The Unfaithfulness of the Manturov-Nikonov Map for k > 5

The representation theory of classical braids is well-established, with key examples including the Burau representation (unfaithful for n≥5), the Temperley–Lieb representation, and the faithful Lawrence–Krammer–Bigelow representation. In contrast, virtual knots exhibit distinct properties such as parity. The Manturov-Nikonov (M-N) map bridges these domains by embedding classical braids into virtual braids. In this talk, we prove that the M-N map is unfaithful for k>5. By analyzing the kernel of the Burau representation, we explicitly construct elements in the kernel of the M-N map, revealing new obstructions to faithfulness in virtual braid representations.