Graphoids

TL;DR

This paper explores invariants of virtual graphoids, providing topological interpretations and applications to knotted proteins and efficient virtual spatial graph representations.

Contribution

It introduces topological interpretations of virtual graphoids and demonstrates their applications in studying knotted proteins and simplifying virtual spatial graph computations.

Findings

Virtual graphoids can model knotted proteins with open ends.

They offer a way to represent virtual spatial graphs with fewer crossings.

Applications include new methods for computing invariants.

Abstract

We study invariants of virtual graphoids, which are virtual spatial graph diagrams with two distinguished degree-one vertices modulo graph Reidemeister moves applied away from the distinguished vertices. Generalizing previously known results, we give topological interpretations of graphoids. There are several applications to virtual graphoid theory. First, virtual graphoids are suitable objects for studying knotted graphs with open ends arising in proteins. Second, a virtual graphoid can be thought of as a way to represent a virtual spatial graph without using as many crossings, which can be advantageous for computing invariants.