Solving Infinite Kolam in Knot Theory

TL;DR

This paper explores the mathematical properties of infinite Kolam patterns in knot theory, demonstrating how to efficiently determine their existence for specific dot grid patterns, challenging the NP-completeness assumption.

Contribution

It introduces a knot-theoretic approach to analyze infinite Kolam patterns, providing a method to find solutions for specific grid patterns and arguing that the problem is not NP complete.

Findings

Knot theory can describe infinite Kolam patterns effectively.

The problem of finding infinite Kolam patterns for certain grids is not NP complete.

A method is proposed to determine the existence of infinite Kolam patterns for specific dot arrangements.

Abstract

In south India, there are traditional patterns of line-drawings encircling dots, called ``Kolam'', among which one-line drawings or the ``infinite Kolam'' provide very interesting questions in mathematics. For example, we address the following simple question: how many patterns of infinite Kolam can we draw for a given grid pattern of dots? The simplest way is to draw possible patterns of Kolam while judging if it is infinite Kolam. Such a search problem seems to be NP complete. However, it is certainly not. In this paper, we focus on diamond-shaped grid patterns of dots, (1-3-5-3-1) and (1-3-5-7-5-3-1) in particular. By using the knot-theory description of the infinite Kolam, we show how to find the solution, which inevitably gives a sketch of the proof for the statement ``infinite Kolam is not NP complete.'' Its further discussion will be given in the final section.