Some Corollaries of Manturov's projection Theorem

TL;DR

This paper demonstrates that certain virtual knot invariants coincide with classical invariants for all classical knots by utilizing Manturov's projection theorem, establishing a key link between virtual and classical knot theory.

Contribution

The paper proves that virtual canonical genus and virtual bridge number invariants equal their classical counterparts for all classical knots using Manturov's projection theorem.

Findings

Virtual invariants match classical invariants for all classical knots.

Manturov's projection preserves key knot invariants.

Establishes a bridge between virtual and classical knot invariants.

Abstract

In our works with Stoimenow, Vdovina and with Byberi, we introduced the virtual canonical genus $g_{vc}(K)$ and the virtual bridge number $vb(K)$ invariants of virtual knots. One can see from the definitions that for an classical knot $K$ the values of these invariants are less or equal than the classical canonical genus $g_c(K)$ and the bridge number $b(K)$ respectively. We use Manturov's projection from the category of virtual knot diagrams to the category of classical knot diagrams, to show that for every classical knot type $K$ we have $g_{vc}(K)=g_c(K)$ and $vb(K)=b(K)$.