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 a one-parameter family of knots ${\left[K\right]}_p$ derived from tangle surgery on Manolescu’s quasi-alternating knot diagrams. Within topological string dualities, we have verified Marino’s integrality conjecture for these families of knots up to the Young diagram representation $\mathbf{R}$, with $\mathrm{\mid }\mathbf{R}\mathbf{\mid }\le 2$. Furthermore, through our analysis, we have conjectured the closed structure of extremal refined BPS integers for the torus knots ${\left[{\mathbf{3}}_{\mathbf{1}}\right]}_{2p+1}$ and ${\left[{\mathbf{8}}_{\mathbf{20}}\right]}_{2p+1}$, $p\in {\mathbb{Z}}_{\ge 0}$. 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 a 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.