A Cable Knot and BPS-Series II

This is a companion paper to earlier work of the author, which generalizes toan infinite family of $(2,2w+1)$-cabling of the figure eight knot ($|w|>3$) andproposes general formulas for the two-variable series invariant of the familyof the cable knots. The formulas provide an insight into the cabling operation.We verify the conjecture through explicit examples using the recursion method,which also provide a strong evidence for the $q$-holonomic property of theseries invariant. This result paves a road for computation of the WRT invariantof a 3-manifold obtained from Dehn surgery on the cable knots via a certain$q$-series. We also analyze and conjecture formulas for $(3,3w+1)$-cabling($|w|>3$).