On coxeter mapping classes and fibered alternating links

Abstract

Alternating-sign Hopf plumbing along a tree yields fibered alternating links whose homological monodromy is, up to a sign, conjugate to some alternating-sign Coxeter transformation. Exploiting this tie, we obtain results about the location of zeros of the Alexander polynomial of the fibered link complement implying a strong case of Hoste's conjecture, the trapezoidal conjecture, bi-orderability of the link group, and a sharp lower bound for the homological dilatation of the monodromy of the fibration. The results extend to more general hyperbolic fibered 3-manifolds associated to alternating-sign Coxeter graphs.

Minimal dilatation in Penner's construction

“…Indeed, the union of the constructed multicurves fills the surface (which has boundary, along which we glue in discs to land in the setting we are considering) and components intersect at most once, hence their intersection is minimal. Secondly, the given matrix describing the homological action of φ equals the corresponding matrix product M φ of Penner's construction described in Section 1.1, see [4]. We deduce that for such a mapping class φ realising the Coxeter transformation associated to (Γ, ±), the dilatation equals the spectral radius of the Coxeter transformation associated to (Γ, ±).…”

“…On the other hand, Hironaka and the author showed that the eigenvalues λ i of the Coxeter transformation corresponding to the Coxeter tree (Γ, ±) with alternating signs are related to the eigenvalues α i of A(Γ) by the equation [4]. Combined, we get that the µ i and the λ i are related by…”

“…The transition from the Coxeter transformation to the homological action of the associated Coxeter mapping class requires the addition of a minus sign, see [2,4]. We slightly abuse notation and again write µ i and λ i for the eigenvalues of the homological actions of the associated Coxeter mapping classes.…”

Minimal dilatation in Penner’s construction

“…Such a pair (Γ, s) is called a mixed-sign Coxeter graph; cf. [3]. To such a pair, we will associate certain products of reflections.…”

“…In the case of classical Coxeter trees (Γ, +), A'Campo realised the Coxeter transformation, up to a sign, as the homological action of a mapping class given by the product of two positive Dehn twists along multicurves that intersect each other with the pattern of Γ; see [2]. Similarly, in the case of alternating-sign Coxeter trees (Γ, ±), Hironaka and the author realised the Coxeter transformation, up to a sign, as the homological action of a mapping class given by the product of two Dehn twists of opposite signs along multicurves that intersect each other with the pattern of Γ; see [4]. We call these mapping classes Coxeter mapping classes.…”