The intersection polynomials of a long virtual knot I: Definitions and properties

TL;DR

This paper introduces twelve new polynomial invariants called intersection polynomials for long virtual knots, extending previous invariants by analyzing intersection numbers on surfaces and exploring their algebraic properties.

Contribution

It defines a new family of intersection polynomial invariants for long virtual knots, providing a detailed study of their properties and relations to existing invariants.

Findings

All invariants are finite-type of degree two under crossing changes.

They are not finite-type under virtualizations.

The invariants' behavior under symmetries and concatenation is characterized.

Abstract

We introduce twelve polynomial invariants for long virtual knots, called intersection polynomials, extending and refining the three intersection polynomials for virtual knots. They are defined via intersection numbers of cycles on a closed surface, considering the order of over- and under-crossings. We study their fundamental properties including behavior under symmetries, crossing changes, and concatenation products. All are finite-type invariants of degree two under crossing changes, but not under virtualizations, and we examine their relation to the closure and the values at $t=1$ of their derivatives.