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Link homology and equivariant gauge theory

Prayat Poudel and Nikolai Saveliev

Algebraic & Geometric Topology 17 (2017) 2635–2685

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.

Floer homology, equivariant gauge theory, knots, links, Khovanov homology
Mathematical Subject Classification 2010
Primary: 57M27
Secondary: 57R58
Received: 15 June 2015
Accepted: 23 June 2017
Published: 19 September 2017
Prayat Poudel
Department of Mathematics & Statistics
McMaster University
Nikolai Saveliev
Department of Mathematics
University of Miami
Coral Gables
United States