A New Bound on Odd Multicrossing Numbers of Knots and Links

TL;DR

This paper establishes a new lower bound on odd multicrossing numbers of knots and links, providing insights into their complexity and improving bounds for specific classes like torus knots and certain knots with low crossing numbers.

Contribution

It introduces a novel inequality relating odd crossing numbers to genus and components, generalizes previous results, and offers improved bounds for specific knot types and classes.

Findings

Proves a new inequality for odd crossing numbers involving genus and components.

Provides a new upper bound on 5-crossing numbers of 2-torus knots and links.

Improves lower bounds on 5- and 7-crossing numbers for knots with low crossing numbers.

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 $k \in \mathbb{N}$ and all links $L$, the inequality $$c_{2k+1}(L) \geq \frac{2g(L) + r(L)-1}{k^2}$$ holds, where $c_{2k+1}(L)$, $g(L)$, and $r(L)$ are the $(2k+1)$-crossing number, $3$-genus, and 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. 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 $79$ knots with $2$-crossing number $ \leq 12$. Finally, we improve the lower bounds on the $7$-crossing numbers of $5$ knots with $2$-crossing number $\leq 12$.