Minimal diagram connectivity for handlebody-knots with 2-sum spines

Ascertain whether every handlebody-knot that admits a spine obtained as a 2-sum of spatial graphs possesses a minimal diagram whose underlying plane graph has edge-connectivity 2.

Background

The paper also considers handlebody-knots whose spines are constructed as 2-sums of spatial graphs. A 2-sum is a vertex-connected sum along edges of spatial graphs, and such spines naturally suggest diagrams with connectivity 2.

The enumeration strategy partitions minimal diagrams by connectivity, independent of whether a handlebody-knot’s spine arises via a 2-sum. It remains unresolved whether the existence of a 2-sum spine ensures the existence of a minimal diagram with connectivity 2, making the link between spine decomposition and minimal-diagram connectivity an open problem.