A Cable Knot and BPS-Series II

This is a companion paper to earlier work of the author, which generalizes to an infinite family of $(2,2w+1)$-cabling of the figure eight knot ($|w|>3$) and proposes general formulas for the two-variable series invariant of the family of 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 the series invariant. This result paves a road for computation of the WRT invariant of 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 ($|w|>3$).