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April, 2021 Knotted surfaces as vanishing sets of polynomials
Benjamin BODE, Seiichi KAMADA
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J. Math. Soc. Japan Advance Publication 1-34 (April, 2021). DOI: 10.2969/jmsj/84618461

Abstract

We present an algorithm that takes as input any element $B$ of the loop braid group and constructs a polynomial $f:\mathbb{R}^5 \to \mathbb{R}^2$ such that the intersection of the vanishing set of $f$ and the unit 4-sphere contains the closure of $B$. The polynomials can be used to create real analytic time-dependent vector fields with zero divergence and closed flow lines that move as prescribed by $B$. We also show how a family of surface braids in $\mathbb{C} \times S^1 \times S^1$ without branch points can be constructed as the vanishing set of a holomorphic polynomial $f:\mathbb{C}^3 \to \mathbb{C}$ on $\mathbb{C} \times S^1 \times S^1 \subset \mathbb{C}^3$. Both constructions allow us to give upper bounds on the degree of the polynomials.

Funding Statement

This work is supported by JSPS KAKENHI Grant Numbers JP18F18751 and JP19H01788. The first author is supported as a JSPS International Research Fellow.

Citation

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Benjamin BODE. Seiichi KAMADA. "Knotted surfaces as vanishing sets of polynomials." J. Math. Soc. Japan Advance Publication 1 - 34, April, 2021. https://doi.org/10.2969/jmsj/84618461

Information

Received: 9 April 2020; Revised: 28 June 2020; Published: April, 2021
First available in Project Euclid: 4 May 2021

Digital Object Identifier: 10.2969/jmsj/84618461

Subjects:
Primary: 57K45
Secondary: 57K10

Rights: Copyright ©2021 Mathematical Society of Japan

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