Distance two links (Q2634842)

A sphere \(S\) in \(S^3\) is a bridge sphere for a link \(L\) if \(L\) intersects each of the balls \(B_1\) and \(B_2\) bounded by \(S\) in trivial tangles. That is, \(L\cap B_i\) consists of boundary-parallel arcs. The bridge number of \(L\) is the minimal number of such arcs required in \(B_1\). Compressing discs for \(S\setminus L\) in \(B_i\setminus L\) define two sets of essential simple closed curves on \(S\setminus L\). The distance of the bridge splitting is the minimal distance between these two sets in the curve complex. This paper studies links with bridge number at least \(3\) and distance at most \(2\). It shows that all such links match at least one of three descriptions. This is achieved by building suitable surfaces and isotopies. Links with distance \(1\) are known to have an essential meridional surface in their complement, while those with distance at least \(3\) share various nice properties and are often collectively known as `high distance'. The paper includes a discussion of what a converse statement to the theorem would look like, and finishes with a series of outstanding questions.