Maps from knots in the cylinder to flat-virtual knots

TL;DR

This paper develops a method to map knots in cylinders and tori to virtual-flat knots, enabling the transfer of invariants from virtual knot theory to these more complex surfaces.

Contribution

It introduces a construction that adds 'invisible' crossings to diagrams, creating a formal immersion that facilitates mapping to virtual-flat knots.

Findings

Allows transfer of virtual knot invariants to knots in cylinders and tori.

Enables analysis of knots in higher genus surfaces using virtual knot invariants.

Provides a new framework for studying knots in thickened surfaces.

Abstract

In the present paper, we address the problem how to get a map from knots in the cylinder and on the thickened torus to some (generalisation of) virtual knots called virtual-flat knots. The main construction takes a diagram on a cylinder (torus) and adds some ``invisible'' crossings which gives rise to a diagram which can be formally immersed but not embedded (drawn) on the cylinder (torus) and living comfortably in thickened surfaces of higher genera. This allows one to ``pull back'' invariants of virtual theory to the theory of knots in the thickened cylinder (torus) where the parity bracket and other picture-valued invariants are not strong enough.