| In this work, we develop an approach to understanding (1, 1)-decompositions of (1, 1)-knots, called torus leveling. The minimum number of levels needed for a torus leveling is an invariant of an equivalence class of (1, 1)-decompositions. We show that this number equals a Hempel-type distance invariant defined using a certain arc complex. We also give an algebraic description of the torus level number, using a description of the (1, 1)-decomposition as a collection of words in the 2-braid group on the torus. We give some explicit computations for 2-bridge knots, and use our algebraic description to obtain upper bounds for their torus level numbers. |
