A finite type invariant of order at most 4 for genus 2 handlebody-knots is a constant map

We show that a finite type invariant of order at most 4 for genus 2 handlebody-knots is a constant map. For this purpose, we give a concrete basis for the vector space of all finite type invariants of order at most 4 for spatial theta-curves, which makes a correction to Koikeʼs result. Each invariant of the basis is derived from the HOMFLYPT polynomial of the associated 3-component link of a spatial theta-curve.