Khovanov homology is an unknot-detector

Kronheimer, P. B.; Mrowka, Tomasz

doi:10.1007/s10240-010-0030-y

Cited by 221 publications

(481 citation statements)

References 32 publications

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“…However, given a pair (Y, α), the Floer groups associated to any two sets of auxiliary choices are related by a canonical isomorphism which is well-defined up to sign (cf. [14,Section 4]). In particular, the pair (Y, α) defines a projectively transitive system of C-modules, which we will denote by I * (Y ) α .…”

Section: Instanton Floer Homologymentioning

confidence: 99%

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Instanton Floer homology and contact structures

Baldwin

Sivek

2015

Sel. Math. New Ser.

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Abstract. We define an invariant of contact 3-manifolds with convex boundary using Kronheimer and Mrowka's sutured instanton Floer homology theory. To the best of our knowledge, 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 [1]. IntroductionFloer-theoretic invariants of contact manifolds have been responsible for many important results in low-dimensional topology. Notable examples include the invariants of closed contact 3-manifolds defined by Kronheimer and Mrowka [10] and by Ozsváth and Szabó [18] in monopole and Heegaard Floer homology, respectively. Also important is the work in [8], where Honda, Kazez, and Matić extend Ozsváth and Szabó's construction, using sutured Heegaard Floer homology to define an invariant of sutured contact manifolds, which are triples of the form (M, Γ, ξ) where (M, ξ) is a contact 3-manifold with convex boundary and Γ ⊂ ∂M is a multicurve dividing the characteristic foliation of ξ on ∂M . Recently, we defined an analogous invariant of sutured contact manifolds in Kronheimer and Mrowka's sutured monopole Floer homology theory [1].The goal of this paper is to define an invariant of sutured contact manifolds in Kronheimer and Mrowka's sutured instanton Floer homology (SHI). To the best of our knowledge, this is the first invariant of contact manifolds-with or without boundary-defined in the instanton Floer setting. Like the Heegaard Floer invariants but in contrast with the monopole invariants, our instanton Floer contact invariant is defined using the full relative Giroux correspondence. Its construction is inspired by a reformulation of the monopole Floer invariant in [1] which was used there to prove that the monopole invariant is well-defined.A unique feature of the instanton Floer viewpoint is the central role played by the fundamental group. Along these lines, we conjecture a means by which our contact invariant in SHI might be used to relate the fundamental group of a closed contact 3-manifold to properties of its Stein fillings, a relationship which has been largely unexplored to this point. Below, we sketch the construction of our contact invariant, describe some of its most important properties, state some conjectures, and discuss plans for future work which include using the constructions in this paper to define invariants of bordered manifolds in the instanton Floer setting.1.1. A contact invariant in SHI. Suppose (M, Γ) is a balanced sutured manifold. Roughly speaking, a closure of (M, Γ) is formed by gluing on some auxiliary piece a...

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Section: Instanton Floer Homologymentioning

confidence: 99%

“…But There are several ways to see this; it follows easily, for instance, from the work of Hedden, Herald, and Kirk [6]. On the other hand, Kronheimer and Mrowka proved in [14] that…”

Section: Stein Fillings and The Fundamental Groupmentioning

confidence: 99%

Instanton Floer homology and contact structures

Baldwin

Sivek

2015

Sel. Math. New Ser.

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“…Kronheimer and Mrowka [10] constructed an abelian group called the framed instanton Floer homology FI .K/ of knots K in the 3-sphere, and then later introduced an equivalent theory I # .K/ in [12] that is there called "unreduced singular knot Floer homology" of K , and a reduced version I \ .K/. These are Morse homologies of a Chern-Simons functional CS defined on a space of connections on an open 3-manifold obtained Y from K .…”

Section: Relation To Instanton Knot Floer Homologymentioning

confidence: 99%

Representation spaces of pretzel knots

Zentner

2011

Algebr. Geom. Topol.

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We study the representation spaces $R(K;\bf{i})$ as appearing in Kronheimer and Mrowka's framed instanton knot Floer homology, for a class of pretzel knots. In particular, for pretzel knots $P(p,q,r)$ with $p, q, r$ pairwise coprime, these appear to be non-degenerate and comprise representations in SU(2) that are not binary dihedral.Comment: 30 pages, 3 figures. This is the published version. It is more general than the previous version regarding the values $p,q,r$. Furthermore, rewritten according after referee's remark

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“…One might believe this to be true because Khovanov homology, a knot invariant of bigraded abelian groups whose graded Euler characteristic gives the Jones polynomial, has been shown by Kronheimer and Mrowka to detect the unknot [KM10]. Dasbach and Hougardy [DH97] found no counterexample in all knots up to seventeen crossings, including over two million prime knots, and this was extended to eighteen crossings by Yamada [Yam00].…”

Section: Introductionmentioning

confidence: 99%

Random knots using Chebyshev billiard table diagrams

Cohen

Krishnan

2015

Topology and its Applications

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Abstract. We use the Chebyshev knot diagram model of Koseleff and Pecker in order to introduce a random knot diagram model by assigning the crossings to be positive or negative uniformly at random. We give a formula for the probability of choosing a knot at random among all knots with bridge index at most 2. Restricted to this class, we define internal and external reduction moves that decrease the number of crossings of the diagram. We make calculations based on our formula, showing the numerics in graphs and providing evidence for our conjecture that the probability of any knot K appearing in this model decays to zero as the number of crossings goes to infinity.

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Khovanov homology is an unknot-detector

Cited by 221 publications

References 32 publications

Instanton Floer homology and contact structures

Instanton Floer homology and contact structures

Representation spaces of pretzel knots

Random knots using Chebyshev billiard table diagrams

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