2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/215512A knot K in 1-bridge position with respect to a genus-g Heegaard surface in a 3-manifold can be moved by isotopy through knots in 1-bridge position until it lies in a union of n parallel genus-g surfaces tubed together by n-1 straight tubes, with K intersecting each tube in two arcs connecting the ends. We prove that the minimum n for which this is possible is equal to a Hempel-type distance invariant defined using an arc complex of the two holed genus-g surface8 pages, 4 figures, to appear in Proc. Amer. Math. SocGeometric Topology57M25text