Legendrian isotopy

Question

I think the definition of Legendrian isotopy is that you can find an isotopy of the ambient manifold such that it takes a Legendrian knot to another through Legendrian knots. What I cannot figure out is that why it is important or relevant to think about such isotopies.

Answer 1

Here's a definition of Legendrian isotopy that better generalizes isotopy of knots in other categories: A contact isotopy of $(M,\xi)$ is a family of diffeomorphisms $\Phi_t: M \to M \,$ such that $(\Phi_t)_*(\xi)=\xi$ for all $t\in [0,1]$. Two Legendrian knots are Legendrian isotopic if there is a contact isotopy of the ambient manifold taking one to the other. (Note that the image of the knot remains Legendrian throughout the contact isotopy.)

So, why think about the isotopies you described (i.e. a path of Legendrian knots)? You can use Gray's theorem to prove that a map $\phi: S^1 \times [0,1] \to M$ where each $\phi_t(S^1)$ is Legendrian extends to a contact isotopy $\Phi: M \times [0,1] \to M$. (This is a good exercise.) So your definition is equivalent to the one above. Note that transverse knots in contact manifolds have an analogously defined notion of knot equivalence.

There are certainly cases where we choose to think about the Legendrian representatives without concern for their Legendrian isotopy type. This frequently happens for invariants of smooth knot types which are defined using Legendrian representatives. For example, the Thurston-Bennequin number of a Legendrian knot $L$ in $(\mathbb{R}^3,\xi_{\operatorname{std}})$, denoted $tb(L)$, is a Legendrian isotopy invariant, but we derive an invariant for smooth knots in $\mathbb{R}^3$ by defining $$\overline{tb}(K)=\max\{tb(L): L \text{ is a Legendrian representative of } K\}.$$ The invariant $\overline{tb}$ would be unaffected even if $tb$ was not a Legendrian isotopy invariant.

More generally, Legendrian knots (and their higher-dimensional analogues) are central to contact geometry as a whole, serving as tools for detecting and manipulating properties of contact manifolds. So anytime you ask questions "up to contact isotopy", Legendrian knots become identified with their Legendrian isotopy classes. But if you're not particularly interested in contact geometry (or closely-related symplectic geometry) for its own sake, you could look at some of the applications of contact geometry and Legendrian knot theory to the wider world of knot theory and low-dimensional topology. Here are a few:

• Kronheimer and Mrowka's proof of Property P for nontrivial knots
• Chekanov and Pushkar's proof of Arnold’s Four Cusp Conjecture
• relationships with smooth knot invariants, e.g. bounds on slice genus of a knot $K \hookrightarrow S^3 = \partial D^4$