The intersection polynomials of a long virtual knot II: Two supporting genera and characterizations

Abstract

We develop the study of the twelve intersection polynomials of long virtual knots, previously introduced in our preceding paper. We define two geometric invariants, the $1$- and $2$-supporting genera, using two distinct surface realizations. These genera yield a natural filtration of the set of long virtual knots, and we analyze the behavior of the intersection polynomials for long virtual knots with small supporting genera. Moreover, we investigate virtual $2$-string tangles, analyzing how their sums with long virtual knots affect the intersection polynomials through right closures. As an application, we provide complete realizability criteria for all twelve intersection polynomials.