Constructing a polynomial whose nodal set is any prescribed knot or link. (English) Zbl 1411.57010
Summary: We describe an algorithm that for every given braid \(B\) explicitly constructs a function \(f : \mathbb C^2 \rightarrow \mathbb C\) 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.
MSC:
| 57M25 | Knots and links in the \(3\)-sphere (MSC2010) |
Keywords:
applied topology; real algebraic knot theory; braids; Morse-Novikov number; knotted fields; constructive approach to knot theoryReferences:
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