[Paper Review] Homology cylinders in knot theory

Hiroshi Goda, Takuya Sakasai|arXiv (Cornell University)|Jul 25, 2008

Geometric and Algebraic Topology20 references12 citations

TL;DR

This paper establishes a connection between sutured manifolds and homology cylinders by studying homologically fibered knots—knots whose complement admits a sutured manifold structure with homology cylinder invariants. By leveraging invariants of homology cylinders, the authors derive new fibering obstructions, compute Reidemeister torsions, and determine handle numbers for these knots, advancing the understanding of knot invariants through finite-type invariants and homology cobordism theory.

ABSTRACT

Sutured manifolds defined by Gabai are useful in the geometrical study of knots and 3-dimensional manifolds. On the other hand, homology cylinders are in an important position in the recent theory of homology cobordisms of surfaces and finite-type invariants. We study a relationship between them by focusing on sutured manifolds associated with a special class of knots which we call {\it homologically fibered knots}. Then we use invariants of homology cylinders to give applications to knot theory such as fibering obstructions, Reidemeister torsions and handle numbers of homologically fibered knots.

Motivation & Objective

• To investigate the relationship between sutured manifolds and homology cylinders in the context of knot theory.
• To define and study a special class of knots—homologically fibered knots—whose complements support sutured manifold structures compatible with homology cylinders.
• To apply invariants of homology cylinders to extract new knot-theoretic invariants such as fibering obstructions, Reidemeister torsions, and handle numbers.
• To extend the framework of finite-type invariants and homology cobordism to the setting of knot complements via sutured manifold techniques.

Proposed method

• Construct sutured manifolds from the complements of homologically fibered knots, leveraging their homological fibered structure.
• Utilize the theory of homology cylinders to extract invariants from the sutured manifold structures of these knot complements.
• Apply known invariants of homology cylinders—such as finite-type invariants and torsion invariants—directly to the knot complements.
• Use the resulting invariants to derive obstructions to fibering, compute Reidemeister torsions, and determine handle numbers of the knots.
• Establish a bridge between 3-manifold topology (via sutured manifolds) and the theory of homology cobordisms (via homology cylinders).
• Employ algebraic and geometric techniques to relate the homology cylinder invariants to topological invariants of the knots.

Experimental results

Research questions

• RQ1 How can sutured manifolds associated with homologically fibered knots be used to extract new invariants in knot theory?
• RQ2 What specific knot invariants—such as fibering obstructions, Reidemeister torsions, and handle numbers—can be derived from invariants of homology cylinders?
• RQ3 In what ways do homology cylinder invariants refine or extend existing invariants in the study of knot complements?
• RQ4 How does the structure of homologically fibered knots relate to the algebraic and geometric properties of homology cylinders?
• RQ5 Can the interplay between sutured manifolds and homology cylinders lead to new obstructions for a knot to be fibered?

Key findings

• The paper introduces homologically fibered knots as a class of knots whose complements support sutured manifold structures compatible with homology cylinder invariants.
• It derives new fibering obstructions for knots using invariants of homology cylinders, providing a novel algebraic-topological criterion for non-fiberedness.
• Reidemeister torsions of homologically fibered knots are computed via invariants of associated homology cylinders, linking torsion invariants to finite-type invariants.
• The handle number of a homologically fibered knot is determined through the structure of the corresponding homology cylinder, offering a new computational method.
• The framework successfully connects sutured manifold theory with homology cobordism and finite-type invariants, enabling new applications in knot theory.
• The results demonstrate that homology cylinder invariants can detect subtle topological features of knot complements, such as fibering and torsion properties.