Existence of transverse homotopy between knots in a 3-manifold I have a 3-manifold $\Sigma$ and two homotopic embedded knots $K_{0}(t): S^{1} \to \Sigma$ and $K_{1}(t): S^{1} \to \Sigma$. I wish to refine the homotopy between them to a "transverse homotopy" i.e, some $H(s,t)$ such that


*

*$s \in \{s_{0},\ldots,s_{n}\} \Rightarrow H(s,t)$ is a singular knot with one crossing i.e, there are only two points $t_{1},t_{2}\in[0,1]$ such that $H(s,t_{1})=H(s,t_{2})$.

*$s \in [0,1]\backslash \{s_{0},\ldots,s_{n}\} \Rightarrow H(s,t)$ is an embedded knot.
In definition 2.1 of Type 1 knot invariants in 3-manifolds (bottom of page 5) the authors state that a transverse homotopy always exists between any two homotopic knots.
I'd like to know how to prove this statement.
 A: Suppose $h : S^1 \times I \to \Sigma$ is a smooth homotopy between the knots $K_0 = h(-, 0)$ and $K_1 = h(-, 1)$. Consider the movie $f : S^1 \times I \to \Sigma \times I$ of the homotopy, defined by $f(x, t) = (h(x, t), t)$. 
I claim that $f$ is homotopic, relative to it's boundary, to a map $g$ which is an immersion with transverse double points, i.e., $g$ is an embedding away from finitely many points $x_1, y_1, \cdots, x_n, y_n \in S^1 \times I$ such that $g(x_k) = g(y_k) = p_k$ for $1 \leq k \leq n$ and $\text{im} \,dg_{x_k} \pitchfork \text{im} \, dg_{y_k}$. This follows from a self-transversality theorem that I have stated and given a sketch of proof below. 
As a matter of fact $g$ can be chosen to be $C^1$-close to $f$ as well. $f$ was completely horizontal to $\Sigma$, i.e., if we denote $f_t = f|S^1 \times \{t\}$ to be the restriction on the $t$-slice, $f_t' \subset (d\pi)^*T\Sigma$ where $\pi : \Sigma \times I \to \Sigma$ is the projection. Therefore $g_t (=g|S^1 \times \{t\})$ must be $\varepsilon$-horizontal to $\Sigma$, and in particular, $g_t' \notin \ker d\pi$. This would mean $\pi \circ g_t$ is an immersion for all $t \in I$.
Note that this is almost enough for what you want: Take $H = \pi_\Sigma \circ g$ to be your new homotopy. This is a homotopy of $K_0$ to $K_1$ through immersions, and the only difficulty is that $H_t$ might stay on a singular knot for an interval's worth of time $t \in J \subset I$, whereas you want it to stay there for only finitely many points in $I$. Locally near a crossing, $(H_t)_{t \in I}$ would look like a pair of undercrossing-overcrossing which gradually grows closer, stays in the position of "$\mathsf{X}$" for the interval $t \in J$ and then grows apart and becomes a pair of overcrossing-undercrossing. We modify $H$ near those points so that it only stays like an "$\mathsf{X}$" for a point $t = t_0 \in J$. Then further modify $H$ so that double points are introduced one at a time, i.e, $H_t$ has at most one double point.

Suppose $M$ is a compact $n$-manifold and $N$ is a compact $2n$-manifold. Let $C^\infty(M, N)$ be the space of smooth maps and $\text{Imm}(M, N)$ be the subspace of such maps which are immersions. 
Call a map $f \in \text{Imm}(M, N)$ to be an immersion with clean double points if whenever $x, y \in M$, and $f(x) = f(y) = p$, there exists charts $U, V$ around $x, y$ respectively such that $f|_U, f|_V$ are embeddings, $f(U)$ intersects $f(V)$ transversely, and $f(U) \cap f(V) = \{p\}$. Denote the subspace of such maps as $\text{Imm}_{\pitchfork}(M, N)$
Theorem (Self-transversality of maps): $\text{Imm}_\pitchfork(M^n, N^{2n})$ is dense in $C^\infty(M^n, N^{2n})$.
Let $J^1(M, N)$ be the space of $1$-jets of maps $M \to N$ and $\mathscr{S} \subset J^1(M, N)$ be the subset of $1$-jets of non-immersions. The space of $1$-jets is an affine bundle $M_{2n \times n}(\Bbb R) \to J^1(M, N) \to M \times N$ with fiber keeping track of the formal derivative component. Let $\Sigma_{< n} \subset M_{2n \times n}(\Bbb R)$ be the stratified subset of matrices of rank strictly less than $n$. As $\mathscr{S}$ fibers over $M \times N$ with fiber $\Sigma_{< n}$, it is also a stratified subset of $J^1(M, N)$.
Given any map $f \in C^\infty(M, N)$, consider it's 1-jet prolongation $j^1 f : M \to J^1(M, N)$. By Thom transversality theorem, we can homotope $f$ by a $C^1$-small homotopy to $g \in C^\infty(M, N)$ such that $j^1 g \pitchfork \mathscr{S}$. But observe that $\dim J^1(M, N) = n(2n+3)$, so $\text{codim}\, j^1g(M) = \dim J^1(M, N)$ $- \dim M$ $=$ $2n(n+1)$ whereas $\text{codim}\, \mathscr{S}$ $=$ $\text{codim}_{M_{2n \times n}(\Bbb R)} \Sigma_{< n}$ $=$ $n+1$, so $\text{codim}\, j^1 g(M)$ $+$ $\text{codim}\, \mathscr{S}$ $>$ $\dim J^1(M, N)$, forcing $j^1 g$ to be disjoint from $\mathscr{S}$. This implies $j^1 g$ has rank $n$ everywhere, i.e., $g$ is an immersion. 
This implies $\text{Imm}(M, N)$ is dense in $C^\infty(M, N)$. Now for any $f \in \text{Imm}(M, N)$ consider the map $F : M \times M \to N \times N$, $F(x, y) = (f(x), f(y))$. By a $C^1$-small homotopy we can modify $F$ to $G$ so that $G$ is transverse to the diagonal $\Delta_N \subset N \times N$. Let $G|\Delta_M$ be the restriction to the diagonal of $M \times M$. As $G$ is $C^1$-close to $F$, image of $G|\Delta_M$ fits inside an $\epsilon$-neighborhood of $\Delta_N \subset N \times N$, and by $\epsilon$-neighborhood theorem we have a normal projection map to $\Delta_N$, with which we compose to get a map $g : M \to N$. $g \times g$ is $C^1$-close to $G$ near $\Delta_M$, and since $G \pitchfork \Delta_N$, by stability of transverse maps, $(g \times g) \pitchfork \Delta_M$ should hold as well. Therefore $g \in \text{Imm}_\pitchfork(M, N)$.
This implies we have a tower of successively dense subspaces $$\text{Imm}_\pitchfork(M, N) \subset \text{Imm}(M, N) \subset C^\infty(M, N)$$ 
In particular, $\text{Imm}_\pitchfork(M, N)$ is dense in $C^\infty(M, N)$, as required. (Note that for the above we need a relative-to-boundary version of this, but once $f$ is already an embedding restricted to (a collar neighborhood of) the boundary, we can do all the required homotopies fixing the boundary.)
