Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

BPS invariants from framed links

Kai Wang, Shengmao Zhu

TL;DR

The paper extends the Garoufalidis-Kucharski-Sulkowski framework from unframed knots to framed knots and links by introducing framing-aware framed colored HOMFLYPT invariants and a framing-change formula for the dual A-polynomial. It shows how to extract extremal and general BPS invariants from the associated algebraic varieties, and provides explicit computations for framed unknot, framed twist knots, Whitehead links, and Borromean rings, with rigorous integrality checks. The work reinforces the large-N duality picture for framed objects, connects open string disk counts to BPS degeneracies, and generalizes the OV integrality to framed links, while outlining promising directions such as higher-genus recursions and a full proof of integrality. Overall, the paper broadens the bridge between knot invariants, algebraic curves, and string-theoretic BPS data in the framing context, offering concrete formulas and test cases for framed structures.

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.

BPS invariants from framed links

TL;DR

The paper extends the Garoufalidis-Kucharski-Sulkowski framework from unframed knots to framed knots and links by introducing framing-aware framed colored HOMFLYPT invariants and a framing-change formula for the dual A-polynomial. It shows how to extract extremal and general BPS invariants from the associated algebraic varieties, and provides explicit computations for framed unknot, framed twist knots, Whitehead links, and Borromean rings, with rigorous integrality checks. The work reinforces the large-N duality picture for framed objects, connects open string disk counts to BPS degeneracies, and generalizes the OV integrality to framed links, while outlining promising directions such as higher-genus recursions and a full proof of integrality. Overall, the paper broadens the bridge between knot invariants, algebraic curves, and string-theoretic BPS data in the framing context, offering concrete formulas and test cases for framed structures.

Abstract

In this article, we investigate the BPS invariants associated with framed links. We extend the relationship between the algebraic curve (i.e. dual -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 -polynomial of a framed knot, we give several explicit formulas for the extremal -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.

Paper Structure

This paper contains 17 sections, 6 theorems, 140 equations, 3 figures, 14 tables.

Table of Contents

  1. Introduction
  2. BPS invariants of framed knots
  3. Large $N$ duality for framed knots
  4. BPS invariants for framed knots
  5. String theory interpretation of the BPS invariants $b_{r,i}(\mathcal{K}^\tau)$
  6. Dual $A$-polynomial and extremal invariants for framed knots
  7. Computations of the dual $A$-polynomial $\mathcal{A}_{\mathcal{K}^\tau}(x,y,a)$
  8. Extremal invariants and extremal $A$-polynomial
  9. Examples
  10. Framed unknot
  11. Framed twist knots
  12. Integrality
  13. BPS invariants of framed links
  14. General formulas
  15. Framed Whitehead link $W^{(\tau_1,\tau_2)}$
  16. ...and 2 more sections

Key Result

Proposition 1.1

(1) The dual $A$-polynomial for framed unknot $U^\tau$ is given by (2) The BPS invariant for $U^\tau$ is given by where $\mu(d)$ is the Möbius function and

Figures (3)

  • Figure 4.1: Twist knot $\mathcal{K}_p$
  • Figure 5.1: Whitehead link $W$
  • Figure 5.2: Borromean ring $\mathbf{B}$

Theorems & Definitions (11)

  • Proposition 1.1
  • Proposition 1.2
  • Proposition 1.3
  • Theorem 1.4
  • Conjecture 1.5
  • Remark 4.1
  • Lemma 4.2
  • Remark 4.3
  • Theorem 4.4
  • Conjecture 5.1
  • ...and 1 more