Research ArticleNo Access

A new bound on odd multicrossing numbers of knots and links

Department of Mathematics, Yale University, New Haven, CT 06511, USA

https://doi.org/10.1142/S0218216522500080Cited by:0 (Source: Crossref)

Abstract

An $n$ -crossing projection of a link $L$ is a projection of $L$ onto a plane such that $n$ points on $L$ are superimposed on top of each other at every crossing. We prove that for all $k \in ℕ$ and all links $L$ , the inequality

c2k+1(L)≥2g(L)+r(L)−1k2<math display="block" altimg="eq-00008.gif"><mrow><msub><mrow><mi>c</mi></mrow><mrow><mn>2</mn><mi>k</mi><mo stretchy="false" class="MathClass-bin">+</mo><mn>1</mn></mrow></msub><mo class="MathClass-open" stretchy="false">(</mo><mi>L</mi><mo class="MathClass-close" stretchy="false">)</mo><mo class="MathClass-rel">≥</mo><mfrac><mrow><mn>2</mn><mi>g</mi><mo class="MathClass-open" stretchy="false">(</mo><mi>L</mi><mo class="MathClass-close" stretchy="false">)</mo><mo stretchy="false" class="MathClass-bin">+</mo><mi>r</mi><mo class="MathClass-open" stretchy="false">(</mo><mi>L</mi><mo class="MathClass-close" stretchy="false">)</mo><mo stretchy="false" class="MathClass-bin">−</mo><mn>1</mn></mrow><mrow><msup><mrow><mi>k</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfrac></mrow></math>

holds, where

$c_{2 k + 1} (L)$ ,

$g (L)$ and

$r (L)$ are the

$(2 k + 1)$ -crossing number,

$3$ -genus, and the number of components of

$L,$ respectively. This result is used to prove a new bound on the odd crossing numbers of torus knots and generalizes a result of Jablonowski (see [M. Jabłonowski, Triple-crossing number, the genus of a knot or link and torus knots, Topology Appl.285 (2020) 107389]). We also prove a new upper bound on the

$5$ -crossing numbers of the 2-torus knots and links. Furthermore, we improve the lower bounds on the

$5$ -crossing numbers of

$82$ knots with

$2$ -crossing number at most

$12$ . Finally, we improve the lower bounds on the

$7$ -crossing numbers of

$5$ knots with

$2$ -crossing number at most

$12$ .

Keywords:

AMSC: 57K10

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