# A regular Morse map $f:C_K\to S^1$

A circle-valued Morse map $$f:C_K \to S^1$$ on the complement of a knot $$K$$ is said to be regular if there is a $$C^\infty$$ trivialisation $$\Phi : N(K)\to K\times B^2(0,\epsilon)$$ of a tubular neighbourhood of $$K$$ such that the restriction of $$f$$ to $$N(K)\setminus K$$ satisfies $$f\circ\Phi^{-1}(x,z) = \frac{z}{|z|}$$.

Would anyone be able to give me a better intuitive way of understanding what exactly a regular Morse map is, why for example is it "regular"? What kind of obstructions could there be to stop a Morse function from being regular? Is it the knot itself which affects whether a Morse map is regular or not?

Thanks!

The condition of regularity implies that the image of a meridian through $$f$$ winds exactly once around $$S^1$$.
Every $$\mathbb{R}$$-valued Morse function can be turned into an $$S^1$$-valued Morse function by including $$\mathbb{R}$$ along some open arc of $$S^1$$. Such Morse functions are not regular.
Another thing you can do is post-compose an $$S^1$$-valued Morse function with the degree-$$n$$ map $$\theta\mapsto n\theta$$. This is still a Morse function (when $$n\neq 0$$), but it fails to be regular when $$n\neq\pm 1$$.
Regular Morse functions are ones that, when restricted to $$S^3-N(K)\to S^1$$, can then be extended to $$S^3\to D^2$$, with $$S^1$$ being the boundary of the disk $$D^2$$.
Recall that $$H^1(S^3-K;\mathbb{Z})$$ is equivalent to homotopy classes of maps $$S^3-K\to S^1$$. Regular Morse functions are generators for $$H^1$$. (So, given a curve $$\gamma:S^1\to S^3-K$$, the image $$[\gamma]\in H_1(S^3-K;\mathbb{Z})\cong\mathbb{Z}$$ is given by the degree of the composition $$f\circ\gamma:S^1\to S^1$$.)
Every knot has a regular Morse function. This is essentially from using smooth approximation and transversality arguments, starting from a continuous map that represents a generator for $$H^1$$.