Flat-virtual knot: introduction and some invariants

TL;DR

This paper introduces flat-virtual knots, defines their properties, and develops invariants such as Alexander-like polynomials and Kauffman brackets, extending classical knot concepts to a new virtual setting.

Contribution

It presents the first definitions and invariants for flat-virtual knots, establishing a framework for mapping classical knots to virtual counterparts.

Findings

Defined flat-virtual knots and their properties.

Developed Alexander-like polynomial invariant.

Constructed a picture-valued Kauffman bracket.

Abstract

The motivation for this work is to construct a map from classical knots to virtual ones. What we get in the paper is a series of maps from knots in the full torus (thickened torus) to flat-virtual knots. We give definition of flat-virtual knots and presents Alexander-like polynomial and (picture-valued) Kauffman bracket for them.