Independence of volume and genus 𝑔 bridge numbers

Abstract

Abstract. A theorem of Jorgensen and Thurston implies that the volume of a hyperbolic 3-manifold is bounded below by a linear function of its Heegaard genus. Heegaard surfaces and bridge surfaces often exhibit similar topological behavior; thus it is natural to extend this comparison to ask whether a (g, b)-bridge surface for a knot K in S 3 carries any geometric information related to the knot exterior. In this paper, we show that -unlike in the case of Heegaard splittings -hyperbolic volume and genus g bridge numbers are completely independent. That is, for any g, we construct explicit sequences of knots with bounded volume and unbounded genus g bridge number, and explicit sequences of knots with bounded genus g bridge number and unbounded volume.

On the pathwidth of hyperbolic 3-manifolds

“…The proof of Theorem 1 rests on Theorem 11, a folklore result according to which the Heegaard genus of a closed, orientable, hyperbolic 3-manifold can be upper-bounded in terms of its volume (see, e.g., [56, p. 336-337] or [46]).…”

Bridge numbers and meridional ranks of knotted surfaces and welded knots

“…The minimum number n such that K admits an n-bridge splitting with respect to a genus g Heegaard splitting is the genus g-bridge number. Purcell and Zupan asked the following question in [32].…”

“…In general, Purcell and Zupan showed that for any g, b ≥ 1, there does not exist C such that for any hyperbolic knot Cβ g (K) ≤ vol(K), where β g is the genus g-bridge number [32]. Blair et al showed that the analogue of Jørgensen and Thurston's result does hold if one restricts to the collection of prime alternating knots in S 3 with respect to the genus 0 Heegaard splitting, and set the universal constant C = 1 6 v tet , where v tet ≈ 1.01 is the volume of a regular ideal tetrahedron [5,Theorem 1.5].…”