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Folding Branched Covers of the $3$-Sphere Branched over Knots

J. Scott Carter, Seonmi Choi, Byeorhi Kim

TL;DR

This work develops a diagrammatic framework for folding and braiding branched covers of $S^m$, focusing on cyclic and dihedral covers and their realizations as maps into $S^m\times[0,n+1]$ (and into higher-dimensional spaces for cyclic cases). By pairing braid charts with permutation charts and introducing dihedral charts for $D_5$, the authors construct explicit foldings, notably a detailed folding of the $D_5$-cover of $S^3$ branched along the torus knot $T(2,5)$, which is itself $S^3$. The core contributions include a category-based justification for the invertibility of diagrammatic moves, a systematic method to convert dihedral colorings of knot exteriors into folded embeddings, and general pathways to extend these results to arbitrary finite groups via presentation theory. The results advance concrete, constructive techniques to realize folded branched covers, offering tools that may yield new knot invariants and liftability criteria in 3- and 4-dimensional topology.

Abstract

A folding of a branched cover of the 3-sphere that is branched over a knot is a continuous map of the cover into the product of the sphere with a disk that has the property that the projection onto the sphere factor induces the covering. Moreover, away from the branch set, the map is a general position immersion. Cyclic branched covers can be folded so that the map is an embedding when the disk factor is 2-dimensional. Dihedral branched covers can also be folded. In as much as possible, the foldings that are presented are quite detailed. In particular, the paper focuses upon a folding of the dihedral cover of the 3-sphere that is branched along a torus knot of type (2,5). The cover also is homeomorphic to the $3$-sphere.

Folding Branched Covers of the $3$-Sphere Branched over Knots

TL;DR

This work develops a diagrammatic framework for folding and braiding branched covers of , focusing on cyclic and dihedral covers and their realizations as maps into (and into higher-dimensional spaces for cyclic cases). By pairing braid charts with permutation charts and introducing dihedral charts for , the authors construct explicit foldings, notably a detailed folding of the -cover of branched along the torus knot , which is itself . The core contributions include a category-based justification for the invertibility of diagrammatic moves, a systematic method to convert dihedral colorings of knot exteriors into folded embeddings, and general pathways to extend these results to arbitrary finite groups via presentation theory. The results advance concrete, constructive techniques to realize folded branched covers, offering tools that may yield new knot invariants and liftability criteria in 3- and 4-dimensional topology.

Abstract

A folding of a branched cover of the 3-sphere that is branched over a knot is a continuous map of the cover into the product of the sphere with a disk that has the property that the projection onto the sphere factor induces the covering. Moreover, away from the branch set, the map is a general position immersion. Cyclic branched covers can be folded so that the map is an embedding when the disk factor is 2-dimensional. Dihedral branched covers can also be folded. In as much as possible, the foldings that are presented are quite detailed. In particular, the paper focuses upon a folding of the dihedral cover of the 3-sphere that is branched along a torus knot of type (2,5). The cover also is homeomorphic to the -sphere.

Paper Structure

This paper contains 16 sections, 6 theorems, 30 equations, 27 figures.

Table of Contents

  1. Introduction
  2. Main Results
  3. Presentation of the material
  4. Braid and Permutation Charts
  5. Diagrammatics of the dihedral group
  6. The reflections in $D_5$ as conjugates
  7. Dihedral covers of $S^2$ branched over two points
  8. A schematic description of a dihedral cover
  9. Digression: Moves to Dihedral Charts / Moves to Permutation Charts
  10. A category that is associated to a group presentation
  11. Lemmas
  12. Choosing reflections
  13. Proof of Theorem \ref{['Dihedral2']}
  14. Proof of Theorem \ref{['DihedralThm']}
  15. Proof of Theorem \ref{['CyclicThm']}
  16. ...and 1 more sections

Key Result

Theorem 1

In the notation above, let $D_n \stackrel{\phi}{\longleftarrow} \pi_1(E(K)) = F_{\ell}$ denote a group homomorphism into the dihedral group of order $2n$. Let $M$ denote the associated branched cover of $S^2$ that is branched along $K$. There is an immersed braiding In particular, the restriction of continuous, general position map $f$ to the exterior $E(K)$ is an immersion, and the composition

Figures (27)

  • Figure 1: Types of vertices in a braid chart
  • Figure 2: A chart and the associated braided (or folded) surface
  • Figure 3: Projections of braids to permuations
  • Figure 4: A choice of generators for the dihedral group
  • Figure 5: Schematics of the relations in the dihedral group $D_5$
  • ...and 22 more figures

Theorems & Definitions (6)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Lemma 4
  • Corollary 5
  • Lemma 6