We show that the set of augmentations Chekanov-Eliashberg algebra a Legendrian link underlies structure unital A-infinity category.This differs from non-unital category constructed in [BC14], but is related to it same way cohomology compactly supported cohomology.The existence such was predicted by [STZ17], who moreover conjectured its equivalence sheaves on front plane with singular support meeting infinity knot.After showing augmentation forms sheaf over x-line, we are able prove this...

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

We investigate the question of existence a Lagrangian concordance between two Legendrian knots in R 3 .In particular, we give obstructions to from an arbitrary knot standard unknot, terms normal rulings.We also place strong restrictions on that have concordances both and unknot construct infinite family with nonreversible unknot.Finally, use our present complete list up 14 crossings representatives are slice.

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

There has been a great deal of interest in understanding which knots are characterized by their Dehn surgeries. We study 4-dimensional version this question: determined traces? prove several results that stark contrast with what is known about characterizing surgeries, most notably the 0-trace detects every L-space knot. Our proof combines tools Heegaard Floer homology surface homeomorphisms and dynamics. also consider nonzero traces, proving for instance each positive torus knot its...

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

Abstract We prove that all rational slopes are characterizing for the knot , except possibly positive integers. Along way, we classify Dehn surgeries on knots in produce Brieskorn sphere and study which large integral almost L‐spaces.

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

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, includes a new decomposition theorem for cobordism maps framed homology akin to \operatorname{Spin}^c decompositions other theories. As our main application, we (modulo mild nondegeneracy condition) r positive rational number K nontrivial knot 3 -sphere, there exists an irreducible homomorphism \pi_1(S^3_r(K)) \to SU(2)...

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

We use the contact invariant defined in [2] to construct a new of Legendrian knots Kronheimer and Mrowka's monopole knot homology theory (KHM), following prescription Stipsicz V\'ertesi. Our improves upon an earlier KHM by second author several important respects. Most notably, ours is preserved negative stabilization. This fact enables us define transverse via approximation. It also makes our more likely candidate for Floer analogue "LOSS" homology. Like its predecessor, behaves...

We study knots in $S^3$ 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. show for nontrivial knot $K$, its set slopes is bounded and a unique limit point, both rational number boundary slope $K$. also prime have instanton L‑space surgeries. Our methods include application holonomy perturbation techniques to homology, using strengthening recent work by second author.

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

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

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

We define two concordance invariants of knots using framed instanton homology. These $\nu^\sharp$ and $\tau^\sharp$ provide bounds on slice genus maximum self-linking number, the latter is a homomorphism which agrees in all known cases with $\tau$ invariant Heegaard Floer use to compute homology nonzero rational Dehn surgeries on: 20 35 nontrivial prime through 8 crossings, infinite families twist pretzel knots, L-space knots; 19 first closed hyperbolic manifolds Hodgson--Weeks census. In...

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 isomorphism between Heegaard which identifies our the class by Honda, Kazez Mati\'c latter One consequence Legendrian invariants knot behave functorially respect to Lagrangian concordance. In particular, these provide computable effective obstructions existence such concordances. Our work also provides first proof does not...

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

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

We prove that any link in S 3 whose Khovanov homology is the same as of a Hopf must be isotopic to link.This holds for both reduced and unreduced homology, with coefficients either Z or Z/2Z.Khovanov [Kho00] associates each L ⊂ bigraded group Kh * , (L), graded Euler characteristic recovers Jones polynomial V (q), well variant red (L) [Kho03].It known detect unknot [KM11], n-component unlink all n [BS15], trefoils [BS18].In this note we links H ± which are oriented so two components have...

Abstract 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 that are not hyperbolic. As an application, we give examples hyperbolic do admit degree-1 maps to any Seifert Fibered manifold other than $S^3$ a lens space. also produce infinitely many one-cusped manifolds with at least four Dehn fillings, one more number fillings allowed by surgery theorem.

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