Finite-Type Invariants of order one for long virtual knots

TL;DR

This paper extends the theory of order-one finite-type invariants to long virtual knots, introducing new invariants and concepts that relate to long flat virtual knots, building on previous work in the field.

Contribution

It introduces three order-one Vassiliev invariants for long virtual knots, including a universal invariant, and explores their relation to based matrices and long flat virtual knots.

Findings

Defined three order-one Vassiliev invariants for long virtual knots

Established the universality of one of the invariants

Linked invariants to based matrices and long flat virtual knots

Abstract

We extend to the long virtual knot case the constructions first presented by A. Henrich and later generalized by the author to the framed virtual knot case. These consist of three Vassiliev invariants of order one, including a universal one, as well as the notions of a based matrix and a singular based matrix and their relation to long flat virtual knots.