An infinite family of links with critical bridge spheres

Abstract

A closed, orientable, splitting surface in an oriented 3-manifold is a topologically minimal surface of index n if its associated disk complex is (n−2)-connected but not (n−1)-connected. A critical surface is a topologically minimal surface of index 2. In this paper, we use an equivalent combinatorial definition of critical surfaces to construct the first known critical bridge spheres for nontrivial links.

Critical Bridge Spheres for Links with Arbitrarily Many Bridges

“…This implies that the unknot in 3-bridge position has a corresponding bridge sphere whose index is at most 2, and in fact it is critical. In [13], we constructed a family of 4-bridge links and showed that their corresponding bridge spheres are critical, demonstrating the existence of critical bridge spheres for nontrivial multicomponent links. This paper generalizes that result, showing that for every b ≥ 3, there is an infinite family of b-bridge links whose corresponding bridge spheres are critical.…”

“…Under the isotopy of F taking F from level 1 to level h, the loop ∂B becomes incredibly convoluted. In [13], the author describes a convenient way to represent the image of ∂B under this isotopy using the unlink diagram in Figure 4. To obtain Figure 4, we use the fact that the signs of the rows of twist regions alternate, but we refer readers to [13] for more detail.…”

“…In [13], the author describes a convenient way to represent the image of ∂B under this isotopy using the unlink diagram in Figure 4. To obtain Figure 4, we use the fact that the signs of the rows of twist regions alternate, but we refer readers to [13] for more detail. In this unlink diagram, each circle represents a number N j of parallel copies of that circle.…”

“…This paper should be understood as a companion to [13], of which this is a generalization. Many of the steps we take here are straightforward generalizations of steps taken there, and for the sake of brevity, we have here omitted several of the details and proofs that are essentially identical to those found in [13].…”

“…This paper should be understood as a companion to [13], of which this is a generalization. Many of the steps we take here are straightforward generalizations of steps taken there, and for the sake of brevity, we have here omitted several of the details and proofs that are essentially identical to those found in [13]. Every link L has a plat position, which is an embedding into S 3 such that L is a union of vertical strands, twist regions, and bridges, arranged as in Figure 1.…”