A Cable Knot and BPS-Series II

Abstract

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 ($\left|w\right|>3$) 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 a Dehn surgery on the cable knots via a certain q-series. We also analyze and conjecture formulas for $(3,3w+1)$-cabling ($\left|w\right|>3$).

KEYWORDS:

• cable knot
• knot complement
• quantum invariant
• q-series

Acknowledgments

I would like to thank the referee for helpful suggestions.

Notes

1 These other approaches only apply to particular classes of links.

2 A generalization to links is given in [17]

3 Implicitly, there is a choice of group; originally, the group used is $\text{SU}(2)$.

4 They can be polynomials for monic Alexander polynomial of K (See Section 3.2)

5 There is an overall factor of 2 difference due to a different convention of $f_m(q)$ in F_K

6 Surgery along a positive slope may not produce a power series; there has been some progress on positive surgery in [16].