On the support genus of Legendrian knots

Abstract

In this paper, we show that any topological knot or link in $$S^1 \times S^2$$ sits on a planar page of an open book decomposition whose monodromy is a product of positive Dehn twists. As a consequence, any knot or link type in $$S^1 \times S^2$$ has a Legendrian representative having support genus zero. We also show this holds for some knots and links in the lens spaces L(p, 1).