BPS invariants from framed links

TL;DR

This paper explores the relationship between framed links and BPS invariants, providing explicit formulas and numerical calculations for various framed links, extending previous work on unframed knots.

Contribution

It extends the dual A-polynomial and BPS invariant relationship to framed knots and links, offering explicit formulas and numerical verifications.

Findings

Derived explicit formulas for extremal A-polynomials and BPS invariants of framed knots.

Performed numerical calculations for Whitehead links and Borromean rings.

Verified the integrality property of BPS invariants for these links.

Abstract

In this article, we investigate the BPS invariants associated with framed links. We extend the relationship between the algebraic curve (i.e. dual $A$-polynomial) and the BPS invariants of a knot investigated in \cite{GKS} to the case of a framed knot. With the help of the framing change formula for the dual $A$-polynomial of a framed knot, we give several explicit formulas for the extremal $A$-polynomials and the BPS invariants of framed knots. As to the framed links, we present several numerical calculations for the Ooguri-Vafa invariants and BPS invariants for framed Whitehead links and Borromean rings and verify the integrality property for them.