follow the same lines as in previous work. The key difference is that in honor of Tom Mrowka's 60th birthday, the arguments here use instanton Floer homology rather than knot Floer homology.

We prove that all rational slopes are characterizing for the knot 5₂, except possibly for positive integers. Along the way, we classify the Dehn surgeries on knots in S³ that produce the Brieskorn sphere Σ(2,3,11), and we study knots on which large integral surgeries are almost L-spaces.

We prove for the first time that knot Floer homology and Khovanov homology can detect non-fibered knots, and that HOMFLY homology detects infinitely many such knots; these theories were previously known to detect a mere six knots, all fibered. These results rely on our main technical theorem, which gives a complete classification of genus-1 knots in the 3-sphere whose knot Floer homology in the top Alexander grading is 2-dimensional. We discuss applications of this classification to problems in Dehn surgery which are carried out in two sequels. These include a proof that 0-surgery characterizes infinitely many knots, generalizing results of Gabai from his 1987 resolution of the Property R conjecture.

We continue our study of the integer-valued knot invariants ν^#(K) and r₀(K), which together determine the dimensions of the framed instanton homologies of all nonzero Dehn surgeries on K. We first establish a "conjugation" symmetry for the decomposition of cobordism maps constructed in our earlier work, and use this to prove, among many other things, that ν^#(K) is always either zero or odd. We then apply these technical results to study linear independence in the homology cobordism group, to define an instanton Floer analogue ε^#(K) of Hom's ε-invariant in Heegaard Floer homology, and to the problem of characterizing a given 3-manifold as Dehn surgery on a knot in S³.

We prove that the knot Floer complex of a fibered knot detects whether the monodromy of its fibration is right-veering. In particular, this leads to a purely knot Floer-theoretic characterization of tight contact structures, by the work of Honda, Kazez, and Matić. Our proof makes use of the relationship between the Heegaard Floer homology of mapping tori and the symplectic Floer homology of area-preserving surface diffeomorphisms. We describe applications of this work to Dehn surgeries and taut foliations.

We prove that the fundamental group of 3-surgery on a nontrivial knot in S³ always admits an irreducible SU(2)-representation. This answers a question of Kronheimer and Mrowka dating from their work on the Property P conjecture. An important ingredient in our proof is a relationship between instanton Floer homology and the symplectic Floer homology of genus-2 surface diffeomorphisms, due to Ivan Smith. We use similar arguments at the end to extend our main result to infinitely many surgery slopes in the interval [3,5).

We prove that Khovanov homology with coefficients in ℤ/2ℤ detects the (2,5) torus knot. Our proof makes use of a wide range of deep tools in Floer homology, Khovanov homology, and Khovanov homotopy. We combine these tools with classical results on the dynamics of surface homeomorphisms to reduce the detection question to a problem about mutually braided unknots, which we then solve with computer assistance.

We prove that the 3-manifold obtained by gluing the complements of two nontrivial knots in homology 3-sphere instanton L-spaces, by a map which identifies meridians with Seifert longitudes, cannot be an instanton L-space. This recovers the recent theorem of Lidman, Pinzón-Caicedo, and Zentner that the fundamental group of every closed, oriented, toroidal 3-manifold admits a nontrivial SU(2)-representation, and consequently Zentner's earlier result that the fundamental group of every closed, oriented 3-manifold besides the 3-sphere admits a nontrivial SL(2,C)-representation.

We show that if Y is the boundary of an almost-rational plumbing, then the framed instanton Floer homology I^#(Y) is isomorphic to the Heegaard Floer homology HF^{^}(Y;ℂ). This class of 3-manifolds includes all Seifert fibered rational homology spheres with base orbifold S² (we establish the isomorphism for the remaining Seifert fibered rational homology spheres—with base ℝℙ²—directly). Our proof utilizes lattice homology, and relies on a decomposition theorem for instanton Floer cobordism maps recently established by Baldwin and Sivek.

We define two concordance invariants of knots using framed instanton homology. These invariants ν^# and τ^# provide bounds on slice genus and maximum self-linking number, and the latter is a concordance homomorphism which agrees in all known cases with the τ invariant in Heegaard Floer homology. We use ν^# and τ^# to compute the framed instanton homology of all nonzero rational Dehn surgeries on: 20 of the 35 nontrivial prime knots through 8 crossings, infinite families of twist and pretzel knots, and instanton L-space knots; and of 19 of the first 20 closed hyperbolic manifolds in the Hodgson–Weeks census. In another application, we determine when the cable of a knot is an instanton L-space knot. Finally, we discuss applications to the spectral sequence from odd Khovanov homology to the framed instanton homology of branched double covers, and to the behaviors of τ^# and τ under genus-2 mutation.

