*Forma,* Vol. 22 (No. 1), pp. 15-30, 2007

*Original Paper*

## Solving Infinite Kolam in Knot Theory

Yukitaka Ishimoto

Okayama Institute for Quantum Physics, 1-9-1 Kyouyama, Okayama 700-0015, Japan

E-mail address: ishimoto@yukawa.kyoto-u.ac.jp

(Received November 3, 2006; Accepted August 14, 2007)

**Keywords: **
Kolam, Knot Theory, Morse Link Presentation, Temperley-Lieb Algebra

**Abstract. **
In southern 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: almost all cases should be examined for a solution. However, it is certainly not. In this paper, we focus on diamondshaped 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.

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