# Enumeration of planar Tangles

## TL;DR

This paper introduces a method to enumerate planar Tangles by analyzing their dual graphs, revealing exponential growth patterns and establishing growth constants through comparison with polyominoes.

## Contribution

It provides a novel enumeration approach for planar Tangles using dual graph analysis and compares their growth to polyominoes, establishing growth constants.

## Key findings

- Number of Tangles grows exponentially with length or area.
- Existence of growth constants for Tangles is proven.
- Enumeration method links Tangles to polysticks and polyominoes.

## Abstract

A planar Tangle is a smooth simple closed curve piecewise defined by quadrants of circles with constant curvature. We can enumerate Tangles by counting their dual graphs, which consist of a certain family of polysticks. The number of Tangles with a given length or area grows exponentially, and we show the existence of their growth constants by comparing Tangles to two families of polyominoes.