# Virtual Links with Finite Medial Bikei

## TL;DR

This paper develops an algorithm to determine the finite fundamental medial bikei of virtual knots, enabling classification and invariants computation for knots with up to four crossings.

## Contribution

It introduces a novel algorithm for completing medial bikei presentation matrices and computing their invariants for virtual knots.

## Key findings

- Computed fundamental medial bikei for all prime virtual knots up to four crossings
- Established link invariants based on medial bikei properties
- Demonstrated the algorithm's effectiveness on knot classification

## Abstract

We consider the question of which virtual knots have finite fundamental medial bikei. We describe and implement an algorithm for completing a presentation matrix of a medial bikei to an operation table, determining both the cardinality and isomorphism class of the fundamental medial bikei, each of which are link invariants. As an example, we compute the fundamental medial bikei for all of the prime virtual knots with up to four classical crossings as listed in the knot atlas.