#### Solving Infinite Kolam in Knot Theory

##### Yukitaka Ishimoto

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.

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