Extending homeomorphisms from 2-punctured surfaces to handlebodies
  Let F a closed connected orientable surface bounding a genus g handlebody H. In this paper we find a finite set of generators for the subgroup E(2,g) of the pure mapping class group of the twice punctured torus PMCG(2,g), consisting of the isotopy classes of homeomorphisms of F which admit an extension to H keeping a properly embedded trivial arc fixed. This subgroup turns out to be important for the study of knots in closed 3-manifolds via (g,1)-decomposition. In fact, the knots represented by the isotopy classes belonging to the same left cosets of E(2,g) in PMCG(2,g) are equivalent. 
Alessia Cattabriga, Michele Mulazzani