The intersection polynomials of a virtual knot I: Definitions and calculations

TL;DR

This paper introduces three new invariants called intersection polynomials for virtual knots, based on intersection numbers on surfaces, with calculations up to four crossings and analysis of their properties.

Contribution

It defines three types of intersection polynomials for virtual knots and provides explicit calculations and property analyses, advancing knot invariant theory.

Findings

Defined three intersection polynomials as invariants of virtual knots.

Calculated intersection polynomials for knots with up to four crossings.

Explored properties and potential applications of these invariants.

Abstract

We introduce three kinds of invariants of a virtual knot called the first, second, and third intersection polynomials. The definition is based on the intersection number of a pair of curves on a closed surface. The calculations of intersection polynomials are given up to crossing number four. We also study several properties of intersection polynomials.