On extended graphical calculus for categorified quantum $sl(n)$

TL;DR

This paper advances the graphical calculus for categorified quantum sl(n), proving key Reidemeister moves involving strands of various thicknesses and colors, which are crucial for understanding the structure of this mathematical framework.

Contribution

It provides rigorous proofs of Reidemeister 2 and 3-like moves for extended graphical calculus with arbitrary strand thicknesses and colors, building on prior conjectures.

Findings

Proved Reidemeister 2-like moves for arbitrary strands.

Established Reidemeister 3-like moves involving different strand colors.

Enhanced the understanding of graphical calculus in categorified quantum sl(n).

Abstract

We study the properties of the extended graphical calculus for categorified quantum $sl(n)$. The main results include proofs of Reidemeister 2 and Reidemeister 3-like moves involving strands corresponding to arbitrary thicknesses and arbitrary colors -- the results that were anounced in [M. Stosic: Indecomposable objects and Lusztig's canonical basis, Math. Res. Lett. 22, no. 1 (2015), 245-278].