On a Generalization of Alexander Polynomial for Long Virtual Knots

TL;DR

This paper introduces a new polynomial invariant for long virtual knots, generalizing the Alexander polynomial, which helps estimate virtual crossing numbers and study minimal diagrams.

Contribution

It presents a novel invariant polynomial for long virtual knots, extending the Alexander polynomial and providing tools for analyzing virtual crossing complexity.

Findings

The $z$-polynomial estimates the virtual crossing number.

Application to minimal long virtual diagrams.

Provides a new method for studying virtual knot complexity.

Abstract

We construct new invariant polynomial for long virtual knots. It is a generalization of Alexander polynomial. We designate it by $\zeta$ meaning an analogy with $\zeta$-polynomial for virtual links. A degree of $\zeta$-polynomial estimates a virtual crossing number. We describe some application of $\zeta$-polynomial for the study of minimal long virtual diagrams with respect number of virtual crossings.