A generating set of Reidemeister moves of oriented virtual knots

TL;DR

This paper identifies a minimal set of four oriented virtual Reidemeister moves that can generate all other moves, simplifying the process of verifying invariants in virtual knot theory.

Contribution

It establishes a four-move generating set for oriented virtual Reidemeister moves, extending classical results to virtual knots and aiding invariant verification.

Findings

Enumerated 17 distinct virtual Reidemeister moves

Proved a four-element subset generates all moves

Facilitates simplified invariant verification in virtual knots

Abstract

In oriented knot theory, verifying a quantity is an invariant involves checking its invariance under all oriented Reidemeister moves, a process that can be intricate and time-consuming. A generating set of oriented moves simplifies this by requiring verification for only a minimal subset from which all other moves can be derived. While generating sets for classical oriented Reidemeister moves are well-established, their virtual counterparts are less explored. In this study, we enumerate the oriented virtual Reidemeister moves, identifying seventeen distinct moves after accounting for redundancies due to rotational and combinatorial symmetries. We prove that a four-element subset serves as a generating set for these moves. This result offers a streamlined approach to verifying invariants of oriented virtual knots and lays the groundwork for future advancements in virtual knot theory, particularly in the study of invariants and their computational properties.