Cabling Conjecture for Small Bridge Number

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 $5$.