The tunnel number and the cutting number with constituent handlebody-knots

TL;DR

This paper establishes lower bounds for the tunnel number and cutting number of knots and handlebody-knots, introduces conditions for constituent handlebody-knots using quandle colorings, and constructs examples with arbitrary numbers.

Contribution

It provides new lower bounds and necessary conditions for handlebody-knot invariants, and constructs handlebody-knots with specified tunnel and cutting numbers.

Findings

Lower bounds for tunnel number and cutting number established

Necessary conditions for constituent handlebody-knots derived

Construction of handlebody-knots with arbitrary tunnel and cutting numbers

Abstract

We give lower bounds for the tunnel number of knots and handlebody-knots. We also give a lower bound for the cutting number, which is a "dual" notion to the tunnel number in the handlebody-knot theory. We provide necessary conditions for constituent handlebody-knots by using $G$-family of quandles colorings. The above two evaluations are obtained as the corollaries. Furthermore, we construct handlebody-knots with arbitrary tunnel number and cutting number.