Heegaard genus is a concept in 3-manifold topology that describes the minimal number of handles needed to construct a Heegaard surface for a given 3-manifold. This notion is pivotal in distinguishing between different types of knots and links, as it reflects the complexity of the surface that can separate the manifold into two parts. The heegaard genus helps classify manifolds and understand their properties, especially when relating knot groups to the structure of the knots themselves.
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