Constructing a polynomial whose nodal set is any prescribed knot or link

Abstract

We describe an algorithm that for every given braid $B$ explicitly constructs a function $f:{\mathbb{C}}^2\to \mathbb{C}$ such that $f$ is a polynomial in $u$, $v$ and $\overline{v}$ and the zero level set of $f$ on the unit three-sphere is the closure of $B$. The nature of this construction allows us to prove certain properties of the constructed polynomials. In particular, we provide bounds on the degree of $f$ in terms of braid data.

20 pages, 1 figure; updated references, corrected typos, corrected Eq. (32)

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