Georgia Institute of Technology College of Sciences

Taking the double branched cover of $S^3$ over a knot $K$ is natural way to associate $K$ with a 3-manifold, and to study the double branched cover, we often want a Dehn surgery description for it. The Montesinos trick gives a systematic way to get such a description. In this talk, we will go over the broad statement of this trick: that a rational tangle replacement on the knot corresponds to Dehn surgery on the double branched cover. This gives particularly nice descriptions for some satellites of $K$ as surgery on $K \mathrel\# K^r$. We will also discuss an application of the trick which characterizes the 2-bridge knots with unknotting number 1.