[Paper Review] Integrality of the LMOV invariants for framed unknot

Wei Luo, Shengmao Zhu|arXiv (Cornell University)|Jun 5, 2021

Cuban History and Society4 citations

TL;DR

This paper establishes the integrality of LMOV invariants for the framed unknot across all genera using a unified algebraic method based on Möbius inversion and quantum integers. It proves that genus-zero invariants $ n_{m,l}( au) $ and two-part boundary invariants $ n_{(m_1,m_2)}( au) $ are integers, and extends this to higher genera via a generating function $ g_m(q,a) $, showing all coefficients lie in $ \mathbb{Z} $, thus confirming the integrality conjecture for this class of invariants.

ABSTRACT

The Labastida-Marinõ-Ooguri-Vafa (LMOV) invariants are the open string BPS invariants which are expected to be integers based on the string duality conjecture from M-theory. Several explicit formulae of LMOV invariants for framed unknot have been obtained in the literature. In this paper, we present a unified method to deal with the integrality of such explicit formulae. Furthermore, we also prove the integrality of certain LMOV invariants for framed unknot in higher genera.

Motivation & Objective

• To prove the integrality of explicit formulae for LMOV invariants of the framed unknot, which are conjectured to be integers based on M-theory duality.
• To unify and rigorously establish the integrality of genus-zero LMOV invariants $ n_{m,l}( au) $ via Möbius inversion on binomial and signed binomial expressions.
• To extend the integrality proof to higher-genus LMOV invariants by analyzing the generating function $ g_m(q,a) $.
• To confirm the integrality of special cases such as extremal BPS invariants of twist knots, previously conjectured but not rigorously proven.

Proposed method

• Uses Möbius inversion to express genus-zero LMOV invariants $ n_{m,l}( au) $ as a sum over divisors involving $ c_{m,l}( au) $, a signed binomial coefficient expression.
• Defines $ c_{m,l}( au) = -\frac{(-1)^{m\tau+m+l}}{m^2} \binom{m}{l} \binom{m\tau + l - 1}{m - 1} $, which captures the core combinatorial structure of the invariants.
• Applies the Möbius function $ \mu(d) $ to average over symmetric substructures, ensuring integrality through number-theoretic cancellation.
• Introduces the generating function $ g_m(q,a) = \sum_{g \geq 0} \sum_Q n_{m,g,Q}(\tau) z^{2g-2} a^Q $ with $ z = q^{1/2} - q^{-1/2} $, linking invariants to quantum group invariants.
• Uses the identity $ g_m(q,a) = \sum_{d|m} \mu(d) \mathcal{Z}_{m/d}(q^d, a^d) $, where $ \mathcal{Z}_m(q,a) $ is a rational function in quantum integers and $ a $-deformed $ q $-binomials.
• Proves that $ \{m\}\{m\tau\} \mathcal{Z}_m(q,a) \in \mathbb{Z}[q^{\pm 1/2}, a^{\pm 1/2}] $, ensuring the coefficients of $ g_m(q,a) $ are integral after scaling.

Experimental results

Research questions

• RQ1 Are the explicit formulae for genus-zero LMOV invariants of the framed unknot with boundary type $ (m) $ and parameter $ Q $ integers?
• RQ2 Does the integrality of the genus-zero invariants extend to higher genera for the framed unknot?
• RQ3 Can the extremal BPS invariants of twist knots, previously conjectured to be integers, be rigorously proven to be so?
• RQ4 Is the generating function $ g_m(q,a) $ for higher-genus LMOV invariants integral in the ring $ z^{-2} \mathbb{Z}[z^2, a^{\pm 1/2}] $?

Key findings

• The genus-zero LMOV invariants $ n_{m,l}( au) $ are integers for all $ m \geq 1 $, $ l \geq 0 $, and $ \tau \in \mathbb{Z} $, as proven via Möbius inversion on signed binomial coefficients.
• The two-part boundary invariants $ n_{(m_1,m_2)}(\tau) $ are integers for all $ m_1, m_2 \geq 1 $, $ \tau \in \mathbb{Z} $, and satisfy $ n_{(m_1,m_2)}(\tau) = n_{(m_2,m_1)}(\tau) $.
• The generating function $ g_m(q,a) $ lies in $ z^{-2} \mathbb{Z}[z^2, a^{\pm 1/2}] $, proving that all higher-genus LMOV invariants $ n_{m,g,Q}(\tau) $ are integers and vanish for large $ g $ or $ Q $.
• The formulae for extremal BPS invariants of twist knots in [8] are confirmed to yield integers, as they are special cases of the general formula for $ n_{m,l}(\tau) $.
• The expression $ \{m\}\{m\tau\} \mathcal{Z}_m(q,a) $ lies in $ \mathbb{Z}[q^{\pm 1/2}, a^{\pm 1/2}] $, which is key to proving integrality of the generating function.