[Paper Review] Homology cylinders and sutured manifolds for homologically fibered knots

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

Geometric and Algebraic Topology31 references9 citations

TL;DR

This paper establishes a connection between homology cylinders and sutured manifolds in the context of homologically fibered knots—knots with minimal genus Seifert surfaces and monic Alexander polynomials of degree twice the genus. By showing that such knots yield homology cylinders via their complementary sutured manifolds, the authors apply invariants like Magnus representations and Reidemeister torsion to derive fibering obstructions, factorization formulas for torsion, and handle number bounds, including examples with arbitrarily large handle numbers via the Nakanishi index.

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 characterize homologically fibered knots as those with minimal genus Seifert surfaces and monic Alexander polynomials of degree twice the genus.
• To apply invariants of homology cylinders—particularly Magnus representations and Reidemeister torsions—to knot-theoretic problems.
• To derive lower bounds on handle numbers of sutured manifolds using invariants like the Nakanishi index.
• To demonstrate that homologically fibered knots can yield non-product homology cylinders, extending beyond fibered knots.

Proposed method

• Constructing homology cylinders from the complementary sutured manifold of a homologically fibered knot via Seifert surface decomposition.
• Using the Magnus representation of the fundamental group of the knot exterior to analyze the structure of the homology cylinder.
• Applying noncommutative Reidemeister torsion invariants to the homology cylinder to derive factorization formulas for the torsion of the knot exterior.
• Employing product decompositions of sutured manifolds to verify when a sutured manifold is a product, aiding in handle number estimation.
• Utilizing the Nakanishi index as a lower bound for the handle number of doubled knots with specific Seifert surfaces.
• Mapping the homology cylinder's structure via matrices over the group ring ℤ[s±,t±], relating them to the Alexander matrix of the knot group.

Experimental results

Research questions

• RQ1 When does the complementary sutured manifold of a knot with a minimal genus Seifert surface form a homology cylinder?
• RQ2 How can invariants of homology cylinders, such as Magnus representations and Reidemeister torsion, be used to obstruct fibering in homologically fibered knots?
• RQ3 What factorization formulas can be derived for the Reidemeister torsion of the exterior of a homologically fibered knot using homology cylinder invariants?
• RQ4 How do handle numbers of sutured manifolds relate to invariants like the Nakanishi index in the context of doubled knots?
• RQ5 Can homologically fibered knots yield non-product homology cylinders, and what does this imply for the classification of 3-manifolds?

Key findings

• Homologically fibered knots are characterized as those with minimal genus Seifert surfaces and monic Alexander polynomials of degree twice the genus.
• The complementary sutured manifold of a homologically fibered knot naturally yields a homology cylinder, extending the classical monodromy construction beyond fibered knots.
• Magnus representations and Reidemeister torsions of homology cylinders provide effective fibering obstructions for homologically fibered knots.
• Factorization formulas for the Reidemeister torsion of the knot exterior are derived using the structure of the homology cylinder's homology and the Alexander matrix.
• There exist homologically fibered knots with genus-1 Seifert surfaces and arbitrarily large handle numbers, as shown via the Nakanishi index.
• For the pretzel knot P(3,-3,3), the handle number of its doubled knot is exactly 2, and this bound is sharp, confirmed by product decomposition techniques.