We provide infinitely many rational homology 3-spheres with weight-one fundamental groups which do not arise from Dehn surgery on knots in S³. In contrast with previously known examples, our proofs do not require any gauge theory or Floer homology. Instead, we make use of the SU(2) character variety of the fundamental group, which for these manifolds is particularly simple: they are all SU(2)-cyclic, meaning that every SU(2) representation has cyclic image.

We give a new, conceptually simpler proof of the fact that knots in S³ with positive L-space surgeries are fibered and strongly quasipositive. Our motivation for doing so is that this new proof uses comparatively little Heegaard Floer-specific machinery and can thus be translated to other forms of Floer homology. We carried this out for instanton Floer homology in our recent article "Instantons and L-space surgeries", and used it to generalize Kronheimer and Mrowka's results on SU(2) representations of fundamental groups of Dehn surgeries.

We prove that instanton L-space knots are fibered and strongly quasipositive. Our proof differs conceptually from proofs of the analogous result in Heegaard Floer homology, and includes a new decomposition theorem for cobordism maps in framed instanton Floer homology akin to the Spin^c decompositions of cobordism maps in other Floer homology theories. As our main application, we prove (modulo a mild nondegeneracy condition) that for r a positive rational number and K a nontrivial knot in the 3-sphere, there exists an irreducible homomorphism π₁(S³_r(K)) → SU(2) unless r ≥ 2g(K)-1 and K is both fibered and strongly quasipositive, broadly generalizing results of Kronheimer and Mrowka. We also answer a question of theirs from 2004, proving that there is always an irreducible homomorphism from the fundamental group of 4-surgery on a nontrivial knot to SU(2). In another application, we show that a slight enhancement of the A-polynomial detects infinitely many torus knots, including the trefoil.

We classify SU(2)-cyclic and SU(2)-abelian 3-manifolds, for which every representation of the fundamental group into SU(2) has cyclic or abelian image respectively, among geometric 3-manifolds which are not hyperbolic. As an application, we give examples of hyperbolic 3-manifolds which do not admit degree-1 maps to any Seifert fibered manifold other than S³ or a lens space. We also produce infinitely many one-cusped hyperbolic manifolds with at least four SU(2)-cyclic Dehn fillings, one more than the number of cyclic fillings allowed by the cyclic surgery theorem.

We prove that any link in S³ whose Khovanov homology is the same as that of a Hopf link must be isotopic to that Hopf link. This holds for both reduced and unreduced Khovanov homology, and with coefficients in either ℤ or ℤ/2ℤ.

We study an A_∞ category associated to Legendrian links in ℝ³ whose objects are n-dimensional representations of the Chekanov-Eliashberg differential graded algebra of the link. This representation category generalizes the positive augmentation category and we conjecture that it is equivalent to a category of sheaves of microlocal rank n constructed by Shende, Treumann, and Zaslow. We establish the cohomological version of this conjecture for a family of Legendrian (2,m) torus links.

We prove that Khovanov homology detects the trefoils. Our proof incorporates an array of ideas in Floer homology and contact geometry. It uses open books; the contact invariants we defined in the instanton Floer setting; a bypass exact triangle in sutured instanton homology, proven here; and Kronheimer and Mrowka's spectral sequence relating Khovanov homology with singular instanton knot homology. As a byproduct, we also strengthen a result of Kronheimer and Mrowka on SU(2) representations of the knot group.

We study knots in S³ with infinitely many SU(2)-cyclic surgeries, which are Dehn surgeries such that every representation of the resulting fundamental group into SU(2) has cyclic image. We show that for every such nontrivial knot K, its set of SU(2)-cyclic slopes is bounded and has a unique limit point, which is both a rational number and a boundary slope for K. We also show that such knots are prime and have infinitely many instanton L-space surgeries. Our methods include the application of holonomy perturbation techniques to instanton knot homology, using a strengthening of recent work by the second author.

We show that the problem of recognizing that a knot diagram represents a specific torus knot, or any torus knot at all, is in the complexity class NP ∩ co-NP, assuming the generalized Riemann hypothesis. We also show that satellite knot detection is in NP under the same assumption, and that cabled knot detection and composite knot detection are unconditionally in NP. Our algorithms are based on recent work of Kuperberg and of Lackenby on detecting knottedness.

