Small genus knots in lens spaces have small bridge number

Abstract

In a lens space X of order r a knot K representing an element of the fundamental group 1 X Š ‫=ޚ‬r ‫ޚ‬ of order s Ä r contains a connected orientable surface S properly embedded in its exterior X N .K/ such that @S intersects the meridian of K minimally s times. Assume S has just one boundary component. Let g be the minimal genus of such surfaces for K , and assume s 4g 1. Then with respect to the genus one Heegaard splitting of X , K has bridge number at most 1. 57M27; 57M25 Statement of resultsAny knot K in a lens space X D L.r; q/, r > 0, is rationally nullhomologous, ie OEK D 0 2 H 1 .X I ‫/ޑ‬ Š 0. We say r is the order of the lens space X , and we say the smallest positive integer s such that sOEK D 0 2 H 1 .X I ‫/ޚ‬ Š ‫=ޚ‬r ‫ޚ‬ is the order of the knot K . Note s Ä r . The exterior X N .K/ of K thus contains a connected properly embedded orientable surface S such that when S is oriented @S is coherently oriented on @ x N .K/ and intersects the meridian  @ x N .K/ of K minimally s times, ie j @S j D s . Such a surface S is an analogue of a Seifert surface for a knot in S 3 . We refer to the genus of a knot K in X as the minimal genus of these "rational" Seifert surfaces for K . For this article we will restrict our attention to knots with rational Seifert surfaces that have just one boundary component.In this paper we prove the following theorem. Theorem 1.1 Let K be a genus g knot of order s in a lens space X whose Seifert surfaces have one boundary component. If s 4g 1 then, with respect to the Heegaard torus of X , K has bridge number at most 1. Theorem 1.1 may be curiously rephrased as saying small genus knots in lens spaces have small bridge number.In [1] Berge shows that double-primitive knots (ie simple closed curves that lie on a genus 2 Heegaard surface in S 3 and represent a generator of 1 for each handlebody)

Once-punctured tori and knots in lens spaces

“…Assume s ≥ 3. Theorem 1.1 of [Bak06] states that a knot in a lens space with an essential once-punctured genus g surface properly embedded in its exterior such that the boundary of the surface is distance at least 4g − 1 from the meridian (i.e. 4g − 1 ≤ s) then the knot is 1-bridge with respect to the genus one Heegaard splitting of the lens space.…”

“…Since K t,1 = H t,1 ∩ K is the only arc of K −T not contained in A, we may use A to thin K unless t = 2. See §4 of [Bak06].…”

Characterizing slopes for torus knots

“…In one case, the knots he constructed are all genus 2 fibred knots, one of which is the connected sum of two copies of the figure-8 knot. Teragaito [38] constructed infinitely many knots on which the 4-surgery yields the same Seifert fibred space over the base orbifold S 2 (2,6,7). One of these knots is 9 42 , again a genus 2 fibred knot.…”

“…Our aim in this paper is to focus on torus knots, trying to find their sets of characterizing slopes. In this regard, Rasmussen [35] proved, using a result of Baker [2], that (4n + 3) is a characterizing slope for T 2n+1,2 . It was also known, as a consequence of a result of Greene [15], that rs is a characterizing slope for T r,s .…”

Bounds on alternating surgery slopes

“…It turns out that whenever this bound is realised the resulting alternating surgery yields a lens space. Hence, work of Baker shows that the T 2,n torus knots are the only knots achieving achieving equality in Theorem 1.1 [1,Theorem 1.2].…”