Polynomial link invariants and quantum algebras

Abstract

The definition of the Jones polynomial in the 80's gave rise to a large family of so-called quantum link invariants, based on quantum groups. These quantum invariants are all controlled by the same two-variable invariant (the HOMFLY-PT polynomial), which also specializes to the older Alexander polynomial. Building upon quantum Schur-Weyl duality and variants of this phenomenon, I will explain an algebraic setup that allows for global definitions of these quantum polynomials, and discuss extensions of these quantum objects designed to encompass all of the mentioned invariants, including the HOMFLY-PT polynomial. IV-1Hoel Queffelec words better replacements, for the categories of representations. Furthermore, these objects, contrarily to the categories of representations, can be extended to encompass the HOMFLY-PT polynomial as well, which I'll briefly illustrate at the end of these notes.Organization: Section 1 is devoted to brief definitions of the notions of knots, links and tangles, and to their invariants. These invariants are defined using the so-called quantized oriented Brauer category. Section 2 relates this to the original Reshetikhin-Turaev approach by defining a functor between the quantized oriented Brauer category and the category of representations of the quantum group U q (gl m|n ). Finally, Section 3 is devoted to Schur-Weyl and skew Howe dualities and the role they recently played in knot theory.The presentation given in these notes is far from being historically accurate: I have rather tried to define all invariants in a unified way and use this definition to recover several quantum constructions. In many situations, the story actually went the other way around. However, I hope that the presentation I chose to follow will make the relations and the central role that quantum algebras play in this picture more apparent, and that these notes can serve as a motivation and a general illustration before reading more involved references.Acknowledgements: These lecture notes follow a lecture series given at Winterbraids IX in Reims in March 2019. I would like to warmly thank the organizers (Paolo Bellingeri, Vincent Florens, Jean-Baptiste Meilhan, Loïc Poulain d'Andecy, Emmanuel Wagner) for their invitation, their support, and more generally for putting together these great winter schools year after year. Many thanks also to Antonio Sartori for his comments on a preliminary version of these notes and for teaching me most of what's now in these notes, to Baptiste Loreau Unger for carefully reading them, and to the anonymous referee for her/his helpful comments.