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Counterexamples to the quadrisecant approximation conjecture

Sheng Bai

LMAM, School of Mathematical Sciences, Peking University, Beijing 100871, P. R. China

E-mail Address: barries@163.com

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Chao Wang

Jonsvannsveien 87B, H0201, 7050 Trondheim, Norway

E-mail Address: chao_wang_1987@126.com

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, and

Jiajun Wang

http://orcid.org/0000-0002-6852-771X

LMAM, School of Mathematical Sciences, Peking University, Beijing 100871, P. R. China

E-mail Address: wjiajun@pku.edu.cn

Corresponding author.

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https://doi.org/10.1142/S0218216518500220Cited by:0 (Source: Crossref)

Abstract

A quadrisecant line of a knot $K$ is a straight line which intersects $K$ in four points, and a quadrisecant is a 4-tuple of points of $K$ which lie in order along the quadrisecant line. If $K$ has a finite number of quadrisecants, take $W$ to be the set of points of $K$ which are in a quadrisecant. Replace each subarc of $K$ between two adjacent points of $W$ along $K$ with the straight line segment between them. This gives the quadrisecant approximation of $K$ . It was conjectured that the quadrisecant approximation is always a knot with the same knot type as the original knot. We show that every knot type contains two knots, the quadrisecant approximation of one knot has self-intersections while the quadrisecant approximation of the other knot is a knot with a different knot type.

Keywords:

AMSC: 57M25

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