Finite-Type Invariants of Order One for Framed Virtual Knots

TL;DR

This paper extends the concept of finite-type invariants of order one from virtual knots to framed virtual knots, introducing new notions like virtual strings and based matrices to facilitate this generalization.

Contribution

It develops a framework for finite-type invariants of order one specifically for framed virtual knots, building on Henrich's work for unframed virtual knots.

Findings

Construction of universal finite-type invariant for framed virtual knots

Introduction of virtual strings and based matrices for framed knots

Extension of existing invariants to a new class of knots

Abstract

A. Henrich proved the existence of the universal finite-type invariant of order one for virtual knots. We extend the construction and the methods of her paper to framed virtual knots. To do so, we introduce the notions of virtual strings and based matrices for framed flat virtual knots.