On two categorifications of the arrow polynomial for virtual knots

TL;DR

This paper introduces two new homology theories that categorify the arrow polynomial, an extension of the Kauffman bracket polynomial for virtual knots, providing tools for analyzing virtual knot properties.

Contribution

It presents novel gradings and homology theories based on arrow numbers, extending Khovanov homology to virtual knots with applications to virtual crossing number and surface genus estimation.

Findings

New homology theories for virtual knots

Applications to virtual crossing number estimation

Applications to surface genus estimation

Abstract

Two categorifications are given for the arrow polynomial, an extension of the Kauffman bracket polynomial for virtual knots. The arrow polynomial extends the bracket polynomial to infinitely many variables, each variable corresponding to an integer {\it arrow number} calculated from each loop in an oriented state summation for the bracket. The categorifications are based on new gradings associated with these arrow numbers, and give homology theories associated with oriented virtual knots and links via extra structure on the Khovanov chain complex. Applications are given to the estimation of virtual crossing number and surface genus of virtual knots and links. Key Words: Jones polynomial, bracket polynomial, extended bracket polynomial, arrow polynomial, Miyazawa polynomial, Khovanov complex, Khovanov homology, Reidemeister moves, virtual knot theory, differential, partial differential, grading, dotted grading, vector grading.