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Equivalence of two definitions of set-theoretic Yang–Baxter homology and general Yang–Baxter homology

Józef H. Przytycki

The George Washington University, Washington, DC 20052, USA

University of Gdańsk, Poland

E-mail Address: przytyck@gwu.edu

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and

Xiao Wang

The George Washington University, Washington, DC 20052, USA

E-mail Address: wangxiao@gwu.edu

Corresponding author.

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https://doi.org/10.1142/S0218216518410134Cited by:9 (Source: Crossref)

This article is part of the issue:

Special Issue — AMS Special Session on Inverse Problems; Guest Editors: Hanna E. Makaruk and Robert M.Owczarek

Abstract

In 2004, Carter, Elhamdadi and Saito defined a homology theory for set-theoretic Yang–Baxter operators (we will call it the “algebraic” version in this paper). In 2012, Przytycki defined another homology theory for pre-Yang–Baxter operators which has a nice graphic visualization (we will call it the “graphic” version in this paper). We show that they are equivalent. The “graphic” homology is also defined for pre-Yang–Baxter operators, and we give some examples of its one-term and two-term homologies. In the two-term case, we have found torsion in homology of Yang–Baxter operator that yields the Jones polynomial.

Keywords:

AMSC: 57M25, 18G60

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