Flags and tangles

TL;DR

This paper demonstrates the equivalence of two braided monoidal categories derived from topological Legendrian tangles and algebraic chain complexes, connecting knot theory with algebraic structures like Iwahori--Hecke algebras.

Contribution

It establishes a deep equivalence between topological and algebraic constructions of braided monoidal categories, linking Legendrian tangles to algebraic chain complexes.

Findings

Equivalence between topological and algebraic braided categories.

Identification of Iwahori--Hecke algebras within the category.

Unification of Legendrian tangle theory and algebraic chain complexes.

Abstract

We show that two constructions yield equivalent braided monoidal categories. The first is topological, based on Legendrian tangles and skein relations, while the second is algebraic, in terms of chain complexes with complete flag and convolution-type products. The category contains Iwahori--Hecke algebras of type $A_n$ as endomorphism algebras of certain objects.