Diagrams realizing prescribed sublink diagrams for virtual links and welded links

TL;DR

This paper explores how prescribed sublink diagrams can be realized within larger links, establishing a positive result for welded links but not for virtual links, thus advancing understanding of link diagram constructions.

Contribution

The paper extends Jin and Lee's result to welded links, showing such diagram realizations are possible, but demonstrates this does not hold for virtual links, highlighting differences between these link types.

Findings

Realization theorem holds for welded links.

Realization theorem does not hold for virtual links.

Differentiates properties of virtual and welded links.

Abstract

Jin and Lee proved the following: Suppose that $D_1, \dots, D_n$ are link diagrams. Given a link $L$ which is partitioned into sublinks $L_1, \dots, L_n$ admitting diagrams $D_1, \dots, D_n$ respectively, there is a diagram $D$ of $L$ whose restrictions to $L_1, \dots, L_n$ are isotopic to $D_1, \dots, D_n$, respectively. In this paper we show that a similar result does hold for welded links and does not for virtual links.