Cut-disks for level spheres in link and tangle complements

TL;DR

This paper investigates the properties of thin spheres in prime links and certain tangles, showing they lack vertical cut-disks under specific conditions, which advances understanding of the topology of link and tangle complements.

Contribution

It extends Wu's results by proving that the lowest-width thin sphere in prime links and certain tangles has no vertical cut-disks, revealing new structural insights.

Findings

Lowest-width thin sphere in prime links has no vertical cut-disks

Results apply to specific tangles in S^2 x [-1,1]

Enhances understanding of essential surfaces in link complements

Abstract

Wu has shown that if a link or a knot $L$ in $S^3$ in thin position has thin spheres, then the thin sphere of lowest width is an essential surface in the link complement. In this paper we show that if we further assume that $L \subset S^3$ is prime, then the thin sphere of lowest width also does not have any vertical cut-disks. We also prove the result for a specific kind of tangles in $S^2 \times [-1,1]$.