Abstract. Let K be a knot in 1-bridge position with respect to a genusg Heegaard surface that splits a 3-manifold M into two handlebodies V and W . One can move K by isotopy keeping K ∩ V in V and K ∩ W in W so that K lies in a union of n parallel genus-g surfaces tubed together by n − 1 straight tubes, and K intersects 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 the arc complex of the two holed genus-g surface.

We show that if a Heegaard splitting is obtained by gluing a splitting of Hempel distance at least 4 and the genus-1 splitting of S 2 × S 1 , then the Goeritz group of the splitting is finitely generated. To show this, we first provide a sufficient condition for a full subcomplex of the arc complex for a compact orientable surface to be contractible, which generalizes the result by Hatcher that the arc complexes are contractible. We then construct infinitely many Heegaard splittings, including the above-mentioned Heegaard splitting, for which suitably defined complexes of Haken spheres are contractible.

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Any knot K in genus-1 1-bridge position can be moved by isotopy to lie in a union of n parallel tori tubed by n − 1 tubes so that K intersects each tube in two spanning arcs, which we call a leveling of the position. The minimal n for which this is possible is an invariant of the position, called the level number. In this work, we describe the leveling by the braid group on two points in the torus, which yields a numerical invariant of the position, called the (1, 1)-length. We show that the (1, 1)-length equals the level number. We then find braid descriptions for (1, 1)-positions of all 2-bridge knots providing upper bounds for their level numbers, and also show that the (−2, 3, 7)-pretzel knot has level number two.

Any knot [Formula: see text] in genus-[Formula: see text] [Formula: see text]-bridge position can be moved by isotopy to lie in a union of [Formula: see text] parallel tori tubed by [Formula: see text] tubes so that [Formula: see text] intersects each tube in two spanning arcs, which we call a leveling of the position. The minimal [Formula: see text] for which this is possible is an invariant of the position, called the level number. In this work, we describe the leveling by the braid group on two points in the torus, which yields a numerical invariant of the position, called the [Formula: see text]-length. We show that the [Formula: see text]-length equals the level number. We then find braid descriptions for [Formula: see text]-positions of all [Formula: see text]-bridge knots providing upper bounds for their level numbers and also show that the [Formula: see text]-pretzel knot has level number two.