Nozaki, Yuta Every Lens space contains a genus one homologically fibered knot. (English) Zbl 1409.57017 Ill. J. Math. 62, No. 1-4, 99-111 (2018). Using the Chebotarev density theorem and a well-known fact about binary quadratic forms, this paper constructs a surface whose complement is a requisite homology cobordism to prove that every lens space contains a genus one homologically fibered knot, despite the fact that some lens spaces contain no genus one fibered knot. It also characterizes homologically fibered knots in a rational homology 3-sphere in terms of their Alexander polynomial, using linking numbers [L. Rozansky, Adv. Math. 134, No. 1, 1–31 (1998; Zbl 0949.57006)] to determine the Seifert matrix with respect to curves on the surface depicted in Figure 2. The author acknowledges helpful comments from T. Sakasai, G. Massuyeau, M. Somekawa, I. Nagamachi, J. Ueki and, of course, “the referees”. Reviewer: Kenneth A. Perko Jr. (New York) Cited in 1 ReviewCited in 3 Documents MSC: 57M27 Invariants of knots and \(3\)-manifolds (MSC2010) 11R45 Density theorems Keywords:lens spaces; cobordism; homologically fibred knots; linking numbers; Chebotarev density theorem; Alexander polynomials; Seifert matrices; binary quadratic forms Citations:Zbl 0949.57006 × Cite Format Result Cite Review PDF Full Text: DOI arXiv Euclid References: This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.