Solving Infinite Kolam in Knot Theory
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.
- Publication:
-
arXiv e-prints
- Pub Date:
- October 2007
- DOI:
- 10.48550/arXiv.0710.1976
- arXiv:
- arXiv:0710.1976
- Bibcode:
- 2007arXiv0710.1976I
- Keywords:
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- Computer Science - Discrete Mathematics;
- Condensed Matter - Statistical Mechanics
- E-Print:
- 13 pages, 2 figures, the final version for FORMA with typo fixed