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English

Δ-groupoids in knot theory

Published inGeometriae Dedicata, vol. 150, no. 1, p. 105-130
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  • Open Access - Licence nationale Springer 
Publication date2011
Abstract

A Δ-groupoid is an algebraic structure which axiomatizes the combinatorics of a truncated tetrahedron. It is shown that there are relations of Δ-groupoids to rings, group pairs, and (ideal) triangulations of three-manifolds. In particular, we describe a class of representations of group pairs H⊂G into the group of upper triangular two-by-two matrices over an arbitrary ring R, and associate to that group pair a universal ring so that any representation of that class factorizes through a respective ring homomorphism. These constructions are illustrated by two examples coming from knot theory, namely the trefoil and the figure-eight knots. It is also shown that one can associate a Δ-groupoid to ideal triangulations of knot complements, and a homology of Δ-groupoids is defined.

Citation (ISO format)
KASHAEV, Rinat Mavlyavievich. Δ-groupoids in knot theory. In: Geometriae Dedicata, 2011, vol. 150, n° 1, p. 105–130. doi: 10.1007/s10711-010-9496-5
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ISSN of the journal1572-9168
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