# $F$-polynomials of tabulated virtual knots

## TL;DR

This paper computes the $F$-polynomials, a family of invariants generalizing the Affine Index Polynomial, for all oriented virtual knots with up to four classical crossings, providing explicit values.

## Contribution

It provides explicit calculations of $F$-polynomials for small virtual knots, expanding the understanding of these invariants and their applications.

## Key findings

- Explicit $F$-polynomial values for knots with up to four crossings.
- $F$-polynomials generalize the Affine Index Polynomial.
- $F$-polynomials are invariants of virtual knots.

## Abstract

A sequence of $F$-polynomials $\{ F^n_K (t, \ell)\}_{n=1}^{\infty}$ of virtual knots $K$ was defined by Kaur, Prabhakar, and Vesnin in 2018. These polynomials have been expressed in terms of index value of crossing and $n$-writhe of $K$. By the construction, $F$-polynomials are generalizations of the Kauffman's Affine Index Polynomial, and are invariants of virtual knot $K$. We present values of $F$-polynomials of oriented virtual knots having at most four classical crossings in a diagram.