| A natural question in knot theory is to ask how certain properties of a knot behave under satellite operations. We will focus on the satellite operation of cabling, and on Heegaard Floer-theoretic properties. In particular, we will give a formula for the Ozsvath-Szabo concordance invariant tau of iterated cables of a knot K in terms of the cabling parameters, tau( K), and a new concordance invariant, epsilon(K). We show that, in many cases, epsilon gives better bounds on the 4-ball genus of a knot that tau alone, and discuss further applications of epsilon. We will also completely classify when the iterated cable of a knot admits a positive L-space surgery. |
