Jones polynomials of long virtual knots

TL;DR

This paper introduces new versions of the Jones polynomial and Khovanov homology for long virtual knots using Gauss diagram maps, demonstrating their distinct behavior from classical invariants through example calculations.

Contribution

It develops alternative formulations of Jones polynomial and Khovanov homology for virtual knots, highlighting differences from traditional invariants.

Findings

New Jones polynomial and Khovanov homology versions for virtual knots

Examples show these versions differ from classical invariants

Behavior of these invariants varies from original ones

Abstract

This paper defines versions of the Jones polynomial and Khovanov homology by using several maps from the set of Gauss diagrams to its variant. Through calculation of some examples, this paper also shows that these versions behave differently from the original ones.