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On alternating closed braids

María de los Angeles Guevara-Hernández

OCAMI, Osaka City University, Sugimoto-cho, Sumiyoshi-ku, Osaka 558-8585, Japan

E-mail Address: guevarahernandez.angeles@gmail.com

Corresponding author.

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and

Akio Kawauchi

OCAMI, Osaka City University, Sugimoto-cho, Sumiyoshi-ku, Osaka 558-8585, Japan

E-mail Address: kawauchi@sci.osaka-cu.ac.jp

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https://doi.org/10.1142/S0218216521500176Cited by:0 (Source: Crossref)

Abstract

We introduce a numerical invariant called the braid alternation number that measures how far a link is from being an alternating closed braid. This invariant resembles the alternation number, which was previously introduced by the second author. However, these invariants are not equal, even for alternating links.

We study the relation of this invariant with others and calculate this invariant for some infinite knot families. In particular, we show arbitrarily large gaps between the braid alternation number and the alternation and unknotting numbers. Furthermore, we estimate the braid alternation number for prime knots with nine crossings or less.

Keywords:

AMSC: 57K10

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