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

Benjamin Bode; Mark Dennis

doi:10.1142/S0218216518500827

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

Benjamin Bode, Mark Dennis

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

18 Citations (Scopus)

148 Downloads (Pure)

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.

Original language	English
Article number	1850082
Number of pages	31
Journal	Journal of Knot Theory and its Ramifications
Volume	28
Issue number	1
Early online date	10 Jan 2019
DOIs	https://doi.org/10.1142/S0218216518500827
Publication status	Published - Jan 2019

Research Groups and Themes

SPOCK

Keywords

applied topology
real algebraic knot theory
braids
Morse-Novikov number
knotted fields
constructive approach to knot theory

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10.1142/S0218216518500827

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AU - Bode, Benjamin

AU - Dennis, Mark

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N2 - 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.

AB - 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.

KW - applied topology

KW - real algebraic knot theory

KW - braids

KW - Morse-Novikov number

KW - knotted fields

KW - constructive approach to knot theory

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JO - Journal of Knot Theory and its Ramifications

JF - Journal of Knot Theory and its Ramifications

IS - 1

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ER -

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

Abstract

Research Groups and Themes

Keywords

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