The two one-forbidden quotients of UV_n(k) are isomorphic and the unrestricted virtual braid group UUV_n(k) inherits structural properties from the universal setting, with consequences for multi-virtual braid groups.
On Universal Virtual and Welded Braid Groups and Their Linear Representations
2 Pith papers cite this work. Polarity classification is still indexing.
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abstract
We introduce linear representations of the universal virtual braid group $UV_n(c)$, where $n\geq 2$ and $c\geq 1$, which is a unifying framework for braid-type groups with multiple types of crossings. We classify and study its complex homogeneous $2$-local representations for all $n\geq 3$ and $c\geq 1$ (unique up to equivalence) and complex homogeneous $3$-local representations for all $n\geq 4$ and $c=2$ (four distinct families). We then introduce the universal welded braid group $UW_n(c)$ as a quotient of $UV_n(c)$ by the welded relations. This group recovers all known welded-type groups as quotients. We prove that $UW_n(c)$ has abelianization $\mathbb{Z}^c \oplus \mathbb{Z}_2$, perfect commutator subgroup for $n \geq 5$, trivial center, and $S_n$ as its smallest non-abelian finite quotient. Finally, we classify and study the complex homogeneous $2$-local representations of $UW_n(c)$ for all $n\geq 3$ and $c\geq 1$, obtaining three distinct families.
years
2026 2verdicts
UNVERDICTED 2representative citing papers
Introduces multi-welded twin groups M_kWT_n, establishes quotient maps and structural properties including abelianization and perfect commutator subgroup for n≥5, and classifies homogeneous 2-local and 3-local representations.
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Forbidden relations in universal virtual braid groups
The two one-forbidden quotients of UV_n(k) are isomorphic and the unrestricted virtual braid group UUV_n(k) inherits structural properties from the universal setting, with consequences for multi-virtual braid groups.
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Multi-welded twin groups
Introduces multi-welded twin groups M_kWT_n, establishes quotient maps and structural properties including abelianization and perfect commutator subgroup for n≥5, and classifies homogeneous 2-local and 3-local representations.