Minimal diagram connectivity for 1-sum handlebody-knots

Determine whether every handlebody-knot that is a 1-sum V1 -- V2 of two handlebody-knots admits a minimal diagram whose underlying plane graph has edge-connectivity 1.

Background

The paper enumerates genus two handlebody-knots with seven crossings by classifying minimal diagrams according to the edge-connectivity of their underlying plane graphs. A 1-sum of handlebody-knots is obtained by gluing along disks, producing a new handlebody-knot whose genus is the sum of the genera of the summands.

The authors’ approach relies on the connectivity of minimal diagrams rather than the decomposition type. However, it remains unresolved in general whether the operation of taking a 1-sum guarantees that some minimal diagram of the resulting handlebody-knot has connectivity 1. Resolving this would clarify the relationship between decomposition operations and minimal-diagram connectivity.