On the Unfaithfulness of the Manturov-Nikonov Map

The representation theory of braid groups has advanced significantly beyond that of classical knots, with several foundational linear representations established. These include the Burau representation (which is known to be unfaithful for Bn when n≥5), the Temperley–Lieb representation (closely related to the Jones polynomial), and the Lawrence–Krammer–Bigelow representation (faithful for all n≥1). In 2022, Professor Manturov and Professor Nikonov introduced a map from classical braids to virtual braids, extending the framework of braid group representations into the domain of virtual knot theory. This talk will demonstrate that the Manturov-Nikonov map is unfaithful by constructing explicit counterexamples.