Some properties of knotted patterns

Let us consider a knotted pattern (chikku kOlam or melikala muggu) with r points in the row and c points in the columns. This is thus a rxc kOlam.

1) For an rxc pattern, the number of lines is the highest common denominator of r and c. If r and c are relatively prime, i.e., they are divisible by only unity (the number 1), then there is only one line. As an example, if it is a 3x6 pattern, there will be three lines as shown in the picture. If it is 3x7 pattern, there is only one line.

2) If r = c, then it is a square pattern and there will be r lines.

3) The total number of gaps between the lines is equal to rxc + (r-1) (c-1). If its is a 3x6 pattern , there will be 3x6 + 2x5 = 18+10 = 28 gaps. These are indicated by the 18 white dots and 10 blue dots. If r = c, then we have r^2 + (r-1)^2 gaps. If r = 5, then there will be 25 + 16 = 41 gaps.

4) The point where two lines (or strings) meet is called a cross-over. The number of cross-overs is one less than the number of gaps. For the 3x6 pattern, it will be 28 - 1 = 27 (5 + 6 + 5 + 6 + 5). Count these in the figure.

5) In the above figure, if we take the blue dots arranged in a staggered fashion, the dot count would be 6-5-6-5-6. Let us consider an 8x8 square. If we take the staggered dots also, the dot count will be (8-7) seven times and finally 8 dots. Now if we rotate this by 45 degrees, we get 1 to 15 to 1 straight triangular dots. That is why I always call the 1 to 15 dot pattern as 8x8 pattern. As mentioned by Mrs Mahalakshmi, the number of dots in the central row in the straight triangular arrangement is (2xr-1) where is the number of dots in any row or column of the square. For r = 8, it is 15, for r = 10, it is 19 and so on.

6) The number of patterns for a given number of cross-overs varies. For 3 and 4 cross-overs it is 1, for 5 it is 2, for 6 it is 3, for 7 it is 7, for 8 it is 21, for 9 it is 49, for 1o it is 65 and so on.

Regards! - mOhana