The hyperbolic complex line arrangement realization conjecture for quasipositive Seifert surfaces

From papers

Let FS3F\subset S^3 be a quasipositive Seifert surface. A proper arc in FF is an arc with endpoints on F\partial F and interior in the interior of FF. Let A(O,1)A(O,-1) be the surface used for plumbing, and let FF' denote the surface obtained by plumbing copies of A(O,1)A(O,-1) to FF along parallels of pairwise disjoint proper arcs α1,,αμF\alpha_1,\dots,\alpha_\mu\subset F. Hyperbolic complex line arrangement realization conjecture. There exists a system of pairwise disjoint proper arcs α1,,αμF\alpha_1,\dots,\alpha_\mu\subset F such that, for all sufficiently large k1,,kμNk_1,\dots,k_\mu\in\mathbb N, if FF' is obtained by plumbing kik_i copies of A(O,1)A(O,-1) along parallels of αi\alpha_i for i=1,,μi=1,\dots,\mu, then F\partial F' is isotopic to the link at infinity of a hyperbolic C\mathbb C-line arrangement. This would generalize the preceding theorem by realizing a broad class of links, obtained from quasipositive Seifert surfaces by sufficiently many such plumbings, as links at infinity of hyperbolic complex line arrangements; the source indicates that similar techniques may establish it, but does not provide a proof.

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Sources & referencesView supporting material

Primary source

Lee Rudolph, “Some fibered and non-fibered links at infinity of hyperbolic complex line arrangements”, arXiv:math/9912238 (2000).

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