On Free Knots and Links

TL;DR

This paper introduces invariants for simplified virtual knots, derived from Gauss diagrams, which help analyze minimality and non-invertibility issues by reducing knot equivalence to Reidemeister 2 moves.

Contribution

It constructs new invariants for simplified virtual knots and applies them to minimality and non-invertibility problems, extending to orientable cases.

Findings

Invariants distinguish simplified virtual knots effectively.

Applications to minimality problems in virtual knots.

Insights into non-invertibility and graph-link questions.

Abstract

Both classical and virtual knots arise as formal Gauss diagrams modulo some abstract moves corresponding to Reidemeister moves. If we forget about both over/under crossings structure and writhe numbers of knots modulo the same Reidemeister moves, we get a dramatic simplification of virtual knots, which kills all classical knots. However, many virtual knots survive after this simplification. We construct invariants of these objects and present their applications to minimality problems of virtual knots as well as some questions related to graph-links. One can easily generalize these results for the orientable case and apply them for solving non-invertibility problems. The main idea behind these invariants is some geometrical construction which reduces the general equivalence to the equivalence only modulo Reidemeister - 2 move.