Every lens space contains a genus one homologically fibered knot

Abstract

We prove that every lens space contains a genus one homologically fibered knot, which is contrast to the fact that some lens spaces contain no genus one fibered knot. In the proof, the Chebotarev density theorem and binary quadratic forms in number theory play a key role. We also discuss the Alexander polynomial of homologically fibered knots.

The existence of homologically fibered links and solutions of some equations

“…The author [11] prove that L(p, q) has a homologically fibered link whose homological fiber is homeomorphic to Σ 0,3 for every pair (p, q) by solving (1) following the Nozaki's argument. Thanks to Nozaki [10], we see that hc(M (A p (q))) = 1 for every (p, q). In this section, we determine hc(•) for the other generators for linkings, M (E k 0 ) and M (E k 1 ), we state as a proposition, proved at Subsections 6.1 and 6.2: Proposition 6.1.…”

“…Though we have Theorem 1.1, it is difficult in general to find a solution of (1) for given manifold and homeomorphic type of a surface. Nozaki [10] prove that L(p, q) has a homologically fibered link whose homological fiber is homeomorphic to Σ 1,1 for every pair (p, q) by solving some equations (, which is equivalent to (1)) using the density theorem. The author [11] prove that L(p, q) has a homologically fibered link whose homological fiber is homeomorphic to Σ 0,3 for every pair (p, q) by solving (1) following the Nozaki's argument.…”

Lens spaces which are realizable as closures of homology cobordisms over planar surfaces

“…We will show that for all (p, q), there exist integers satisfying (3) by using the following two facts, which and whose use are fully explained in Section 3 of [5]: From these we have the following:…”

“…In [5], it was proved that for any 3-manifold M whose first homology group is non-trivial and generated by one element, hc(M )= 1. However in general for a 3-manifold M whose first homology group has a non-trivial torsion, the computation of hc(M ) becomes difficult.…”

“…Recently, Nozaki [5] showed that every lens space contains a genus one homologically fibered knot by solving some algebraic equations. In this paper, like [5], we give the relation between the existence of homologically fibered link of a given number of components whose Seifert surface is a planar surface in a given lens space and some algebraic equation as follows. We denote by L(p, q) a lens space of type (p, q), and Σ g,n a surface of genus g with n boundary components.…”