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

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.