Invariants of knots and links at roots of unity

TL;DR

This paper classifies knot and link invariants derived from irreducible representations of quantum groups at roots of unity, revealing new invariants linked to nilpotent representations beyond standard HOMFLY-PT invariants.

Contribution

It introduces a comprehensive classification of invariants at roots of unity, highlighting the role of nilpotent representations in generating non-trivial invariants.

Findings

Standard HOMFLY-PT invariants relate to highest and lowest weight representations.

Non-trivial invariants are associated with nilpotent representations with parameters.

Relations between new invariants and standard invariants at specific parameter values are discussed.

Abstract

We present a comprehensive classification of invariants of knots and links associated with irreducible representations of \uqslN{}, when the parameter of quantization $q$ is a root of unity. We demonstrate that, besides the standard HOMFLY-PT invariants, which are associated with representations with highest and lowest weights, non-trivial invariants can be associated only with nilpotent representations with parameters. We define the corresponding invariants and discuss their relations with standard invariants at particular values of parameters.