Constructing links of isolated singularities of polynomials R4→R2

Benjamin Bode

doi:10.1142/S0218216519500093

Constructing links of isolated singularities of polynomials R4→R2

Benjamin Bode

Research output: Contribution to journal › Article (Academic Journal) › peer-review

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Abstract

We show that if a braid B can be parametrized in a certain way, then the previous work (B. Bode and M. R. Dennis, Constructing a polynomial whose nodal set is any prescribed knot or link, arXiv:1612.06328) can be extended to a construction of a polynomial f: R4 → R2 with the closure of B as the link of an isolated singularity of f, showing that the closure of B is real algebraic. In particular, we prove that closures of squares of strictly homogeneous braids and certain lemniscate links are real algebraic. We also show that the constructed polynomials satisfy the strong Milnor condition, providing an explicit fibration of the complement of the closure of B over S1.

Original language	English
Article number	1950009
Number of pages	21
Journal	Journal of Knot Theory and its Ramifications
Volume	28
Issue number	1
DOIs	https://doi.org/10.1142/S0218216519500093
Publication status	Published - 21 Jan 2019

Structured keywords

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Keywords

homogeneous braids
Isolated singularities
Milnor fibration
real algebraic links

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10.1142/S0218216519500093

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N2 - We show that if a braid B can be parametrized in a certain way, then the previous work (B. Bode and M. R. Dennis, Constructing a polynomial whose nodal set is any prescribed knot or link, arXiv:1612.06328) can be extended to a construction of a polynomial f: R4 → R2 with the closure of B as the link of an isolated singularity of f, showing that the closure of B is real algebraic. In particular, we prove that closures of squares of strictly homogeneous braids and certain lemniscate links are real algebraic. We also show that the constructed polynomials satisfy the strong Milnor condition, providing an explicit fibration of the complement of the closure of B over S1.

AB - We show that if a braid B can be parametrized in a certain way, then the previous work (B. Bode and M. R. Dennis, Constructing a polynomial whose nodal set is any prescribed knot or link, arXiv:1612.06328) can be extended to a construction of a polynomial f: R4 → R2 with the closure of B as the link of an isolated singularity of f, showing that the closure of B is real algebraic. In particular, we prove that closures of squares of strictly homogeneous braids and certain lemniscate links are real algebraic. We also show that the constructed polynomials satisfy the strong Milnor condition, providing an explicit fibration of the complement of the closure of B over S1.

KW - homogeneous braids

KW - Isolated singularities

KW - Milnor fibration

KW - real algebraic links

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DO - 10.1142/S0218216519500093

M3 - Article (Academic Journal)

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SN - 0218-2165

VL - 28

JO - Journal of Knot Theory and its Ramifications

JF - Journal of Knot Theory and its Ramifications

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ER -

Constructing links of isolated singularities of polynomials R4→R2

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