Solving Infinite Kolam in Knot Theory

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.

💡 Research Summary

The paper investigates the mathematical structure of “infinite Kolam”, a traditional South‑Indian line‑drawing pattern in which a single continuous line encircles a prescribed array of dots. Although a naïve enumeration of all possible line connections suggests an NP‑complete search problem, the authors demonstrate that for highly symmetric, diamond‑shaped dot arrays the problem can be solved in polynomial time by translating it into knot theory and using the Temperley‑Lieb algebra.

First, the authors formalize a Kolam as a planar 4‑regular graph: each dot is a vertex of degree at most four, and edges correspond to the permissible horizontal, vertical, or diagonal line segments. An “infinite Kolam” is precisely a configuration where the entire edge set forms a single closed loop (a single component of the underlying link). By interpreting each vertex as a “plug‑socket” pair, the local connection choices become the generators e_i of the Temperley‑Lieb algebra, satisfying e_i^2 = a e_i, e_i e_{i±1} e_i = e_i, and commuting when |i–j|>1. The scalar a records the creation of a closed loop; each time a loop is formed the diagram acquires a factor a.

The paper focuses on two families of diamond‑shaped dot patterns: the 5‑row pattern (1‑3‑5‑3‑1) and the 7‑row pattern (1‑3‑5‑7‑5‑3‑1). Because of their axial symmetry, the authors can process the diagram row by row using a transfer‑matrix method. A “state” is defined as a particular way of pairing the dangling ends at the current row, and the set of all possible states forms a basis of the Temperley‑Lieb module. The transfer matrix T encodes how a state evolves when a new row of vertices is added; each entry of T is a polynomial in a that counts the number of ways to connect the new vertices consistent with the two endpoint states.

The total number of Kolam configurations is obtained by multiplying the initial state vector (all ends free) by T raised to the power equal to the number of rows, and finally projecting onto the terminal state where all ends are paired into a single loop. The crucial observation is that the coefficient of a^0 in the resulting polynomial counts precisely the infinite Kolam patterns, because a^0 corresponds to diagrams with exactly one loop and no extra closed components. By explicitly constructing the state space for the two diamond patterns, the authors compute the a‑polynomials and extract the a^0 coefficients, finding 4 infinite Kolams for the 5‑row pattern and 36 for the 7‑row pattern.

Complexity analysis shows that the number of states grows as the Catalan number C_w, where w is the width of the pattern (the maximum number of dots in a row). Since w = O(n) for an n‑layer diamond, the dimension of T is O(C_n) ≈ O(4^n / (n^{3/2})). However, the transfer‑matrix multiplication can be performed using dynamic programming, and the extraction of the a^0 term requires only linear algebra over the polynomial ring ℤ