This is a follow-up post regarding discussions I had with Harold, David, and Steven at the SQuaRE. (Edit: some of this is now obsolete—see new material below.)

Let’s try to count exact Lagrangian fillings of the standard Legendrian $(2,n)$ torus knot for $n$ odd, up to Hamiltonian (Lagrangian?) isotopy. From the cluster picture, there are (at least) $C_n$ distinct fillings, where $C_n = \frac{1}{n+1} {2n\choose n}$ is the usual Catalan number.

The picture of Ekholm-Honda-Kalman (arXiv:1212.1519) also gives $C_n$ fillings (not necessarily distinct). These are determined by permutations of $\{1,2,\ldots,n\}$—the standard $xy$ projection of the Legendrian $(2,n)$ torus knot has $n$ even crossings (and $2$ odd ones), and a permutation specifies an order in which we resolve these $n$ crossings. Two permutations of the form $(…,i,j,…,k,…)$ and $(…,j,i,…,k,…)$ give the same filling if $i<k<j$. When we mod out by this, we get exactly $C_n$ equivalence classes of permutations.

Can we distinguish the resulting $C_n$ fillings using Legendrian contact homology and the technology of EHK? Each filling induces an augmentation of the Chekanov-Eliashberg DGA, and isotopic fillings give equivalent (in this case, identical) augmentations. Mod $2$, though, there are generally fewer than $C_n$ augmentations. By EHK, the collection of $C_n$ fillings does achieve all of the mod $2$ augmentations.

If we want to do better, we can look at the augmentations to $\mathbb{Z}$ rather than $\mathbb{Z}/2$ induced by the fillings. EHK write down a combinatorial description of the augmentation mod $2$ corresponding to a filling, but not the augmentation with signs. However, one can lift their description to $\mathbb{Z}$ and thus produce a $\mathbb{Z}$ augmentation for each filling. I’ve written a computer program that will do this for the fillings of the $(2,n)$ torus knot.

Here are some results. In each line, I’ve given in order: the number of augmentations over $\mathbb{Z}/2$ (i.e., the EHK lower bound on the number of fillings); the number of augmentations over $\mathbb{Z}$ given by the $C_n$ fillings (an improved lower bound using signs); and $C_n$, the number of fillings that we’d hope to distinguish.

• $n=3$: 5, 5, 5
• $n=5$: 21, 35, 42
• $n=7$: 85, 284, 429
• $n=9$: 341, 2494, 4862.

So: augmentations over $\mathbb{Z}$ can distinguish more of the fillings than augmentations over $\mathbb{Z}/2$, but not all of them.

Maybe it’s instructive to see which pairs of fillings give the same augmentations for the (2,5) torus knot: here there are 42 fillings producing 35 augmentations, and the 7 coincidences are:

• 12453, 43251 $\rightarrow (1,0,0,1,0)$; 21543, 23415 $\rightarrow (0,1,0,0,1)$
• 13245, 14532 $\rightarrow (1,-1,1,1,0)$; 21354, 35421 $\rightarrow (0,1,1,-1,1)$
• 21345, 34215 $\rightarrow (0,1,1,0,0)$; 32451, 45321 $\rightarrow (0,0,1,1,0)$
• 32145, 34521 $\rightarrow (0,0,1,0,0)$.

Here I’ve grouped these further by a $\mathbb{Z}/2$ symmetry that reflects the diagram and interchanges crossings 1 and 5 and crossings 2 and 4.

I currently don’t see any good reason why these particular fillings should give the same augmentations over $\mathbb{Z}$, even though they’re distinct fillings. It’s an interesting question though.

Update (July 6):

Following suggestions by David and Harold, I think I understand how to resolve these issues.

The key point is that an exact orientable Lagrangian filling $L$ of $\Lambda$ naturally gives an augmentation from the DGA of $\Lambda$ not just to $\mathbb{Z}$, but to the group ring $\mathbb{Z}[H_1(L)]$. (In fact $H_1(L)$ should be $H_2(W,L)$ where $W$ is the symplectic filling, but in this case $W$ is contractible.) This is because we can do better than just count rigid holomorphic disks with boundary on $L$ and positive puncture at a Reeb chord of $\Lambda$; we can keep track of the homology class of their boundaries as well.

For the $(2,n)$ torus knot, the augmentation maps to

$$\mathbb{Z}[H_1(L)] \cong \mathbb{Z}[s_1^{\pm 1},\ldots,s_{n-1}^{\pm 1}].$$

(Note that this isomorphism depends on an identification of $H_1(L)$ with $\mathbb{Z}^{n-1}$; a different identification, i.e., an element of $GL(n-1,\mathbb{Z})$, acts on the right hand side in the natural way.)

It’s slightly complicated to describe how to lift the combinatorial EHK map to this new ring, so I’ll omit details here, but one can readily add these new coefficients to my computer program. The results are pretty promising. In case someone wants to do further computations, for the $(2,3)$ torus knot, the five fillings give rise to the following augmentations to $\mathbb{Z}[s_1^{\pm 1},s_2^{\pm 1}]$, where an augmentation $\epsilon$ is labeled by where it sends the three degree-0 Reeb chords $a_1,a_2,a_3$, and $1 = \epsilon(a_1) + \epsilon(a_3) + \epsilon(a_1) \epsilon(a_2) \epsilon(a_3)$:

• 123: $\left( s_1,\frac{s_1s_2-1}{s_1},-\frac{s_1-1}{s_1s_2} \right)$
• 132=312: $\left( s_1,-\frac{s_1^2s_2-s_1s_2+1}{s_1},\frac{1}{s_1s_2} \right)$
• 213: $\left( \frac{s_1 s_2-1}{s_2}, s_2, -\frac{s_1s_2-s_2-1}{s_1s_2^2} \right)$
• 231: $\left(\frac{s_1s_2+s_1-1}{s_2}, s_2, -\frac{s_1-1}{s_1s_2} \right)$
• 321: $\left(\frac{s_1s_2-1}{s_2}, -s_2(s_1-1), \frac{1}{s_1s_2} \right)$.

For the $(2,n)$ torus knots with $n=3,5,7$, one can now calculate that the $C_n$ fillings lead to all distinct augmentations to $\mathbb{Z}[s_1^{\pm 1},\ldots,s_{n-1}^{\pm 1}]$. (We have to mod out by the action of $GL(n-1,\mathbb{Z})$; one way to do this is to record all augmentations to $\mathbb{Z}$ obtained by specializing each $s_i$ to $\pm 1$, and using the resulting collection of $2^{n-1}$ augmentations to compare fillings.) So in these cases at least, augmentations distinguish all $C_n$ fillings from each other. I suspect that this may be true for general $n$.

Update (November 15): Yu Pan (my student) can prove that for any $n$, all $C_n$ fillings lead to distinct augmentations over $\mathbb{Z}[s_1^{\pm 1},\ldots,s_{n-1}^{\pm 1}]$ (actually one only needs to work mod $2$). So this gives a new proof that the $C_n$ fillings are all different.