Arc distance equals level number

TL;DR

This paper establishes a relationship between the minimal number of parallel surfaces needed to position a knot and a Hempel-type distance invariant derived from an arc complex, linking geometric and combinatorial properties.

Contribution

It proves that the minimal number of parallel surfaces for a knot in 1-bridge position equals a Hempel-type distance invariant from the arc complex.

Findings

Minimum n equals the Hempel-type distance invariant

Provides a new geometric interpretation of the arc complex distance

Connects knot position complexity with surface topology

Abstract

A 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 surface