Link invariants derived from multiplexing of crossings

TL;DR

This paper introduces a new method called multiplexing of crossings in virtual link diagrams, creating potential for new classical link invariants and insights into the relationship between virtual and classical links.

Contribution

It defines the multiplexing operation on virtual links and explores its implications for invariants and the distinction between virtual and classical links.

Findings

Multiplexing preserves welded isotopy for virtual links.

Potential to derive new classical invariants from welded link invariants.

Multiplexing may produce non-classical links from classical ones.

Abstract

We introduce the multiplexing of a crossing, replacing a classical crossing of a virtual link diagram with multiple crossings which is a mixture of classical and virtual. For integers $m_{i}$ $(i=1,\ldots,n)$ and an ordered $n$-component virtual link diagram $D$, a new virtual link diagram $D(m_{1},\ldots,m_{n})$ is obtained from $D$ by the multiplexing of all crossings. For welded isotopic virtual link diagrams $D$ and $D'$, $D(m_{1},\ldots,m_{n})$ and $D'(m_{1},\ldots,m_{n})$ are welded isotopic. From the point of view of classical link theory, it seems very interesting that $D(m_{1},\ldots,m_{n})$ could not be welded isotopic to a classical link diagram even if $D$ is a classical one, and new classical link invariants are expected from known welded link invariants via the multiplexing of crossings.