Exploring Connections Between Topological Invariants and Computational Complexity: A Modest Investigation

We investigate potential relationships between certain topological invariants of 3-dimensional manifolds and the computational complexity of related algorithmic problems. Specifically, we examine whether properties of fundamental groups correlate with the difficulty of knot recognition algorithms. Through computational experiments on a database of prime knots with up to 10 crossings, we identify preliminary patterns that suggest deeper connections worth further investigation. Our results are modest but concrete, providing a foundation for more systematic study of topology-computation relationships. Classification: Primary 57M25, 68Q17; Secondary 55N10, 68Q15