[Paper Review] Genus one 1-bridge knots and Dunwoody manifolds

Luigi Grasselli, Michele Mulazzani|ArXiv.org|Mar 7, 2000

Geometric and Algebraic Topology18 references4 citations

TL;DR

This paper establishes that all Dunwoody manifolds—3-manifolds constructed from 6-tuples of integers via planar trivalent graphs with cyclic symmetry—are cyclic coverings of lens spaces (including S³), branched over genus one 1-bridge knots. The key contribution is a proof of Dunwoody's conjecture that a wide subclass of these manifolds are cyclic branched coverings of S³ over knots, and it is further shown that all branched cyclic coverings of 2-bridge knots belong to this class, implying their fundamental groups admit geometric cyclic presentations.

ABSTRACT

In this paper we show that all 3-manifolds of a family introduced by M. J. Dunwoody are cyclic coverings of lens spaces (eventually $\bf S^3$), branched over genus one 1-bridge knots. As a consequence, we give a positive answer to the Dunwoody conjecture that all the elements of a wide subclass are cyclic coverings of $\bf S^3$ branched over a knot. Moreover, we show that all branched cyclic coverings of a 2-bridge knot belong to this subclass; this implies that the fundamental group of each branched cyclic covering of a 2-bridge knot admits a geometric cyclic presentation.

Motivation & Objective

• To resolve Dunwoody's conjecture that a wide subclass of Dunwoody manifolds are cyclic branched coverings of S³ over knots.
• To establish a connection between branched cyclic coverings of 2-bridge knots and the class of Dunwoody manifolds.
• To show that the fundamental groups of branched cyclic coverings of 2-bridge knots admit geometric cyclic presentations.
• To demonstrate that all Dunwoody manifolds arise as cyclic coverings of lens spaces (including S³) branched over genus one 1-bridge knots.
• To provide a topological and group-theoretic characterization of Dunwoody manifolds using Heegaard diagrams and cyclic presentations.

Proposed method

• Constructing Dunwoody manifolds via planar, 3-regular graphs with cyclic symmetry defined by admissible 6-tuples (a,b,c,n).
• Using Singer moves of type IC to simplify Heegaard diagrams, reducing the number of vertices and cycles step by step.
• Applying a sequence of Singer moves of type IC to eliminate complementary handle pairs, ultimately reducing to a genus one Heegaard diagram.
• Identifying the final genus one diagram as the canonical diagram of a lens space L(2a+1, 2r), thus showing the original manifold is a cyclic covering of that lens space.
• Analyzing the structure of the resulting diagrams to confirm the branching locus is a genus one 1-bridge knot.
• Establishing that the fundamental group of the resulting manifold admits a cyclic presentation induced by the Heegaard diagram.

Experimental results

Research questions

• RQ1 Are all Dunwoody manifolds cyclic branched coverings of lens spaces, including S³, over genus one 1-bridge knots?
• RQ2 Does the subclass of Dunwoody manifolds defined by certain 6-tuples satisfy Dunwoody's conjecture that they are cyclic coverings of S³ branched over knots?
• RQ3 Do all branched cyclic coverings of 2-bridge knots belong to the class of Dunwoody manifolds?
• RQ4 Can the fundamental group of a branched cyclic covering of a 2-bridge knot be presented as a geometric cyclic presentation?
• RQ5 Is the genus one 1-bridge knot the branching locus for all Dunwoody manifolds?

Key findings

• All Dunwoody manifolds are cyclic coverings of lens spaces (including S³), branched over genus one 1-bridge knots.
• A wide subclass of Dunwoody manifolds are cyclic branched coverings of S³ over knots, confirming Dunwoody's conjecture.
• All branched cyclic coverings of 2-bridge knots are contained within the class of Dunwoody manifolds.
• The fundamental group of each branched cyclic covering of a 2-bridge knot admits a geometric cyclic presentation.
• The final Heegaard diagram after successive Singer moves of type IC is the canonical diagram of the lens space L(2a+1, 2r), proving the covering structure.
• The branching locus in all cases is a genus one 1-bridge knot, which includes both torus knots and 2-bridge knots as special cases.