Knots, sutures, and excision

By P. Kronheimer, T. Mrowka. 2008

189 citations recorded by the source corpus.

We develop monopole and instanton Floer homology groups for balanced sutured manifolds, in the spirit of [12]. Applications include a new proof of Property P for knots.

Subfields: Geometry and Topology, Mathematical Physics, Neurology

View Paper: https://www.semanticscholar.org/paper/46d4cfc4137212d314812af5e91b0412d8524ed1

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Citations in the corpus

A SURVEY OF TWISTED ALEXANDER POLYNOMIALS (2009, unknown): “mplectic, then Nﬁbers over S1. This provides a converse to a theorem of Thurston [Th76]. We refer to [FV06, FV08a, FV08b, FV08c] for details, and we refer to Kutluhan–Taubes [KT08], Kronheimer–Mrowka [KM08] and Ni [Ni08] for an alternative proof in the case that b1(N) = 1. It is natural to ask wheth”
Floer cohomology and pencils of quadrics (2010, result): “This should be compared to similar rank one pieces of monopole or Heegaard Floer homology [78, 57], in that context associated to the “top” Spin-structure.”
Twisted Alexander polynomials detect fibered 3-manifolds (2008, background): “On the other hand it is known, due to the work of Ghiggini, Kronheimer and Mrowka, and Ni that Monopole Floer homology detects fibered 3–manifolds ([Gh08, Ni07, KM08 , Ni08]).”
Instantons and odd Khovanov homology (2014, background): “The group I#(Y ) is isomorphic to the sutured instanton group SHI(M,γ) from [20], where M is the complement of an open 3-ball in Y and γ is a suture on the 2-sphere boundary.”
Instanton Floer homology and the Alexander polynomial (2009, methodology): “We recall the definition of the instanton Floer homology for a balanced sutured manifold, as introduced in [8] with motivation from the Heegaard counterpart defined in [4].”
A combinatorial spanning tree model for knot Floer homology (2011, methodology): “This suggests that our approach might be used to give an axiomatic characterization of knot Floer homology or to prove that ĤFK is isomorphic to other knot homology theories, such Kronheimer and Mrowka’s monopole knot homology [18].”
Khovanov module and the detection of unlinks (2012, methodology): “iginal form detects the unknot remained open. By developing another spectral sequence starting at Khovanov homology - this time abutting to an invariant of knots arising from instanton Floer homology [17] - Kronheimer and Mrowka were able to settle this question in the aﬃrmative: Theorem 1.1. [18, T”
AN INTRODUCTION TO KNOT FLOER HOMOLOGY (2014, unknown): “Nevertheless, some Seiberg-Witten counterparts to knot Floer homology were constructed in [KM10, Kut].”
Tait colorings, and an instanton homology for webs and foams (2015, methodology): “Floer’s excision theorem [2, 13] can be applied in the setting of the SO(3) instanton homology J.”
Obstructions to Lagrangian concordance (2014, methodology): “For nontrivial K, a theorem of Kronheimer and Mrowka [31, Corollary 7.17] says that given a meridian µ of K, there is an irreducible homomorphism ϕ : π1(S 3 rK) → SU(2) such that ϕ(µ) = i. (Here we view SU(2) as the unit quaternions.)”
A link surgery spectral sequence in monopole Floer homology (2009, background): “Note that H̃M•(Y, s) should agree with the sutured monopole Floer homology group SFH(Y − B3, s) relative to the equatorial suture, as defined in [14].”
Khovanov homology is an unknot-detector (2010, methodology): “The generalized eigenspace for +2 is, by definition, the group KHI (Y,K) of [24].”