Link Patterns and the Catalan Tree

TL;DR

This paper introduces a new inductive method for generating link patterns using Temperley-Lieb algebra concepts, revealing the Catalan tree structure and enabling refined enumeration of link patterns.

Contribution

It presents a novel inductive approach based on strand insertion and algebra preimages, linking link patterns to the Catalan tree structure.

Findings

The Catalan tree structure naturally arises in the generation process.

The method allows refined enumeration of link patterns.

It connects algebraic operations with combinatorial structures.

Abstract

We demonstrate that a natural construction based on the two notions of insertion of a strand and finding the preimages of Temperley-Lieb algebra generators give an inductive means to generate all link patterns of a given number of strands. It is shown that the structure of the Catalan tree (as defined by Julian West) arises in the process of this induction, and that it can be exploited to give some refined enumerations of link patterns.