We recently defined invariants of contact 3-manifolds using a version of instanton Floer homology for sutured manifolds. In this paper, we prove that if several contact structures on a 3-manifold are induced by Stein structures on a single 4-manifold with distinct Chern classes modulo torsion then their contact invariants in sutured instanton homology are linearly independent. As a corollary, we show that if a 3-manifold bounds a Stein domain that is not an integer homology ball then its fundamental group admits a nontrivial homomorphism to SU(2). We give several new applications of these results, proving the existence of nontrivial and irreducible SU(2) representations for a variety of 3-manifold groups.

We recently defined an invariant of contact manifolds with convex boundary in Kronheimer and Mrowka’s sutured monopole Floer homology theory. Here, we prove that there is an isomorphism between sutured monopole Floer homology and sutured Heegaard Floer homology which identifies our invariant with the contact class defined by Honda, Kazez and Matić in the latter theory. One consequence is that the Legendrian invariants in knot Floer homology behave functorially with respect to Lagrangian concordance. In particular, these invariants provide computable and effective obstructions to the existence of such concordances. Our work also provides the first proof which does not rely on the relative Giroux correspondence that the vanishing or non-vanishing of Honda, Kazez and Matić’s contact class is a well-defined invariant of contact manifolds.

We introduce a notion of cardinality for the augmentation category associated to a Legendrian knot or link in standard contact ℝ³. This "homotopy cardinality" is an invariant of the category and allows for a weighted count of augmentations, which we prove to be determined by the ruling polynomial of the link. We present an application to the augmentation category of doubly Lagrangian slice knots.

We study the topology of exact and Stein fillings of the canonical contact structure on the unit cotangent bundle of a closed surface Σ_g, where g is at least 2. In particular, we prove a uniqueness theorem asserting that any Stein filling must be s-cobordant rel boundary to the disk cotangent bundle of Σ_g. For exact fillings, we show that the rational homology agrees with that of the disk cotangent bundle, and that the integral homology takes on finitely many possible values: for example, if g−1 is square-free, then any exact filling has the same integral homology and intersection form as DT^*Σ_g.

Quasi-alternating links of determinant 1, 2, 3, and 5 were previously classified by Greene and Teragaito, who showed that the only such links are two-bridge. In this paper, we extend this result by showing that all quasi-alternating links of determinant at most 7 are connected sums of two-bridge links, which is optimal since there are quasi-alternating links not of this form for all larger determinants. We achieve this by studying their branched double covers and characterizing distance-one surgeries between lens spaces of small order, leading to a classification of formal L-spaces with order at most 7.

We show that the set of augmentations of the Chekanov–Eliashberg algebra of a Legendrian link underlies the structure of a unital A-infinity category. This differs from the non-unital category constructed in [BC14], but is related to it in the same way that cohomology is related to compactly supported cohomology. The existence of such a category was predicted by [STZ14], who moreover conjectured its equivalence to a category of sheaves on the front plane with singular support meeting infinity in the knot. After showing that the augmentation category forms a sheaf over the x-line, we are able to prove this conjecture by calculating both categories on sufficiently thin slices of the front plane. In particular, we conclude that every augmentation comes from geometry.

We investigate the question of the existence of a Lagrangian concordance between two Legendrian knots in ℝ³. In particular, we give obstructions to a concordance from an arbitrary knot to the standard Legendrian unknot, in terms of normal rulings. We also place strong restrictions on knots that have concordances both to and from the unknot and construct an infinite family of knots with non-reversible concordances from the unknot. Finally, we use our obstructions to present a complete list of knots with up to 14 crossings that have Legendrian representatives that are Lagrangian slice.

We apply results from both contact topology and exceptional surgery theory to study when Legendrian surgery on a knot yields a reducible manifold. As an application, we show that a reducible surgery on a non-cabled positive knot of genus g must have slope 2g-1, leading to a proof of the cabling conjecture for positive knots of genus 2. Our techniques also produce bounds on the maximum Thurston-Bennequin numbers of cables.

We define an invariant of contact 3-manifolds with convex boundary using Kronheimer and Mrowka's sutured instanton Floer homology theory. This is the first invariant of contact manifolds—with or without boundary—defined in the instanton Floer setting. We prove that our invariant vanishes for overtwisted contact structures and is nonzero for contact manifolds with boundary which embed into Stein fillable contact manifolds. Moreover, we propose a strategy by which our contact invariant might be used to relate the fundamental group of a closed contact 3-manifold to properties of its Stein fillings. Our construction is inspired by a reformulation of a similar invariant in the monopole Floer setting defined by the authors in previous work.

