Instanton knot invariants with rational holonomy parameters and an application for torus knot groups
Year: 2024
Type: article
Abstract: Instanton knot invariants with rational holonomy parameters and an application for torus knot groups HAYATO IMORIThere are several knot invariants in the literature that are defined using singular instantons.Such invariants provide strong tools to study the knot group and give topological applications, for instance, the topology of knots in terms of representations of fundamental groups.In particular, it has been shown that any traceless representation of the torus knot group can be extended to any concordance from the torus knot to another knot.Daemi and Scaduto proposed a generalization that is related to a version of the slice-ribbon conjecture for torus knots.Our results provide further evidence towards the positive answer to this question.We use a generalization of Daemi and Scaduto's equivariant singular instanton Floer theory following Echeverria's earlier work.We also determine the irreducible singular instanton homology of torus knots for all but finitely many rational holonomy parameters as Z=4-graded abelian groups.57R58; 57K18 4. Nondegeneracy of the representation variety 5086 5. Properties of instanton knot invariants and their applications 5101Appendix.The connected sum theorem 5107References 5120Question 4 Can we drop the traceless condition in Theorem 1?Algebraic & Geometric Topology, Volume 24 (2024)Instanton knot invariants with rational holonomy parameters and an application for torus knot groups 5047We will affirmatively solve this question.To explain our strategy, let us describe the technical background of Daemi and Scaduto's work.It mainly consists of three ingredients: singular gauge theory, equivariant Floer theory and the Chern-Simons filtration.Firstly, let us explain the notion of singular connections.Let K Y be a knot in a 3-manifold.Roughly speaking, an SU.2/-singular connection A is an SU.2/-connection defined over the knot complement with the holonomy conditionwhere .r/ is a radius-r meridian of K Y and ˛is a fixed parameter in 0; 1 2 .Here indicates that the two matrices are conjugate in SU.2/.The parameter ˛is called the holonomy parameter of the singular connection A. In particular, a singular flat SU.2/-connection corresponds to an SU.2/-representation of 1 .Y nK/ which sends the meridian K of knot K to an element which is conjugate to the matrix in (1-1).Kronheimer and Mrowka developed a singular version of Yang-Mills gauge theory in [27; 28; 32].These Floer homology theories constructed via singular connections are called singular instanton homology.Singular gauge theory has different features compared to nonsingular.In fact, singular Floer homology cannot be defined over the coefficient ring Z for a general holonomy parameter ˛.To be more precise, singular instanton Floer homology is defined over Z only for ˛D 1 4 .This is called the monotonicity condition.Most of the works in singular instanton homology including [9; 8] impose the monotonicity condition.This is why the statement of Theorem 1 includes the traceless condition.
Source: Algebraic & Geometric Topology
Author: Hayato Imori
Institution: Korea Advanced Institute of Science and Technology
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
OpenAlex ID: W3197594528