Consider $\mathbb{R}^3$ with the standard contact structure $\ker(dz-y\,dx)$. According to the contact version of Weinstein's theorem, any Legendrian knot $L\subset \mathbb{R}^3$ has a tubular neighborhood $U\supset L$ that is contactomorphic to a neighborhood of the zero section $\{0\}\times 0_{T^*L}$ in the 1-jet bundle $J^1(L,\mathbb{R})\approx \mathbb{R}\times T^*L$ with contact structure $\ker (ds - p\,dq)$.

Is anything known regarding how big these tubular neighborhoods can be?

In particular, does there exist a Legendrian knot $L\subset \mathbb{R}^3$ and a Legendrian tubular neighborhood $U\supset L$ such that $\mathbb{R}^3\setminus U$ is "small" (e.g., a stratified subset of positive codimension, or a more general measure zero set)?

Consider $\mathbb{R}^3$ with the standard contact structure $\ker(dz-y\,dx)$. According to the contact version of Weinstein's theorem, any Legendrian knot $L\subset \mathbb{R}^3$ has a tubular neighborhood $U\supset L$ that is contactomorphic to a neighborhood of the zero section $\{0\}\times 0_{T^*L}$ in the 1-jet bundle $J^1(L,\mathbb{R})\approx \mathbb{R}\times T^*L$ with contact structure $\ker (ds - p\,dq)$.

I am wondering what is known regarding how big these tubular neighborhoods can be. Specifically, I have the following

Question. Does there exist a Legendrian knot $L\subset \mathbb{R}^3$ and a Legendrian tubular neighborhood $U\supset L$ such that $\mathbb{R}^3\setminus U$ is a stratified subset of positive codimension, or more generally, a measure zero set?

Consider $\mathbb{R}^3$ with the standard contact structure $\ker(dz-ydx)$. According to the contact version of Weinstein's theorem, any Legendrian knot $L\subset \mathbb{R}^3$ has a tubular neighborhood $U\supset L$ that is contactomorphic to a neighborhood of the zero section $\{0\}\times 0_{T^*L}$ in the 1-jet bundle $J^1(L,\mathbb{R})\approx \mathbb{R}\times T^*L$ with contact structure $\ker (ds - pdq)$.

Is anything known regarding how big these tubular neighborhoods can be?

In particular, does there exist a Legendrian knot $L\subset \mathbb{R}^3$ and a Legendrian tubular neighborhood $U\supset L$ such that $\mathbb{R}^3\setminus U$ is "small" (e.g., a stratified subset of positive codimension, or a more general measure zero set)?

Consider $\mathbb{R}^3$ with the standard contact structure $\ker(dz-y\,dx)$. According to the contact version of Weinstein's theorem, any Legendrian knot $L\subset \mathbb{R}^3$ has a tubular neighborhood $U\supset L$ that is contactomorphic to a neighborhood of the zero section $\{0\}\times 0_{T^*L}$ in the 1-jet bundle $J^1(L,\mathbb{R})\approx \mathbb{R}\times T^*L$ with contact structure $\ker (ds - p\,dq)$.

Is anything known regarding how big these tubular neighborhoods can be?

In particular, does there exist a Legendrian knot $L\subset \mathbb{R}^3$ and a Legendrian tubular neighborhood $U\supset L$ such that $\mathbb{R}^3\setminus U$ is "small" (e.g., a stratified subset of positive codimension, or a more general measure zero set)?