We use the contact invariant defined in previous work to construct a new invariant of Legendrian knots in Kronheimer and Mrowka's monopole knot homology theory (KHM), following a prescription of Stipsicz and Vértesi. Our Legendrian invariant improves upon an earlier Legendrian invariant in KHM defined by the second author in several important respects. Most notably, ours is preserved by negative stabilization. This fact enables us to define a transverse knot invariant in KHM via Legendrian approximation. It also makes our invariant a more likely candidate for the monopole Floer analogue of the "LOSS" invariant in knot Floer homology. Like its predecessor, our Legendrian invariant behaves functorially with respect to Lagrangian concordance. We show how this fact can be used to compute our invariant in several examples. As a byproduct of our investigations, we provide the first infinite family of nonreversible Lagrangian concordances between prime knots.

We define an invariant of contact 3-manifolds with convex boundary using Kronheimer and Mrowka's sutured monopole Floer homology theory (SHM). Our invariant can be viewed as a generalization of Kronheimer and Mrowka's contact invariant for closed contact 3-manifolds and as the monopole Floer analogue of Honda, Kazez, and Matić's contact invariant in sutured Heegaard Floer homology (SFH). In the process of defining our invariant, we construct maps on SHM associated to contact handle attachments, analogous to those defined by Honda, Kazez, and Matić in SFH. We use these maps to establish a bypass exact triangle in SHM analogous to Honda's in SFH. This paper also provides the topological basis for the construction of similar gluing maps in sutured instanton Floer homology, which are used in subsequent work to define a contact invariant in the instanton Floer setting.

We show that sutured embedded contact homology is a natural invariant of sutured contact 3-manifolds which can potentially detect some of the topology of the space of contact structures on a 3-manifold with boundary. The appendix, by C. H. Taubes, proves a compactness result for the completion of a sutured contact 3-manifold in the context of Seiberg–Witten Floer homology, which enables us to complete the proof of naturality.

Kronheimer and Mrowka defined invariants of balanced sutured manifolds using monopole and instanton Floer homology. Their invariants assign isomorphism classes of modules to balanced sutured manifolds. In this paper, we introduce refinements of these invariants which assign much richer algebraic objects called projectively transitive systems of modules to balanced sutured manifolds and isomorphisms of such systems to diffeomorphisms of balanced sutured manifolds. Our work provides the foundation for extending these sutured Floer theories to other interesting functorial frameworks as well, and can be used to construct new invariants of contact structures and (perhaps) of knots and bordered 3-manifolds.

We prove that symplectic 4-manifolds with b⁺>1 have nonvanishing Donaldson invariants, and that the canonical class is always a basic class when b₁=0. We also characterize in many situations the basic classes of a Lefschetz fibration over the sphere which evaluate maximally on a generic fiber.

We use monopole Floer homology for sutured manifolds to construct invariants of unoriented Legendrian knots in a contact 3–manifold. These invariants assign to a knot K ⊂ Y elements of the monopole knot homology KHM(-Y,K), and they strongly resemble the knot Floer homology invariants of Lisca, Ozsváth, Stipsicz, and Szabó. We prove several vanishing results, investigate their behavior under contact surgeries, and use this to construct many examples of nonloose knots in overtwisted 3–manifolds. We also show that these invariants are functorial with respect to Lagrangian concordance.

We show that there exists a Legendrian knot with maximal Thurston-Bennequin invariant whose contact homology is trivial. We also provide another Legendrian knot which has the same knot type and classical invariants but nonvanishing contact homology.

Given a front projection of a Legendrian knot K in ℝ³, which has been cut into several pieces along vertical lines, we assign a differential graded algebra to each piece and prove a van Kampen theorem describing the Chekanov-Eliashberg invariant of K as a pushout of these algebras. We then use this theorem to construct maps between the invariants of Legendrian knots related by certain tangle replacements, and to describe the linearized contact homology of Legendrian Whitehead doubles. Other consequences include a Mayer-Vietoris sequence for linearized contact homology and a van Kampen theorem for the characteristic algebra of a Legendrian knot.

Given a Young diagram λ and the set H^λ of partitions of {1,2,...,|λ|} of shape λ, we analyze a particular S_|λ|-module homomorphism ℚH^λ→ℚH^λ' to show that ℚH^λ is a submodule of ℚH^λ' whenever λ is a hook (n,1,1,...,1) with m rows, n ≥ m, or any diagram with two rows.

We provide several classes of examples to show that Stanley's plethysm conjecture and a reformulation by Pylyavskyy, both concerning the ranks of certain matrices K^λ associated with Young diagrams λ, are in general false. We also provide bounds on the rank of K^λ by which it may be possible to show that the approach of Black and List to Foulkes' conjecture does not work in general. Finally, since Black and List's work concerns K^λ for rectangular shapes λ, we suggest a constructive way to prove that K^λ does not have full rank when λ is a large rectangle.