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

TL;DR

The paper advances the study of twelve intersection polynomials for long virtual knots by introducing the 1- and 2-supporting genera, which create a natural genus-based filtration and enable strict distinctions among early filtration stages. It develops a comprehensive tangle framework with right and left closures to analyze how combining tangles and knots affects the invariants, and provides complete realizability criteria for all twelve polynomials under genus constraints. Through explicit constructions with tangles and closures, it establishes necessary and sufficient conditions (e.g., reciprocity and value constraints at t=1) for realizing each polynomial, linking geometric complexity to algebraic invariants. The results yield a full realization theory for the intersection polynomials and illuminate the interplay between surface realizations, tangles, and long virtual knot invariants.

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.

Figures (15)

• Figure 3.1: A surface realization $(\Sigma_{1,2},D)$
• Figure 3.2: Proof of Lemma \ref{['lem36']}(i)
• Figure 3.3: Proof of Lemma \ref{['lem37']}
• Figure 3.4: A surface realization of $K_n$
• Figure 3.5: A surface realization of $K_n'$

Theorems & Definitions (48)

• Lemma 2.1: NNSW1
• Lemma 2.2: NNSW1
• Lemma 2.3: NNSW1
• Lemma 2.4: NNSW1
• Lemma 2.5: NNSW1
• Lemma 3.1: HigaHNNS4
• Definition 3.2
• Lemma 3.3
• proof
• Lemma 3.4