Temperley-Lieb Categories on Non-Orientable Surfaces

TL;DR

This paper introduces a new skeletal diagram category extending Temperley-Lieb to non-orientable surfaces, incorporating handle decompositions and parameters for simple closed curves, with a tensor product and monoidal generators.

Contribution

It constructs the square with bands category, extending TL diagrams to non-orientable surfaces with new generators and a tensor product, advancing diagrammatic surface topology.

Findings

Defined a handle slide equivalence for diagrams

Extended tensor product to include non-orientable diagrams

Identified monoidal generators including TL and non-orientable diagrams

Abstract

In this paper we present the construction of a skeletal diagram category, which we call the square with bands category. This category extends the Temperley-Lieb (TL) category, where morphisms now include diagrams of embedded curves on (possibly) non-orientable bounded surfaces, and involves three parameters associated to simple closed curves. Such diagrams utilise handle decompositions for surfaces and are considered up to a handle slide equivalence. We define a tensor product on this category, extending the well-known tensor product on the TL category, and a full set of monoidal generators is given, which includes the TL generators, a family of orientable genus one diagrams, and a family of non-orientable diagrams. This document constitutes an initial draft of ongoing research with preliminary reporting of some results in the last section; a subsequent version including a detailed introduction and full proofs will follow.