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Theoretical analysis demonstrates the irreducibility of the extended Lawrence-Bigelow-Krammer representation in the 3-strand virtual braid group, highlighting precise parameter boundaries.
We consider the representation [Formula: see text] which extends the Lawrence-Bigelow-Krammer representation from the braid group [Formula: see text] to the virtual braid group [Formula: see text] on 3 strands. After specializing parameters [Formula: see text], [Formula: see text] to non-zero complex numbers, we study the irreducibility of [Formula: see text]. We prove that [Formula: see text] is irreducible for all non-zero complex numbers [Formula: see text], [Formula: see text] except for the case [Formula: see text].
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Chreif et al. (2026) studied this question.
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