Waist and trunk of knots

Abstract

Let K be a knot in the 3-sphere S3. We define the waist of K as $$waist (K) = \mathop{\rm max}\limits_{F\in\mathcal{F}} \mathop{\rm min}\limits_{D\in\mathcal{D}_{F}} |D \cap K|,$$ where \({\mathcal{F}}\) is the set of all closed incompressible surfaces in S3−K and \({\mathcal{D}_F}\) is the set of all compressing disks for F in S3. We define the trunk of K as $$trunk(K) = \mathop{\rm min}\limits_{h\in\mathcal{H}} \mathop{\rm max}\limits_{t\in\mathbb{R}} |h^{-1}(t) \cap K|,$$ where \({\mathcal{H}}\) is the set of all Morse function \({h : S^3 \to \mathbb{R}}\) with two critical points. We show that $$waist (K) \le \frac{trunk(K)}{3}$$ .