Abstract
In this paper, we characterize all links in \(S^3\) with bridge number at least three that have a bridge sphere of distance two. We show that if a link L has a bridge sphere of distance at most two then it falls into at least one of three categories:
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The exterior of L contains an essential meridional sphere.
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L can be decomposed as a tangle product of a Montesinos tangle with an essential tangle in a way that respects the bridge surface and either the Montesinos tangle is rational or the essential tangle contains an incompressible, boundary-incompressible annulus.
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L is obtained by banding from another link \(L'\) that has a bridge sphere of the same Euler characteristic as the bridge sphere for L but of distance 0 or 1.
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This paper was written as part of an AIM SQuaREs project. The third author was also supported by NSF Grant DMS-1006369 and the fifth author was supported by NSF Grant DMS-1054450. The fourth author was partially supported by a grant from the Natural Sciences Division of Colby College.
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Blair, R., Campisi, M., Johnson, J. et al. Distance two links. Geom Dedicata 180, 17–37 (2016). https://doi.org/10.1007/s10711-015-0088-2
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DOI: https://doi.org/10.1007/s10711-015-0088-2