[Paper Review] The many faces of cyclic branched coverings of 2-bridge knots and links

Michele Mulazzani, Andreĭ Vesnin|ArXiv.org|Jun 19, 2001

Geometric and Algebraic TopologyMathematics68 references22 citations

TL;DR

This paper provides a comprehensive analysis of cyclic branched coverings of 2-bridge knots and links in the 3-sphere, presenting multiple topological descriptions—polyhedral, Heegaard diagrams, Dehn surgery, and colored graphs—and deriving fundamental group and homology group presentations. The key contribution is a decomposition theorem showing that any singly-cyclic branched covering of a 2-bridge link arises as a composition of a meridian-cyclic covering of a derived link and a cyclic covering of a trivial knot.

ABSTRACT

We discuss 3-manifolds which are cyclic coverings of the 3-sphere, branched over 2-bridge knots and links. Different descriptions of these manifolds are presented: polyhedral, Heegaard diagram, Dehn surgery and coloured graph constructions. Using these descriptions, we give presentations for their fundamental groups, which are cyclic presentations in the case of 2-bridge knots. The homology groups are given for a wide class of cases. Moreover, we prove that each singly-cyclic branched covering of a 2-bridge link is the composition of a meridian-cyclic branched covering of a determined link and a cyclic branched covering of a trivial knot.

Motivation & Objective

• To provide a unified topological description of 3-manifolds arising as cyclic branched coverings of 2-bridge knots and links.
• To establish multiple equivalent constructions—polyhedral, Heegaard diagrams, Dehn surgery, and colored graphs—for these manifolds.
• To derive explicit presentations of the fundamental groups and compute homology groups for a broad class of such coverings.
• To prove a structural decomposition of singly-cyclic branched coverings into compositions of simpler coverings.

Proposed method

• Using Fox's theorem on branched coverings, the paper constructs cyclic branched coverings via monodromy representations from the fundamental group of the link complement to the symmetric group.
• The polyhedral construction realizes the covering space as a cell decomposition of the 3-manifold via a branched covering of a 3-sphere with a 1-skeleton as the branching set.
• Heegaard diagrams are constructed by lifting the link to the covering space and analyzing the induced splitting of the 3-manifold into handlebodies.
• Dehn surgery descriptions are derived by identifying the covering space as a result of surgery on a link obtained from the original 2-bridge link.
• Colored graph (gem) constructions are used to encode the 3-manifold via a 4-valent graph with colored edges, representing the cell decomposition.
• The fundamental group is computed via the kernel of a homomorphism from the orbifold group to ℤₙ, and homology is computed using the abelianization of the fundamental group.

Experimental results

Research questions

• RQ1 How can cyclic branched coverings of 2-bridge knots and links be described using multiple topological constructions?
• RQ2 What are the fundamental group and homology group presentations for these coverings, especially in the case of 2-bridge knots?
• RQ3 Can every singly-cyclic branched covering of a 2-bridge link be decomposed into a composition of simpler coverings?
• RQ4 What is the role of the linking number and continued fraction expansion in determining the structure of the covering space?
• RQ5 How do the different constructions (polyhedral, surgery, Heegaard, gem) relate to one another in describing the same 3-manifold?

Key findings

• The fundamental group of a cyclic branched covering of a 2-bridge knot admits a cyclic presentation, which is a strong algebraic invariant of the covering space.
• The homology groups are computed for a wide class of coverings, providing topological invariants that distinguish different manifolds.
• For any singly-cyclic branched covering of a 2-bridge link, the covering space is shown to be equivalent to a composition of a meridian-cyclic covering of a link L(d, α₁/β) and a d-fold cyclic covering of a trivial knot.
• The decomposition is formalized in a commutative diagram of coverings, where the intermediate orbifold has singular set L(d, α₁/β) with index n/d and the final covering is over a trivial knot with index d.
• The construction generalizes previous results on the Whitehead link and provides a systematic method to build and classify such 3-manifolds.
• The paper establishes a one-to-one correspondence between cyclic branched coverings and epimorphisms from the first homology of the link complement to ℤₙ, with the branching data encoded by integers k_j generating ℤₙ.