A Cable Knot and BPS-Series

Abstract

A series invariant of a complement of a knot was introduced recently. The invariant for several prime knots up to ten crossings have been explicitly computed. We present the first example of a satellite knot, namely, a cable of the figure eight knot, which has more than ten crossings. This cable knot result provides nontrivial evidence for the conjectures for the series invariant and demonstrates the robustness of integrality of the quantum invariant under the cabling operation. Furthermore, we find interesting effects of the cabling on the series invariant.

A Connect Sum Formula for the BPS Invariant of Knot Complements

“…This property was used to compute F K for the figure eight knot 4 1 in [11] and was verified for m(5 2 ) in [17]. Moreover, the same method was applied to find F K for a cabling of 4 1 [5].…”

“…Figure1: Minimum powers of q-terms that survived in(5) for the powers of x shown. The upper bound corresponds to the maximum value among the upper bounds in the summations in F T (2,3) # T (2,3) (xq j , q), j = 0, • • • , 3 (see Appendix A for the plots of other powers of x).…”