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Creators/Authors contains: "Kronheimer, Peter B"

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  1. ; (, AxRiV)
    An instanton homology is constructed for webs and foams, using gauge theory with structure group SU (3), adapting previous work of the authors for the SO(3) case. Skein exact triangles are established, and using an eigenspace decomposition arising from operators associated to the edges, it is shown that the dimension of the SU (3) homology counts Tait colorings when theweb is planar. Unlike the SO(3) case, the SU (3) homology is mod-2 graded. Its Euler characteristic can be interpreted as a signed count of Tait colorings, or equivalently as the value at 1 of the Yamada polynomial invariant. Some examples and variants of the construction are also discussed. 
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    Free, publicly-accessible full text available May 5, 2026
  2. ; (, Geometry & Topology)
    Free, publicly-accessible full text available January 1, 2026
  3. ; (, Geometry & topology)
    A deformation of the authors’ instanton homology for webs is constructed by introducing a local system of coefficients. In the case that the web is planar, the rank of the deformed instanton homology is equal to the number of Tait colorings of the web. 
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  4. ; (, Proceedings of the International Congress of Mathematicians (ICM 2018))
    Low-dimensional topology is the study of manifolds and cell complexes in dimensions four and below. Input from geometry and analysis has been central to progress in this field over the past four decades, and this article will focus on one aspect of these developments in particular, namely the use of Yang–Mills theory, or gauge theory. These techniques were pioneered by Simon Donaldson in his work on 4-manifolds, but the past ten years have seen new applications of gauge theory, and new interactions with more recent threads in the subject, particularly in 3-dimensional topology. This is a field where many mathematical techniques have found applications, and sometimes a theorem has two or more independent proofs, drawing on more than one of these techniques. We will focus primarily on some questions and results where gauge theory plays a special role. 
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