#### Volume 17, issue 5 (2017)

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Link homology and equivariant gauge theory

### Prayat Poudel and Nikolai Saveliev

Algebraic & Geometric Topology 17 (2017) 2635–2685
##### Abstract

Singular instanton Floer homology was defined by Kronheimer and Mrowka in connection with their proof that Khovanov homology is an unknot detector. We study this theory for knots and two-component links using equivariant gauge theory on their double branched covers. We show that the special generator in the singular instanton Floer homology of a knot is graded by the knot signature mod $4$, thereby providing a purely topological way of fixing the absolute grading in the theory. Our approach also results in explicit computations of the generators and gradings of the singular instanton Floer chain complex for several classes of knots with simple double branched covers, such as two-bridge knots, some torus knots, and Montesinos knots, as well as for several families of two-component links.

##### Keywords
Floer homology, equivariant gauge theory, knots, links, Khovanov homology
Primary: 57M27
Secondary: 57R58