Crossing numbers of composite knots and spatial graphs

TL;DR

This paper investigates the minimal crossing number of composite knots by relating it to the crossing numbers of associated spatial graphs, establishing a formula for large n and proposing conditions for crossing number additivity.

Contribution

It introduces a novel approach linking composite knot crossing numbers to spatial graph crossing numbers, providing a formula for large n and conditions for additivity.

Findings

For large n, the crossing number of the theta graph equals n times the sum of the crossing numbers of the component knots.

Formulates relations between spatial graph crossing numbers that imply additivity or provide lower bounds.

Establishes a connection between knot theory and spatial graph crossing number properties.

Abstract

We study the minimal crossing number $c(K_{1}\# K_{2})$ of composite knots $K_{1}\# K_{2}$, where $K_1$ and $K_2$ are prime, by relating it to the minimal crossing number of spatial graphs, in particular the $2n$-theta curve $\theta_{K_{1},K_{2}}^n$ that results from tying $n$ of the edges of the planar embedding of the $2n$-theta graph into $K_1$ and the remaining $n$ edges into $K_2$. We prove that for large enough $n$ we have $c(\theta_{K_1,K_2}^n)=n(c(K_1)+c(K_2))$. We also formulate additional relations between the crossing numbers of certain spatial graphs that, if satisfied, imply the additivity of the crossing number or at least give a lower bound for $c(K_1\# K_2)$.