BPS Spectra of complex knots

TL;DR

This paper explores the BPS spectra of complex knots derived from tangle surgery, verifying Marino's conjecture for certain families and analyzing the structure and gaps in BPS integers as knot complexity increases.

Contribution

It verifies Marino's integrality conjecture for specific knot families and conjectures the structure of extremal refined BPS integers for torus knots with increasing complexity.

Findings

Verified Marino's conjecture for knots with Young diagram size ≤ 2

Conjectured the structure of extremal refined BPS integers for certain torus knots

Discovered maximum gaps in BPS spectra grow with knot complexity

Abstract

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 ${\bf R}$, with ${|\bf R|}\leq 2$. Furthermore, through our analysis, we have conjectured the closed structure of extremal refined BPS integers for the torus knots $ \left[{\bf 3_1}\right]_{2p+1}$ and $ \left[{\bf 8_{20}}\right]_{2p+1}$, $p \in \mathbb{Z}_{\geq 0}$. 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.