Unoriented skein exact triangles in equivariant singular instanton Floer theory

TL;DR

This paper extends equivariant singular instanton Floer theory from knots to links with non-zero determinant, establishing unoriented skein exact triangles and defining new link invariants, thereby deepening the understanding of link invariants in gauge theory.

Contribution

It introduces unoriented skein exact triangles in equivariant singular instanton Floer theory for links and constructs cobordism maps in the presence of obstructed reducible instantons.

Findings

Proved unoriented skein exact triangles for links with non-zero determinant.

Defined Fr{ extrm{o}}yshov-type invariants for links.

Suggested a relationship between Heegaard Floer L-space knots and instanton categorifications.

Abstract

Equivariant singular instanton Floer theory is a framework that associates to a knot in an integer homology 3-sphere a suite of homological invariants that are derived from circle-equivariant Morse-Floer theory of a Chern-Simons functional for framed singular $SU(2)$-connections. These invariants generalize the instanton knot homology of Kronheimer and Mrowka. In the present work, these constructions are extended from knots to links with non-zero determinant, and several unoriented skein exact triangles are proved in this setting. As a particular case, a categorification of the behavior of the Murasugi signature for links under unoriented skein relations is established. In addition to the exact triangles, Fr{\o}yshov-type invariants for links are defined, and several computations using the exact triangles are carried out. The computations suggest a relationship between Heegaard Floer L-space knots and those knots whose instanton-theoretic categorification of the knot signature is supported in even gradings. A main technical contribution of this work is the construction of maps for certain cobordisms between links on which obstructed reducible singular instantons are present. These constructions are inspired by recent work of the first author and Miller Eismeier in the setting of non-singular instanton theory for rational homology 3-spheres.