Intrinsically linked graphs and even linking number

Abstract

We study intrinsically linked graphs where we require that every embedding of the graph contains not just a non-split link, but a link that satisfies some additional property. Examples of properties we address in this paper are: a two component link with $lk\left(A,L\right)=k{2}^r,k\ne 0$, a non-split $n$-component link where all linking numbers are even, or an $n$-component link with components $L,A_i$ where $lk\left(L,A_i\right)=3k,k\ne 0$. Links with other properties are considered as well.

For a given property, we prove that every embedding of a certain complete graph contains a link with that property. The size of the complete graph is determined by the property in question.