A Homology Theory For A Special Family Of Semi-groups

TL;DR

This paper introduces a novel homology theory for specific semi-groups with self-distributivity or idempotency, explores its geometric realization, compares it with existing homologies, and links it to knot theory through Temperley-Lieb algebras.

Contribution

It develops a new homology framework for certain semi-groups and connects it to knot theory, expanding the algebraic tools available for topological applications.

Findings

Constructed a homology theory for self-distributive and idempotent semi-groups.

Compared this homology with rack homology theories.

Established connections between the homology theory and knot invariants via Temperley-Lieb algebras.

Abstract

In this paper, we construct a new homology theory for semi-groups satisfying the self distributivity axiom or the idempotency axiom. Next, we consider the geometric realization corresponding to the homology theory. We continue with the comparison of this homology theory with one term and two term (rack) homology theories of self-distributive algebraic structures. Finally, we propose connections between the homology theory and knot theory via Temperley-Lieb algebras.