[Paper Review] Braids and branched coverings of dimension three

J. Scott Carter, Seiichi Kamada|arXiv (Cornell University)|Jun 21, 2012

Geometric and Algebraic Topology12 references3 citations

TL;DR

This paper introduces 2-dimensional charts in S³ to describe 3-dimensional branched coverings and 3-dimensional braids, generalizing classical braid and covering space theory. It establishes that oriented Seifert surfaces of links in S³ can be used as 2-dimensional braid charts to construct simple embedded 3-dimensional braids and branched coverings via monodromy representations, with key results linking quandle homomorphisms to lifts in braid groups.

ABSTRACT

We study simple branched coverings of degree d of the 2- and 3- dimensional sphere branched over oriented links. We demonstrate how to use braid charts to develop embeddings of these into $S^k imes D^2$ for $k=2,3 when $d=2,3$. This is an initial part of our study and represents the manuscript submitted to the RIMS workshop at Intelligence of Low Dimensional Topology.

Motivation & Objective

• To generalize classical braid and covering space theory to higher dimensions, specifically for 3-manifolds.
• To develop a chart-theoretic framework for describing 3-dimensional branched coverings and braids using 2-dimensional subcomplexes in S³.
• To establish a correspondence between oriented Seifert surfaces of links and 2-dimensional braid charts that encode monodromy data.
• To investigate the existence and characterization of lifts of quandle homomorphisms from link quandles to braid group elements.
• To extend the theory of Hurwitz systems and monodromy to 3-dimensional settings via charts and intersection words.

Proposed method

• Use 2-dimensional permutation and braid charts in S³, where faces are labeled by integers in {1,…,d−1} and satisfy local combinatorial conditions around edges.
• Construct monodromy representations π₁(S³∖L, ∗) → S_d (for branched coverings) and π₁(S³∖L, ∗) → B_d (for braids) using intersection words along the chart.
• Define 2-dimensional braid charts as oriented, labeled 2-complexes in S³ that model embedded 3-dimensional braids via motion pictures of surfaces.
• Apply the Riemann-Hurwitz formula and fundamental group analysis to ensure consistency of monodromy data and topological structure.
• Use quandle theory to characterize lifts of homomorphisms f: Q(S³, L) → T_d to A_d, where T_d is the set of transpositions and A_d the conjugates of braid group generators.
• Generalize classical 2D chart constructions (e.g., from Hurwitz and Lüroth) to 3D via Seifert surfaces and their labeled, oriented decompositions.

Experimental results

Research questions

• RQ1 How can 2-dimensional charts in S³ be used to describe 3-dimensional branched coverings and braids?
• RQ2 What conditions ensure that a Seifert surface of a link in S³ induces a well-defined 2-dimensional braid chart?
• RQ3 When does a quandle homomorphism f: Q(S³, L) → T_d admit a lift to a homomorphism into the braid group's conjugacy class set A_d?
• RQ4 How do Hurwitz moves and conjugations relate to equivalence of 3-dimensional braid monodromy systems?
• RQ5 What is the topological significance of nodal curves in 2-dimensional braid charts for immersed 3-dimensional braids?

Key findings

• A 2-dimensional braid chart of degree d in S³ induces a monodromy π₁(S³∖L, ∗) → B_d via intersection words, describing a simple embedded 3-dimensional braid.
• An oriented Seifert surface F of a link L in S³ can be interpreted as a 2-dimensional B_d-chart, yielding a 3-dimensional braid g_F: M³ → D²×S³ ⊂ R⁵.
• A non-oriented Seifert surface F induces a monodromy π₁(S³∖L, ∗) → S_d, describing a simple 2-fold branched covering f_F: M³ → S³ with branch set L.
• For the trefoil knot, a 3-coloring induces a monodromy ρ: π₁(S³∖L) → S₃, which is realized by a 2-dimensional B₃-chart with labeled faces.
• The existence of a lift of a quandle homomorphism f: Q(S³, L) → T_d to A_d is characterized by the existence of a compatible braid group lift, generalizing classical monodromy theory.
• The paper establishes that 2-dimensional charts provide a combinatorial framework for constructing 3-dimensional braids and branched coverings, extending classical 2D chart theory to 3D via Seifert surfaces and motion pictures.