Consider the space $K$ of all immersions of $S^1$ into $\mathbb R^3$. The set of knots with self-intersection is a discriminant in $K$ and divide it into "chambers".

Let $f$ be a knot with $n$ double points. Everybody know that neighbourhood of $f$ in $K$ looks like $\mathbb R^n$, the origin be f, and hyperplanes $x_i=0$ be a knots with self-intersection. (In other words $f$ can be considered as intersection of n planes modelling dicriminant, there are $2^n$ chambers adjacent to $f$, looks like octants in $\mathbb R^n$).

My question: what is it mean? Does there exist a continuous map of neighboorhod $f\in U$ into $\mathbb R^n$ with above prescribed properties?

We can define "tangent" vector for $f$ as piecewise-smooth vector field along $f$. So, does there exist a map from "tangent" vector space of $f$ to $\mathbb R^n$ with above properties? Can we define subspace which is "tangent" to discriminant?