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

2016/12/19 by Benjamin Bode, Bode, Benjamin, Mark R. Dennis +1 · 2 citations
Engineering · Mathematics · Computer Science · #Advanced Numerical Analysis Techniques #Geometric and Algebraic Topology #Computational Geometry and Mesh Generation

paper · doi:10.48550/arxiv.1612.06328

Abstract

We describe an algorithm that for every given braid B explicitly constructs a function f:ℂ²→ℂ such that f is a polynomial in u, v and 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.

Cited by

• Panhandle polynomials of torus links and geometric applications
• Constructing links of isolated singularities of polynomials\n \ℝ⁴\→\ℝ²

Related

• Knotted surfaces as vanishing sets of polynomials
• Polynomial knots
• Passing from plat closure to standard closure of braids in ℝ³, in handlebodies and in thickened surfaces
• Constructing links of isolated singularities of polynomials\n \ℝ⁴\→\ℝ²
• Multivariable knot polynomials from braided Hopf algebras with automorphisms