Embedding a pair of manifolds, torus knots

Question

Let $M^{m}$ be a smooth closed $m$-dimensional manifold ($m\geq1$). Then it is well known that we can embed $M$ into $\mathbb{R}^{2m}$ (this is the strong form of Whitney's embedding theorem).

Suppose now that $M^{m}\subseteq N^{n}$ is a closed submanifold of a larger $n$-dimensional manifold $N$. Define $d(M,N)$ to be the minimal integer $k\in\mathbb{N}$ such that we can embed $N$ into $\mathbb{R}^{k}$ in such a way that there exists an affine subspace $W\cong\mathbb{R}^{l}$ (for some $l\leq k$) such that $N\cap W=M$. What is known about the integer $d(M,N)$? In particular, is it always finite?

As an example, take $N$ to be the 2-torus and $M$ to be an embedded circle in $N$. Then $d(M,N)$ certainly depends on the choice of $M$. For instance, for certain nullhomologous choices of $M$ we have $d(M,N)=3$. Recall now that the torus knot $T_{p,q}$ winds $p$ times around a circle inside the torus, which goes all the way around the torus, and $q$ times around a line through the hole in the torus, which passes once through the hole, (usually drawn as an axis of symmetry). If $M$ is the torus knot $T_{1,0}$ then $d(M,N)=4$. If $M$ is the torus knot $T_{2,3}$ however then $d(M,N)=5$. In fact, $d(T_{p,q},N)\leq5$ for any choice of $p,q$.

Answer 1

Note that if you require some kind of transversality for the intersection of $N$ and $W$ then the bound on $d(M,N)$ is still finite (I think $2n+m+1$ always works) but it can be strictly bigger than $2n+1$. Specifically, if you want to have $T_pN\cap W=T_pM$ for any $p\in M$ then you have that $\nu_M\mathbb R^k$ is the Whitney sum of $\nu_MW$ and a trivial bundle of rank=dim $W^\perp=l-k$ which contains $\nu_MN$ as a subbundle. This of course forces a restriction on $l-k$ depending on the structure on $\nu_MN$. The most extreme case is probably $M=\mathbb{RP}^m\subset N=\mathbb{RP}^{m+1}$. In this case the Whitney embedding bound given by Ryan without transversality assumptions is $2(m+1)+1=2m+3$ for an embedding $\mathbb {RP}^m\subset \mathbb {RP}^{m+1}\subset \mathbb R^{2(m+1)}$. However, taking into account that we must have $\dim W\ge 2m$ and $\dim W^\perp\ge m$ (this is forced by the Stiefel-Whitney class of $\nu_{\mathbb {RP}^{m}}\mathbb {RP}^{m+1}$) we have that $n$ must be at least $3m$.

Answer 2

As stated, yes $d(M,N)$ is always finite -- it's bounded above by $2n+1$ by the strong Whitney embedding theorem. The idea is to use the smooth Urysohn Lemma to write $M$ as $M=f^{-1}(0)$ for some smooth function $f : N \to [0,\infty)$. Embed $N$ in $\mathbb R^{2n}$ then take the graph of $f$ on $N$, this is a subset of $\mathbb R^{2n+1}$ and $M = graph(f) \cap (\mathbb R^{2n} \times \{0\})$.

Is there a particular reason why you're interested in $d(M,N)$ ?