Racks and blocked braids

TL;DR

This paper explores the algebraic structures called racks and iracks to construct braided monoidal categories, leading to new invariants for tangles and blocked braids, and clarifies properties of blocked torsions.

Contribution

It introduces iracks as an extension of racks, providing a framework to analyze blocked braids and torsions, and offers a simplified approach to previous results on blocked double torsions.

Findings

Blocked torsion has order 2 or 4

Iracks generalize racks and enable new tangle invariants

Blocked double torsion is not the identity

Abstract

In the paper Blocked-braid Groups, submitted to Applied Categorical Structures, the present authors together with Davide Maglia introduced the blocked-braid groups BB_n on n strands, and proved that a blocked torsion has order either 2 or 4. We conjectured that the order was actually 4 but our methods in that paper, which involved introducing for any group G a braided monoidal category of tangled relations, were inadequate to demonstrate this fact. Subsequently Davide Maglia in unpublished work investigated exactly what part of the structure and properties of a group G are needed to permit the construction of a braided monoidal category with a tangle algebra and was able to distinguish blocked two-torsions from the identity. In this paper we present a simplification of his answer, which turns out to be related to the notion of rack. We show that if G is a rack then there is a braided monoidal category TRel_G generalizing that of the above paper. Further we introduce a variation of the notion of rack which we call irack which yields a tangle algebra in TRel_G. Iracks are in particular racks but have in addition to the operations abstracting group conjugation also a unary operation abstracting group inverse. Using iracks we obtain new invariants for tangles and blocked braids permitting us to present a proof of Maglia's result that a blocked double torsion is not the identity. This work was presented at the Conference in Memory of Aurelio Carboni, Milan, 24-26 June 2013.