Knot homology groups from instantons

By P. Kronheimer, T. Mrowka. 2008

157 citations recorded by the source corpus.

For each partial flag manifold of SU(N), we define a Floer homology theory for knots in 3‐manifolds, using instantons with codimension‐2 singularities. In the case of SU(2), the resulting Floer homology group for classical knots appears to be related to Khovanov homology.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/4ea24e2ad0cca026b08b790be6fe2267e68309ac

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Citations in the corpus

Tait colorings, and an instanton homology for webs and foams (2015, background): “If we attach cylindrical ends to X̌ and choose auxiliary perturbations in the neighborhoods of the two boundary components [15, 14, 8], then we have moduli spaces of solutions to the perturbed anti-self-duality equations on the bifold:”
Khovanov homology is an unknot-detector (2010, methodology): “) We call these the instanton number and the monopole number respectively: in the case that Σ is orientable, these definitions coincide with those from the authors earlier papers [20, 23].”
Fivebranes and Knots (2011, background): “There have also been attempts [26, 27] to make the three- or fourdimensional symmetry of Khovanov homology manifest by extracting it from a special case or analog of Donaldson-Floer theory in four dimensions.”
Instantons and odd Khovanov homology (2014, background): “The name framed instanton homology comes from [19].”
Grid Homology for Knots and Links (2015, background): “In a different direction, it is also interesting to compare knot Floer homology with analogous invariants coming from gauge theory, especially instanton homology for knots [54, 113].”
Quilted Floer cohomology (2009, methodology): “theoretic invariants: SO(3)-instanton Floer homology, it’s higher rank version (though not strictly deﬁned in the gauge theoretic setting), and the SU(n)-tangle invariants deﬁned by Kronheimer-Mrowka [14] from singular instantons. The same setup is used to give alternative constructions of Heegard F”
The pillowcase and perturbations of traceless representations of knot groups (2013, methodology): “We provide the minimal description needed for our purposes, and refer to the series of articles [23, 24, 25] for details.”
Instanton Floer homology and the Alexander polynomial (2009, background): “As in [7], the corresponding non-degeneracy condition at a flat connection ρ can be interpreted as the condition that the map”