Cabling Conjecture for Small Bridge Number

TL;DR

This paper proves the Cabling Conjecture for knots with bridge number up to 5, showing that certain Dehn surgeries produce reducible manifolds only for specific cable knots.

Contribution

It extends previous work to confirm the Cabling Conjecture for knots with bridge number up to 5, broadening the class of knots for which the conjecture holds.

Findings

Proves the Cabling Conjecture for knots with bridge number ≤ 5

Shows that $ ext{π}$-Dehn surgery yields reducible manifolds only for $(p,q)$-cable knots with slope $pq$

Builds on Hoffman’s work to generalize the conjecture's validity

Abstract

Let $k\subset S^3$ be a nontrivial knot. The Cabling Conjecture of Francisco Gonz\'alez-Acu\~na and Hamish Short posits that $\pi$-Dehn surgery on $k$ produces a reducible manifold if and only if $k$ is a $(p,q)$-cable knot and the surgery slope $\pi$ equals $pq$. We extend the work of James Allen Hoffman to prove the Cabling Conjecture for knots with bridge number up to $5$.