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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!

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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 $H^1$.

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