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

TL;DR

This work defines twelve intersection polynomials for long virtual knots by leveraging intersection numbers of homology cycles on surface realizations and distinguishing crossing types. It establishes their invariance, analyzes behavior under symmetries and knot operations (concatenation and crossing changes), and proves finite-type properties under crossing changes (degree 2 for the intersection polynomials, degree 1 for the writhe polynomials). It also shows the nontrivial influence of virtualizations on these invariants and connects them to the invariants of closures, with explicit examples illustrating their computation. The results lay the groundwork for a second paper that links these polynomials to supporting genera and deeper structural characterizations.

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.

Figures (12)

• Figure 2.1: A long virtual knot diagram
• Figure 2.2: A surface realization $(\Sigma_2,D)$
• Figure 2.3: A surface realization $(\Sigma_1,D)$ with four cycles
• Figure 3.1: A Reidemeister move I
• Figure 4.1: The flat long virtual knot diagram $\Gamma(n)$

Theorems & Definitions (54)

• Theorem 2.1: HNNS2
• Remark 2.2
• Theorem 2.3
• Lemma 2.4
• proof
• Lemma 2.5
• proof
• Lemma 3.1
• proof
• Lemma 3.2