Montesinos knots, Hopf plumbings and L-space surgeries

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. 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.