Algebraic & Geometric Topology

Link homology and equivariant gauge theory

Prayat Poudel and Nikolai Saveliev

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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.

Article information

Source
Algebr. Geom. Topol., Volume 17, Number 5 (2017), 2635-2685.

Dates
Received: 15 June 2015
Accepted: 23 June 2017
First available in Project Euclid: 16 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1510841478

Digital Object Identifier
doi:10.2140/agt.2017.17.2635

Mathematical Reviews number (MathSciNet)
MR3704238

Zentralblatt MATH identifier
06791379

Subjects
Primary: 57M27: Invariants of knots and 3-manifolds
Secondary: 57R58: Floer homology

Keywords
Floer homology equivariant gauge theory knots links Khovanov homology

Citation

Poudel, Prayat; Saveliev, Nikolai. Link homology and equivariant gauge theory. Algebr. Geom. Topol. 17 (2017), no. 5, 2635--2685. doi:10.2140/agt.2017.17.2635. https://projecteuclid.org/euclid.agt/1510841478


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