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

TL;DR

This paper advances the understanding of intersection polynomials in long virtual knots by introducing supporting genera, analyzing their behavior, and establishing realizability criteria for these polynomials.

Contribution

It defines two new geometric invariants, the supporting genera, and provides a complete set of realizability criteria for all twelve intersection polynomials.

Findings

Introduction of 1- and 2-supporting genera as invariants.

Analysis of intersection polynomials for knots with small supporting genera.

Complete criteria for realizing all twelve intersection polynomials.

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.