BPS spectra of complex knots

Marino's Conjecture remains underexplored within the framework of SO(N) string dualities. In this article, we investigated the reformulated invariants of one-parameter families of knots [K]p derived from tangle surgery on Manolescu's quasialternating knot diagrams [C. Manolescu, Math. Res. Lett. 14, 839 (2007)1073-278010.4310/MRL.2007.v14.n5.a11]. Within topological string dualities, we have verified Marino's integrality conjecture for these families of knots up to the Young diagram representation R, with |R|≤2. Furthermore, through our analysis, we have conjectured a closed structure for the extremal refined Bogomol'nyi-Prasad-Sommerfeld (BPS) integers for the torus knots [31]2p+1 and [820]2p+1, p Z≥0. As the parameter p of the knot diagram increases, the total crossing number of a knot exceeds 16, which we describe as a complex knot. Interestingly, we discovered the maximum number of gaps in the BPS spectra associated with complex knot families. Moreover, our observations indicated that as p increases, the size of these gaps also expands.