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Higher degree covering moves for 3-manifolds

Aru Mukherjea

TL;DR

The paper develops a complete set of covering moves for fixed degree $d\ge 4$ to relate colored links describing the same closed 3-manifold as a simple branched cover of $S^3$. It extends prior work by exploiting the liftable subgroup $L^p(B_n)$ and its generators to construct a lifting homomorphism to the mapping class group, enabling a 3-dimensional proof that two such colored links become equivalent after stabilization and a finite sequence of moves (C and N, plus isotopy). It then shows that after $d-2$ stabilizations, moves II, III, and IV can be realized using only moves C and N in the braided setting, and proves a lens-space corollary: the $d$-fold simple branched cover of a $d$-bridge knot is a lens space $L(p,q)$ with $p,q$ computable diagrammatically. The results provide a robust bridge between Kirby-type 4-dimensional cobordisms and 3-dimensional branched-cover moves, with implications for understanding Heegaard splittings, stabilizations, and potential extensions to higher dimensions and braided surfaces.

Abstract

Covering moves relate colored link diagrams appearing as the branch sets of simple branched coverings of $S^3$ by the same 3-manifold. We provide a complete set of covering moves on plat closures of braids in each fixed degree $d \geq 4$, extending prior work of Apostolakis and Piergallini. As a consequence we show that after stabilization to the same degree at least 4, only two local tangle replacements are required to relate any two colored links, recovering Bobtcheva and Piergallini's resolution of a conjecture of Montesinos. We also obtain that in the braided setting, the two local tangle replacements suffice after $d-2$ stabilizations. Lastly, we prove that the $d$-fold simple branched cover of a $d$-bridge knot is a lens space $L(p,q)$ and provide a method for determining $p$ and $q$.

Higher degree covering moves for 3-manifolds

TL;DR

The paper develops a complete set of covering moves for fixed degree to relate colored links describing the same closed 3-manifold as a simple branched cover of . It extends prior work by exploiting the liftable subgroup and its generators to construct a lifting homomorphism to the mapping class group, enabling a 3-dimensional proof that two such colored links become equivalent after stabilization and a finite sequence of moves (C and N, plus isotopy). It then shows that after stabilizations, moves II, III, and IV can be realized using only moves C and N in the braided setting, and proves a lens-space corollary: the -fold simple branched cover of a -bridge knot is a lens space with computable diagrammatically. The results provide a robust bridge between Kirby-type 4-dimensional cobordisms and 3-dimensional branched-cover moves, with implications for understanding Heegaard splittings, stabilizations, and potential extensions to higher dimensions and braided surfaces.

Abstract

Covering moves relate colored link diagrams appearing as the branch sets of simple branched coverings of by the same 3-manifold. We provide a complete set of covering moves on plat closures of braids in each fixed degree , extending prior work of Apostolakis and Piergallini. As a consequence we show that after stabilization to the same degree at least 4, only two local tangle replacements are required to relate any two colored links, recovering Bobtcheva and Piergallini's resolution of a conjecture of Montesinos. We also obtain that in the braided setting, the two local tangle replacements suffice after stabilizations. Lastly, we prove that the -fold simple branched cover of a -bridge knot is a lens space and provide a method for determining and .

Paper Structure

This paper contains 22 sections, 11 theorems, 6 equations, 34 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Braid words and half-twists
  4. Monodromy and coloring
  5. Liftability and the liftable subgroup of $B_n$
  6. Conventions
  7. Proofs
  8. Outline
  9. Proof of Theorem \ref{['thm:moves']}
  10. Normalization of induced Heegaard splittings
  11. Generators for $L^p(B_n)$
  12. Examples
  13. The lifting homomorphism
  14. Equivalence of Heegaard splittings
  15. Normal generators for $\ker L$
  16. ...and 7 more sections

Key Result

Theorem 1.1

Suppose two colored plats $L_1, L_2 \subset S^3$ are such that the degree $d \geq 3$ simple branched covering of $S^3$ over each $L_i$ is the same closed 3-manifold. Then, $L_1$ and $L_2$ can be related by colored isotopy and a finite sequence of moves N, C, II$_i$, III$_{i,j}$, and IV, which preser

Figures (34)

  • Figure 1: The moves in Theorem \ref{['thm:moves']}. The labelling conventions used in these diagrams are described in Section \ref{['bg:conventions']}. The standard coloring is described in Figure \ref{['fig:std']}.
  • Figure 2: Examples of colored knots with the same cover. (a) Given a 3-manifold $M$, take a colored knot that describes $M$ as a 3-fold simple branched cover of $S^3$. Following hil74, this can be written as shown in the figure, for some braid $\beta$. By varying the number of half-twists $n$ in the depicted region, we obtain a family of knots, infinitely many of which are distinct by kms92. (b) This family of knots can be obtained by stabilizing the construction in Figure \ref{['fig:twist1']} and joining components using move C. Twist regions about adjacent strands can be added using move C. Twist regions about non-adjacent strands (not pictured) can be added using move N. We obtain infinite families of distinct knots with the same cover, in each degree. For degree at least 4, the twist regions can be chosen so that the family contains knots of arbitrarily high genus bm19. As degree of a connected covering is a lower bound for bridge number, successive stabilizations (and joining components using move C) provide a family of knots of increasing bridge number that have the same cover.
  • Figure 3: The $i$-th Artin generator $\sigma_i$.
  • Figure 4: Examples of half-twists about different arcs $\gamma$. The positive half-twist is defined on a disk neighbourhood of an arc by rotating it 180 degrees anticlockwise, fixing the arc setwise and exchanging the endpoints.
  • Figure 5: A simple and more complicated Hurwitz system $\alpha_0,\ldots ,\alpha_{n-1}$ and corresponding maximal chain $x_0,\ldots ,x_{n-2}$ for a disk with six punctures.
  • ...and 29 more figures

Theorems & Definitions (21)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Definition 2.1
  • Theorem 2.2: be79, mp01
  • Definition 2.3
  • Definition 2.4
  • Proposition 2.5: ww12 Theorem 31
  • Lemma 3.1
  • ...and 11 more