Waist and trunk of knots

TL;DR

This paper introduces the concepts of waist, trunk, and supertrunk of knots, establishing inequalities among them and relating these invariants to other knot properties like hull number and 3-width.

Contribution

It defines new numerical invariants for knots, proves an optimal inequality between waist and trunk, and relates supertrunk to existing knot invariants.

Findings

Established an inequality between waist and trunk of knots.

Proved the inequality is optimal.

Connected supertrunk to hull number and 3-width.

Abstract

We introduce two numerical invariants, the waist and the trunk of knots. The waist of a closed incompressible surface in the complement of a knot is defined as the minimal intersection number of all compressing disks for the surface in the 3-sphere and the knot. Then the waist of a knot is defined as the maximal waist of all closed incompressible surfaces in the complement of the knot. On the other hand, the trunk of a knot is defined as the minimal number of the intersection of the most thick level 2-sphere and the knot over all Morse positions of the knot. In this paper, we obtain an inequality between the waist and the trunk of knots and show that the inequality is best possible. We also define the supertrunk of a knot and relate it to the hull number and the 3-width.