A funny isotopy on the 3-sphere

In the papers on the conormal torus (I, II), we showed that if two knots $K, K’ \subset \mathbb{R}^3$ have Legendrian isotopic conormal tori $T, T’$ then the knots themselves are isotopic.

Usually the difference between $\mathbb{R}^3$ and $S^3$ is not important in knot theory, but it’s not so clear if that’s the case here: the proofs used in an essential way that in $\mathbb{R}^3$, the projection to the base of an isotopy $\phi_t: T \to T’$ between two tori in $S^* \mathbb{R}^3$ is compact, thus misses a point.

In the sheaf version of the proof — which begins with the observation that, by the Guillermou-Kashiwara-Schapira theorem on sheaf quantization, an isotopy of the conormal tori induces an equivalence of categories

$$\Phi: shv_T(\mathbb{R}^3) \xrightarrow{\sim} shv_{T’}(\mathbb{R}^3)$$

between the categories of sheaves microsupported in these two tori — the role of this point was to rigidify these categories.  For instance, one can tell whether or not a sheaf has support in the complement of the knot by looking at the stalk at this point — one learns right away that $\Phi(\mathbb{Z}_K)$ is supported on $K’$.  (In fact, one can show it’s equal to $\mathbb{Z}_{K’}$).   Similarly, by studying the microlocalization of a nontrivial local system supported on the knot, one learns that the isotopy $\phi_t: T \to T’$ necessarily takes the meridian of $T$ to the meridian of $T’$ (though I cannot rule out the possibility that the orientation of the meridian is reversed).


By contrast, consider a round unknot $U$ (say a Hopf fibre) in $S^3$ carrying the round metric.   Applying the Reeb flow to the conormal torus will take the conormal torus to the conormal torus of the opposite Hopf fibre $U’$, which is after all isotopic to $U$.  This isotopy has the following features:

  1. The projection of the movie of the isotopy to the 3-sphere is surjective.
  2. It carries the longitude and meridian of $T_U$ to the longitude and meridian of $T_{U’}$
  3. The corresponding sheaf quantization carries $\mathbb{Z}_U \to \mathbb{Z}_{S^3 \setminus U’}[2]$.

Corollary: The above isotopy is not isotopic to any isotopy which does not project surjectively to the 3-sphere.  Note you can’t see this just for homological reasons, as the movie of the isotopy is nonorientable.

Remark. A simpler model for this behavior (from which you can recover the above by taking the Hopf fibration) is to take a point in $S^2$ rather than the unknot in $S^3$. Then you are taking the cocircle over the north pole to the cocircle over the south pole, and in the process the area “inside” the circle expands from a point to fill the whole sphere.

I sort-of expect that this can never happen except for the unknot, but note that it will be hard to see this cohomologically: for any knot $K$ one has

$$Hom(\mathbb{Z}_K, \mathbb{Z}_K) = Hom(\mathbb{Z}_{S^3 \setminus K}[2], \mathbb{Z}_{S^3 \setminus K}[2]) = H^*(S^3, \mathbb{Z}_K) = H^*(S^3, \mathbb{Z}_{S^3 \setminus K}[2]) = H^*(S^1, \mathbb{Z})$$

Note sheaf quantization of an isotopy between unions of conormals preserves sheaf cohomology, e.g. because both sides have Serre functors, which thus must be identified, hence the dualizing sheaf is preserved.

Question: find examples of Legendrian subvarieties $L$ such that $shv_L(X)$ has a Serre functor, but $L$ is not isotopic to a union of conormals.


Taking a step back, the following questions are the natural ones in this context, an answer to which will help to prove the result for $S^3$ on the sheaf side.  They’re also kind of interesting in their own right…

  1. Classify semi-orthogonal decompositions of the category $shv_T(S^3)$ for $T$ the conormal torus of a knot.
  2. Classify t-structures on $shv_T(S^3)$.  Maybe they’re all given by the perverse t-structures with respect to the stratification (and maybe this is already known in the literature).
  3. Warmup: do (1) or (2) for a Riemann surface with some marked points (surely this is in the literature — or at least can be extracted from the quiver descriptions).  Possibly, the methods of these guys can upgrade these to an answer to the original questions, i.e., answers to (1) or (2) should be a braid-group-invariant answer to (3).
  4. Classify elements $F \in shv_T(S^3)$ which admit a map from $\mathbb{Z}_{S^3}$ inducing $$H^*(F) = Hom(\mathbb{Z}_{S^3}, F) \xrightarrow{\sim} Hom(F, F) = H^*(S^1, \mathbb{Z})$$

A final philosophical note.  The question of doing this for $S^3$ means working without using the crutch of a base point.  So, it should translate to being able to prove that the original knot contact homology is a complete invariant.

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