Montesinos knots, Hopf plumbings, and L-space surgeries

Abstract

Using Hirasawa-Murasugi's classification of fibered Montesinos knots we classify the L-space Montesinos knots, providing further evidence towards a conjecture of Lidman-Moore that L-space knots have no essential Conway spheres. In the process, we classify the fibered Montesinos knots whose open books support the tight contact structure on S 3 . We also construct L-space knots with arbitrarily large tunnel number and discuss the question of whether L-space knots admit essential tangle decompositions in the context of satellite operations and tunnel number.

Satellite knots and L-space surgeries

“…Regarding necessary conditions for P (K) to be an L-space knot, the author, Lidman, and Vafaee made the following conjecture (cf. [1,Question 22]).…”

“…Since lens spaces are L-spaces, any knot admitting a positive lens space surgery is an L-space knot. However, there are many L-space knots that do not admit lens space surgeries, for example, [1,Proposition 23].…”

L-space surgery and twisting operation

“…Then following Ozsváth and Szabó [47] K n is an L-space knot if n ≥ −3 and thus the twist family {K n } contains infinity many L-space knots. Note that this family, together with a twist family {T 2n+1,2 }, comprise all Montesinos L-space knots; see [33] and [3]. In this example, it turns out that c becomes a Seifert fiber in the lens space K(19) (cf.…”

“…Hedden's cabling construction [25], together with [40], enables us to obtain an L-space knot with tunnel number greater than 1. Actually Baker and Moore [3] have shown that for any integer N , there is an L-space knot with tunnel number greater than N . However, L-space knots with tunnel number greater than one constructed above are all satellite (non-hyperbolic) knots and they ask:…”

Left-orderablity for surgeries on (−2,3,2s + 1)-pretzel knots

“…According to [1], the (−2, 3, 2s + 1)-pretzel knots for integers s ≥ 3 are the only hyperbolic knots with L-space surgeries up to mirroring among all Montesinos knots. It is no wonder these knots draw attention in the literature.…”