Progress in distributive homology: from q-polynomial of rooted trees to Yang-Baxter homology

TL;DR

This paper reviews advances in distributive homology, highlighting the development of Yang-Baxter homology and introducing a new q-polynomial for rooted trees, connecting algebraic structures to low-dimensional topology.

Contribution

It presents recent progress in distributive homology, including the discovery of a q-polynomial for rooted trees and its relation to Yang-Baxter homology.

Findings

Introduction of q-polynomial of rooted trees

Progress in Yang-Baxter homology

Connections to Jones polynomial and Kauffman bracket

Abstract

This is an extended abstract of the talk given at the Oberwolfach Workshop "Algebraic Structures in Low-Dimensional Topology", 25 May -- 31 May 2014. My goal was to describe progress in distributive homology from the previous Oberwolfach Workshop June 3 - June 9, 2012, in particular my work on Yang-Baxter homology; however I concentrated my talk on my recent discovery of q-polynomial of a rooted tree; the appropriate topic as my talk was on May 30, 2014, the 30 anniversary of the Jones polynomial, and the polynomial has its roots in the Kauffman bracket approach to the Jones polynomial.