# Checks of integrality properties in topological strings

## TL;DR

This paper investigates the integrality properties of scalar operators in topological strings on conifold backgrounds, verifying conjectures for various knots and providing extensive data and a new statistical observation of LMOV invariants.

## Contribution

It reviews integrality properties in topological strings, verifies Marino's conjecture for multiple knots, and provides a comprehensive tabulation and analysis of LMOV invariants.

## Key findings

- Verification of Marino's integrality conjecture up to two boxes in Young diagrams.
- Extensive tabulation of integrality properties for over 100 prime knots.
- Observation of Gaussian distribution in LMOV invariants.

## Abstract

Tests of the integrality properties of a scalar operator in topological strings on a resolved conifold background or orientifold of conifold backgrounds have been performed for arborescent knots and some non-arborescent knots. The recent results on polynomials for those knots colored by SU(N) and SO(N) adjoint representations are useful to verify Marino's integrality conjecture up to two boxes in the Young diagram. In this paper, we review the salient aspects of the integrality properties and tabulate explicitly for an arborescent knot and a link. In our knotebook website, we have put these results for over 100 prime knots available in Rolfsen table and some links. The first application of the obtained results, an observation of the Gaussian distribution of the LMOV invariants is also reported.