[Paper Review] The correspondence between augmentations and rulings for Legendrian knots

Lenhard Ng, Joshua M. Sabloff|arXiv (Cornell University)|Mar 8, 2005

Geometric and Algebraic Topology7 references4 citations

TL;DR

This paper establishes a precise correspondence between augmentations of the contact homology DGA and rulings of Legendrian knots in plat position, showing that the number of augmentations yielding a given ruling is determined by the ruling's combinatorics. The key result is a formula linking the total number of $ρ$-graded augmentations to the ruling invariant, enabling computation of augmentation counts from combinatorial data.

ABSTRACT

We strengthen the link between holomorphic and generating-function invariants of Legendrian knots by establishing a formula relating the number of augmentations of a knot's contact homology to the complete ruling invariant of Chekanov and Pushkar.

Motivation & Objective

• To strengthen the connection between holomorphic invariants (contact homology DGA) and combinatorial invariants (ruling invariants) of Legendrian knots.
• To resolve the long-standing conjecture that the information in the linearized contact homology DGA matches that in the ruling invariant.
• To provide a concrete, combinatorially computable formula for the number of $ρ$-graded augmentations of a Legendrian knot’s DGA.
• To show that the total number of augmentations can be derived from the complete ruling invariant, including degree distribution and interlacing data.
• To extend the correspondence beyond the existence of augmentations to a many-to-one mapping with explicit counting rules.

Proposed method

• Define a $ρ$-graded augmentation as an algebra map $ε: \mathcal{A} \to \mathbb{Z}/2$ with $\varepsilon \circ \partial = 0$, $\varepsilon(1) = 1$, and $\varepsilon(a) = 0$ if $\rho \nmid |a|$.
• Use the algorithm from [9] to construct a ruling from a given augmentation, ensuring the correspondence is well-defined for plat-position fronts.
• Introduce the interlacing number of a vertical line in a plat diagram, counting signed interlaced pairs of strands based on Maslov index differences.
• Prove that the interlacing number changes by $\pm 1$ at each non-switch crossing, with the sign determined by the crossing’s degree modulo $\rho$.
• Establish a topological invariant by tracking interlacing number changes from left to right, showing it starts and ends at zero.
• Derive the key identity $\chi^{*}_{\rho}(\mathcal{A}) = s + d + r + c(D)$ for $\rho = 1$, and generalize to $\rho = 0$ and odd $\rho \geq 3$ using signed interlacing contributions.

Experimental results

Research questions

• RQ1 Is there a precise combinatorial formula that relates the number of $\rho$-graded augmentations of a Legendrian knot’s DGA to its ruling invariant?
• RQ2 Can the total number of augmentations be computed solely from the complete ruling data, including switches, returns, and departures?
• RQ3 How does the Maslov index and degree of crossings affect the interlacing behavior of rulings under vertical sweep?
• RQ4 Does the correspondence between augmentations and rulings extend beyond existence to a many-to-one mapping with quantifiable multiplicity?
• RQ5 What is the role of the interlacing number in encoding the algebraic structure of the DGA’s augmentation count?

Key findings

• For any Legendrian knot in plat position, the number of $\rho$-graded augmentations is equal to $s + d + r + c(D)$, where $s$, $d$, and $r$ are the numbers of switches, departures, and returns in the ruling, and $c(D)$ is the number of right cusps.
• When $\rho = 1$, the interlacing number is the count of interlaced pairs, and the number of augmentations is $a_0 = s + d + r + c(D)$, with $d = r$.
• For $\rho = 0$, the interlacing number is signed based on Maslov index differences, and the identity $d - r + \sum_{k>0} (-1)^k \tilde{a}_k + \sum_{k<0} (-1)^{k+1} \tilde{a}_k = 0$ holds, proving the formula.
• The correspondence is many-to-one: multiple augmentations can map to the same ruling, with the number of such augmentations determined by the ruling’s combinatorics.
• The total number of $\rho$-graded augmentations is invariant under stable tame isomorphism and can be computed from the ruling invariant, making it a computable invariant.
• The proof generalizes to odd $\rho \geq 3$ by defining a signed interlacing number based on the value of $m(a_2) - m(b_2) \mod \rho$, preserving the identity.