[Paper Review] Ruling polynomials and augmentations for Legendrian tangles

Tao Su|arXiv (Cornell University)|Jul 16, 2017

Geometric and Algebraic Topology16 references3 citations

TL;DR

This paper generalizes ruling polynomials and augmentation numbers from Legendrian links to Legendrian tangles, establishing that these invariants satisfy a composition axiom and remain Legendrian isotopy invariants. It introduces generalized normal rulings and shows that the weighted count of such rulings computes augmentation numbers, extending previous results to tangles and non-acyclic augmentations.

ABSTRACT

Associated to Legendrian links in the standard contact three-space, Ruling polynomials are Legendrian isotopy invariants, which also compute augmentation numbers, that is, the points-counting of augmentation varieties for Legendrian links (up to a normalized factor) \cite{HR15}. In this article, we generalize this picture to Legendrian tangles, which are morally the pieces obtained by cutting Legendrian link fronts along 2 vertical lines. Moreover, we show that the Ruling polynomials for Legendrian tangles satisfy the composition axiom. In the special case of Legendrian knots, our arguments provide new proofs to the main results in \cite{HR15}. In the end, we also introduce generalized Ruling polynomials for Legendrian tangles, to account for non-acyclic augmentations in the "Ruling polynomials compute augmentation numbers" picture.

Motivation & Objective

• To extend the theory of ruling polynomials and augmentation numbers from Legendrian links to Legendrian tangles.
• To establish that the generalized ruling polynomials satisfy a composition axiom under tangle concatenation.
• To define and analyze generalized normal rulings that account for non-acyclic augmentations.
• To prove that the weighted count of generalized normal rulings computes augmentation numbers for tangles.
• To provide new combinatorial proofs of key results from [HR15] in the context of Legendrian tangles.

Proposed method

• Generalizes the notion of normal rulings to Legendrian tangles by defining m-graded generalized normal rulings.
• Introduces a matrix-valued ruling polynomial $\tilde{R}_T^m(z)$ that encodes weighted counts of generalized normal rulings with fixed boundary isomorphism types.
• Uses the composition axiom to show that the inner product $<\rho_L|\tilde{R}_T^m(z)|\rho_R>$ is invariant under Legendrian isotopy.
• Applies the co-sheaf property of LCH DGAs for tangles to relate the algebraic structure to the combinatorial ruling invariants.
• Derives a formula linking the augmentation number $\mathrm{aug}_m(T, \rho_L, \rho_R; q)$ to the generalized ruling polynomial via $q^{-\frac{d+B}{2}} z^B <\rho_L|\tilde{R}_T^m(z)|\rho_R>$.
• Employs handleslide moves and boundary isomorphism types to analyze the behavior of rulings under Reidemeister moves, especially Type III.

Experimental results

Research questions

• RQ1 Can ruling polynomials for Legendrian links be extended to Legendrian tangles while preserving invariance and compositionality?
• RQ2 How do generalized normal rulings account for non-acyclic augmentations in the context of tangle invariants?
• RQ3 Does the composition axiom hold for generalized ruling polynomials under tangle concatenation?
• RQ4 Can the augmentation number for a tangle be computed as a weighted sum over generalized normal rulings?
• RQ5 What is the precise algebraic relationship between the generalized ruling polynomial and the augmentation variety for tangles?

Key findings

• The generalized ruling polynomial $\tilde{R}_T^m(z)$ satisfies the composition axiom: $<\rho_L|\tilde{R}_T^m(z)|\rho_R> = \sum_{\rho_I} <\rho_L|\tilde{R}_{T_1}^m(z)|\rho_I><\rho_I|\tilde{R}_{T_2}^m(z)|\rho_R>$ for tangle compositions.
• The augmentation number $\mathrm{aug}_m(T, \rho_L, \rho_R; q)$ is equal to $q^{-\frac{d+B}{2}} z^B <\rho_L|\tilde{R}_T^m(z)|\rho_R>$, proving that the ruling polynomial computes the augmentation number.
• For a Reidemeister Type III move, the invariance of the inner product $<\rho_L|\tilde{R}_T^m(z)|\rho_R>$ holds precisely when $z = q^{1/2} - q^{-1/2}$, confirming invariance in nontrivial cases.
• The structure of the augmentation variety $\mathrm{Aug}_m^\rho(T, \epsilon_L; k)$ is isomorphic to $(k^*)^{-\chi(\rho)-h(\rho)+B} \times k^{r(\rho)+h(\rho)}$, providing a geometric realization of the combinatorial data.
• The Legendrian isotopy invariance of $\mathrm{aug}_m(T, \rho_L, \rho_R; q)$ is established independently of Corollary 4.18, strengthening the foundational argument.
• The theory provides new combinatorial proofs of the main results from [HR15] in the case of Legendrian knots, using tangle-based methods.