Aru Mukherjea | Searcharxiv

arXiv2025

Higher degree covering moves for 3-manifolds

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 $q$.

math.GT↗

arXiv2022

From braid groups to mapping class groups

In this paper, we classify homomorphisms from the braid group of $n$ strands to the mapping class group of a genus $g$ surface. In particular, we show that when $g<n-2$, all representations are either cyclic or standard. Our result is sharp in the sense that when $g=n-2$, a generalization of the hyperelliptic representation appears, which is not cyclic or standard. This gives a classification of surface bundles over the configuration space of the complex plane. As a corollary, we partially recover the result of Aramayona-Souto, which classifies homomorphisms between mapping class groups, with a slight improvement.

math.GT↗