Higher degree covering moves for 3-manifolds

2025/07/20 by Aru Mukherjea, Mukherjea, Aru
Mathematics · #Advanced Combinatorial Mathematics #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.2507.15141

openalex publication_date 2025/07/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Covering moves relate colored link diagrams appearing as the branch sets of simple branched coverings of S³ by the same 3-manifold. We provide a complete set of covering moves on plat closures of braids in each fixed degree d ≥ 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.

Citations

• Special moves for open book decompositions of 3-manifolds
• Twist families of L-space knots, their genera, and Seifert surgeries
• Covering moves and Kirby calculus
• Lifting Braids

Related

• On moves between branched coverings of S³: The case of four sheets
• On four-dimensional 2-handlebodies and three-manifolds
• Covering moves and Kirby calculus
• Two Fold Branched Covers
• Algebraic Markov equivalence for links in 3-manifolds